./PaxHeaders.16248/nurbs-1.4.30000644000000000000000000000013214030323741012466 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.823919749 30 ctime=1617012705.823919749 nurbs-1.4.3/0000755000175000017500000000000014030323741014505 5ustar00rvazquezrvazquez00000000000000nurbs-1.4.3/PaxHeaders.16248/CITATION0000644000000000000000000000013214030323741013543 xustar0030 mtime=1617012705.775919014 30 atime=1617012705.775919014 30 ctime=1617012705.823919749 nurbs-1.4.3/CITATION0000644000175000017500000000151314030323741015642 0ustar00rvazquezrvazquez00000000000000 To cite the Octave NURBS package use: [1] M. Spink, D. Claxton, C. de Falco, R. Vazquez, The NURBS toolbox, http://octave.sourceforge.net/nurbs/index.html. [2] C. de Falco, A. Reali, and R. Vazquez. Geopdes: A research tool for isogeometric analysis of pdes. Advances in Engineering Software, 42(12):1020-1034, 2011. BibTeX entries for LaTeX users are: @misc{NT, Author = {Spink, M. and Claxton, D. and de Falco, C. and V{\'a}zquez, R.}, Howpublished = {\url{http://octave.sourceforge.net/nurbs/index.html}}, Title = {The {NURBS} toolbox}} @article{geopdes, Author = {C. de Falco and A. Reali and R. V{\'a}zquez}, Journal = {Advances in Engineering Software}, Number = {12}, Pages = {1020-1034}, Title = {GeoPDEs: A research tool for Isogeometric Analysis of PDEs}, Volume = {42}, Year = {2011}} nurbs-1.4.3/PaxHeaders.16248/src0000644000000000000000000000013214030323741013120 xustar0030 mtime=1617012705.815919627 30 atime=1617012705.823919749 30 ctime=1617012705.823919749 nurbs-1.4.3/src/0000755000175000017500000000000014030323741015274 5ustar00rvazquezrvazquez00000000000000nurbs-1.4.3/src/PaxHeaders.16248/bspeval.cc0000644000000000000000000000013214030323741015140 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/bspeval.cc0000644000175000017500000000707114030323741017244 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include "low_level_functions.h" //#include static bool bspeval_bad_arguments(const octave_value_list& args); DEFUN_DLD(bspeval, args, nargout,"\ BSPEVAL: Evaluate B-Spline at parametric points\n\ \n\ \n\ Calling Sequence:\n\ \n\ p = bspeval(d,c,k,u)\n\ \n\ INPUT:\n\ \n\ d - Degree of the B-Spline.\n\ c - Control Points, matrix of size (dim,nc).\n\ k - Knot sequence, row vector of size nk.\n\ u - Parametric evaluation points, row vector of size nu.\n\ \n\ OUTPUT:\n\ \n\ p - Evaluated points, matrix of size (dim,nu)\n\ ") { octave_value_list retval; if (nargout != 1 || args.length () != 4) print_usage (); if (!bspeval_bad_arguments (args)) { int d = args(0).int_value(); Matrix c = args(1).matrix_value(); RowVector k = args(2).row_vector_value(); NDArray u = args(3).array_value(); octave_idx_type nu = u.numel (); octave_idx_type mc = c.rows(), nc = c.cols(); Matrix p(mc, nu, 0.0); if (nc + d == k.numel () - 1) { //#pragma omp parallel default (none) shared (d, c, k, u, nu, mc, nc, p) { RowVector N(d+1,0.0); int s, tmp1; double tmp2; //#pragma omp for for (octave_idx_type col=0; col. */ #include #include "low_level_functions.h" #include DEFUN_DLD(surfderivcpts, args, nargout,"\ \nSURFDERIVCPTS: Compute control points of n-th derivatives of a NURBS surface.\n \ \n \ \nusage: pkl = surfderivcpts (n, p, U, m, q, V, P, d) \ \n \ \n INPUT: \ \n\ \n n+1, m+1 = number of control points \ \n p, q = spline order \ \n U, V = knots \ \n P = control points \ \n d = derivative order \ \n\ \n OUTPUT: \ \n\ \n pkl (k+1, l+1, i+1, j+1) = i,jth control point \ \n of the surface differentiated k \ \n times in the u direction and l \ \n times in the v direction \ \n \ \n Adaptation of algorithm A3.7 from the NURBS book\n") { //function pkl = surfderivcpts (n, p, U, m, q, V, P, d, r1, r2, s1, s2) octave_value_list retval; if (nargout != 1 || (args.length () != 8 && args.length() != 12)) print_usage (); octave_idx_type n = args(0).idx_type_value (); octave_idx_type p = args(1).idx_type_value (); RowVector U = args(2).row_vector_value (false, true); octave_idx_type m = args(3).idx_type_value (); octave_idx_type q = args(4).idx_type_value (); RowVector V = args(5).row_vector_value (false, true); Matrix P = args(6).matrix_value (); octave_idx_type d = args(7).idx_type_value (); octave_idx_type r1(0), r2 (n), s1 (0), s2 (m); if (args.length () == 12) { r1 = args (8).idx_type_value (); r2 = args (9).idx_type_value (); s1 = args (10).idx_type_value (); s2 = args (11).idx_type_value (); } else if (args.length () > 8) print_usage (); NDArray pkl; surfderivcpts (n, p, U, m, q, V, P, d, r1, r2, s1, s2, pkl); retval(0) = octave_value (pkl); return retval; } /* %!test %! plane = nrbdegelev(nrb4surf([0 0], [0 1], [1 0], [1 1]), [1, 1]); %! %! pkl = surfderivcpts (plane.number(1)-1, plane.order(1)-1, %! plane.knots{1}, plane.number(2)-1, %! plane.order(2)-1, plane.knots{2}, %! squeeze (plane.coefs(1,:,:)), 2); %! %! %! pkl2 = [ 0 0 0 1 0 0 0 0 0 0 0 0 1 0 ... %! 0 0 0 0 0 0 0 1 0 0 0 0 0 0.5 0 ... %! 0 1 0 0 0 0 0 0.5 0 0 1 0 0 0 0 ... %! 0 0.5 0 0 1 0 0 0 0 0 1 0 0 0 0 ... %! 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 ... %! 0 0 0 0 0 0 0]'; %! %! assert (pkl(:),pkl2); */ nurbs-1.4.3/src/PaxHeaders.16248/tbasisfun.cc0000644000000000000000000000013214030323741015502 xustar0030 mtime=1617012705.815919627 30 atime=1617012705.815919627 30 ctime=1617012705.823919749 nurbs-1.4.3/src/tbasisfun.cc0000644000175000017500000002047314030323741017607 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009 Carlo de Falco Copyright (C) 2012, 2020 Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include void onebasisfun__ (double u, octave_idx_type p, RowVector U, double *N) { *N = 0.0; if ((u <= U.min ()) || ( u > U.max ())) return; else if (p == 0) { *N = 1.0; return; } else if (p == 1) { if (u < U(1)) { *N = (u - U(0)) / (U(1) - U(0)); return; } else { *N = (U(2) - u) / (U(2) - U(1)); return; } } else if (p == 2) { double ln = u - U(0); double dn = U(3) - u; double ld = U(2) - U(0); double dd = U(3) - U(1); if (u < U(1)) { *N = ln*ln / (ld * (U(1) - U(0))); return; } else if (u > U(2)) { *N = dn*dn / (dd * (U(3) - U(2))); return; } else { if (ld != 0) *N += ln * (U(2) - u) / ((U(2) - U(1)) * ld); if (dd != 0) *N += dn * (u - U(1)) / ((U(2) - U(1)) * dd); return; } } double ln = u - U(0); double ld = U(U.numel () - 2) - U(0); if (ld != 0) { double tmp; onebasisfun__ (u, p-1, U.extract (0, U.numel () - 2), &tmp); *N += ln * tmp / ld; } double dn = U(U.numel () - 1) - u; double dd = U(U.numel () - 1) - U(1); if (dd != 0) { double tmp; onebasisfun__ (u, p-1, U.extract (1, U.numel () - 1), &tmp); *N += dn * tmp / dd; } return; } void onebasisfun__ (double u, double p, RowVector U, double *N) { onebasisfun__ (u, static_cast (p), U, N); } void onebasisfunder__ (double u, octave_idx_type p, RowVector U, double *N, double *Nder) { double aux; *N = 0.0; *Nder = 0.0; if ((u <= U.min ()) || ( u > U.max ())) return; else if (p == 0) { *N = 1.0; *Nder = 0.0; return; } else { double ln = u - U(0); double ld = U(U.numel () - 2) - U(0); if (ld != 0) { onebasisfun__ (u, p-1, U.extract (0, U.numel () - 2), &aux); aux = aux / ld; *N += ln * aux; *Nder += p * aux; } double dn = U(U.numel () - 1) - u; double dd = U(U.numel () - 1) - U(1); if (dd != 0) { onebasisfun__ (u, p-1, U.extract (1, U.numel () - 1), &aux); aux = aux / dd; *N += dn *aux; *Nder -= p * aux; } } } DEFUN_DLD(tbasisfun, args, nargout,"\ TBASISFUN: Compute a B- or T-Spline basis function, and its derivatives, from its local knot vector.\n\ \n\ usage:\n\ \n\ [N, Nder] = tbasisfun (u, p, U)\n\ [N, Nder] = tbasisfun ([u; v], [p q], {U, V})\n\ [N, Nder] = tbasisfun ([u; v; w], [p q r], {U, V, W})\n\ \n\ INPUT:\n\ u or [u; v] : points in parameter space where the basis function is to be\n\ evaluated \n\ \n\ U or {U, V} : local knot vector\n\ \n\ p or [p q] : polynomial order of the basis function\n\ \n\ OUTPUT:\n\ N : basis function evaluated at the given parametric points\n\ Nder : gradient of the basis function evaluated at the given points\n") { octave_value_list retval; Matrix u = args(0).matrix_value (); RowVector N(u.cols ()); double *Nptr = N.fortran_vec (); if (! args(2).iscell ()) { double p = args(1).idx_type_value (); RowVector U = args(2).row_vector_value (true, true); assert (U.numel () == p+2); if (nargout == 1) for (octave_idx_type ii = 0; ii < u.numel (); ii++) onebasisfun__ (u(ii), p, U, &(Nptr[ii])); if (nargout == 2) { RowVector Nder(u.cols ()); double *Nderptr = Nder.fortran_vec (); for (octave_idx_type ii=0; ii. */ octave_idx_type findspan(int n, int p, double u, const RowVector& U); void basisfun(int i, double u, int p, const RowVector& U, RowVector& N); void basisfunder (int i, int pl, double uu, const RowVector& u_knotl, int nders, NDArray& dersv); int curvederivcpts (octave_idx_type n, octave_idx_type p, const RowVector &U, const NDArray &P, octave_idx_type d, octave_idx_type r1, octave_idx_type r2, Matrix &pk); int surfderivcpts (octave_idx_type n, octave_idx_type p, const RowVector& U, octave_idx_type m, octave_idx_type q, const RowVector& V, const Matrix& P, octave_idx_type d, octave_idx_type r1, octave_idx_type r2, octave_idx_type s1, octave_idx_type s2, NDArray &pkl); int surfderiveval (octave_idx_type n, octave_idx_type p, const RowVector &U, octave_idx_type m, octave_idx_type q, const RowVector &V, const Matrix &P, double u, double v, octave_idx_type d, Matrix &skl); nurbs-1.4.3/src/PaxHeaders.16248/low_level_functions.cc0000644000000000000000000000013214030323741017564 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/low_level_functions.cc0000644000175000017500000003037214030323741021670 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009 Carlo de Falco some functions are adapted from the m-file implementation which is Copyright (C) 2000 Mark Spink, 2007 Daniel Claxton This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include "low_level_functions.h" #include octave_idx_type findspan (int n, int p, double u, const RowVector& U) // Find the knot span of the parametric point u. // // INPUT: // // n - number of control points - 1 // p - spline degree // u - parametric point // U - knot sequence // // RETURN: // // s - knot span // // Note: This is NOT // Algorithm A2.1 from 'The NURBS BOOK' pg68 // as that algorithm only works for nonperiodic // knot vectors, nonetheless the results should // be EXACTLY the same if U is nonperiodic /* Below is the original implementation from the NURBS Book { int low, high, mid; // special case if (u == U(n+1)) return(n); // do binary search low = p; high = n + 1; mid = (low + high) / 2; while (u < U(mid) || u >= U(mid+1)) { if (u < U(mid)) high = mid; else low = mid; mid = (low + high) / 2; } return(mid); } */ { // FIXME : this implementation has linear, rather than log complexity int ret = 0; if (u > U.xelem (U.numel () - 1) || u < U.xelem (0)) error ("Value %g is outside the knot span", u); else while ((ret++ < n) && (U(ret) <= u)) { }; return (ret-1); } void basisfun (int i, double u, int p, const RowVector& U, RowVector& N) // Basis Function. // // INPUT: // // i - knot span ( from FindSpan() ) // u - parametric point // p - spline degree // U - knot sequence // // OUTPUT: // // N - Basis functions vector[p+1] // // Algorithm A2.2 from 'The NURBS BOOK' pg70. { int j,r; double saved, temp; // work space OCTAVE_LOCAL_BUFFER(double, left, p+1); OCTAVE_LOCAL_BUFFER(double, right, p+1); N(0) = 1.0; for (j = 1; j <= p; j++) { left[j] = u - U(i+1-j); right[j] = U(i+j) - u; saved = 0.0; for (r = 0; r < j; r++) { temp = N(r) / (right[r+1] + left[j-r]); N(r) = saved + right[r+1] * temp; saved = left[j-r] * temp; } N(j) = saved; } } void basisfunder (int i, int pl, double u, const RowVector& u_knotl, int nders, NDArray& ders) { // BASISFUNDER: B-Spline Basis function derivatives // // INPUT: // // i - knot span // pl - degree of curve // u - parametric points // k - knot vector // nd - number of derivatives to compute // // OUTPUT: // // ders - ders(n, i, :) (i-1)-th derivative at n-th point // ders = zeros(nders+1,pl+1); Matrix ndu(octave_idx_type(pl+1), octave_idx_type(pl+1), 0.0); // ndu = zeros(pl+1,pl+1); RowVector left(octave_idx_type(pl+1), 0.0); // left = zeros(pl+1); RowVector right(left); // right = zeros(pl+1); Matrix a(2, octave_idx_type(pl+1), 0.0); // a = zeros(2,pl+1); double saved = 0.0, d = 0.0, temp = 1.0; octave_idx_type s1(0), s2(1), rk, pk, j, k, r, j1, j2; ndu(0,0) = 1; // ndu(1,1) = 1; for (j=1; j<=pl; j++) // for j = 1:pl { left(j) = u - u_knotl(i+1-j); // left(j+1) = u - u_knotl(i+1-j); right(j) = u_knotl(i+j) - u; // right(j+1) = u_knotl(i+j) - u; saved = 0.0; // saved = 0; for (r=0; r<=j-1; r++) // for r = 0:j-1 { ndu(j, r) = right(r+1) + left(j-r); // ndu(j+1,r+1) = right(r+2) + left(j-r+1); temp = ndu(r,j-1)/ndu(j,r); // temp = ndu(r+1,j)/ndu(j+1,r+1); ndu(r,j) = saved + right(r+1)*temp; // ndu(r+1,j+1) = saved + right(r+2)*temp; saved = left(j-r)*temp; // saved = left(j-r+1)*temp; } // end ndu(j,j) = saved; // ndu(j+1,j+1) = saved; } // end for (j=0; j<=pl; j++) // for j = 0:pl ders(0,j) = ndu(j,pl); // ders(1,j+1) = ndu(j+1,pl+1); // end for (r=0; r<=pl; r++) // for r = 0:pl { s1 = 0; // s1 = 0; s2 = 1; // s2 = 1; a(0,0) = 1; // a(1,1) = 1; for (k=1; k<=nders; k++) // for k = 1:nders %compute kth derivative { d = 0.0; // d = 0; rk = r-k; // rk = r-k; pk = pl - k; // pk = pl-k; if (r >= k) // if (r >= k) { a(s2, 0) = a(s1, 0)/ndu(pk+1,rk); // a(s2+1,1) = a(s1+1,1)/ndu(pk+2,rk+1); d = a(s2, 0)*ndu(rk,pk); // d = a(s2+1,1)*ndu(rk+1,pk+1); } // end if (rk >= -1) // if (rk >= -1) j1 = 1; // j1 = 1; else // else j1 = -rk; // j1 = -rk; // end if ((r-1) <= pk) // if ((r-1) <= pk) j2 = k-1; // j2 = k-1; else // else j2 = pl-r; // j2 = pl-r; // end for (j=j1; j <= j2; j++) // for j = j1:j2 { a(s2,j) = (a(s1,j) - a(s1,j-1))/ndu(pk+1,rk+j); // a(s2+1,j+1) = (a(s1+1,j+1) - a(s1+1,j))/ndu(pk+2,rk+j+1); d += a(s2,j)*ndu(rk+j,pk); // d = d + a(s2+1,j+1)*ndu(rk+j+1,pk+1); } // end if (r <= pk) // if (r <= pk) { a(s2,k) = -a(s1,k-1)/ndu(pk+1,r); // a(s2+1,k+1) = -a(s1+1,k)/ndu(pk+2,r+1); d += a(s2,k)*ndu(r,pk); // d = d + a(s2+1,k+1)*ndu(r+1,pk+1); } // end ders(k,r) = d; // ders(k+1,r+1) = d; j = s1; // j = s1; s1 = s2; // s1 = s2; s2 = j; // s2 = j; } // end } // end r = pl; // r = pl; for (k=1; k <= nders; k++) // for k = 1:nders { for (j=0; j<=pl; j++) // for j = 0:pl ders(k,j) = ders(k,j)*r; // ders(k+1,j+1) = ders(k+1,j+1)*r; // end r = r*(pl-k); // r = r*(pl-k); } // end } int curvederivcpts (octave_idx_type n, octave_idx_type p, const RowVector &U, const NDArray &P, octave_idx_type d, octave_idx_type r1, octave_idx_type r2, Matrix &pk) { octave_idx_type r = r2 - r1; for (octave_idx_type i(0); i<=r; i++) pk(0, i) = P(r1+i); for (octave_idx_type k (1); k<=d; k++) { octave_idx_type tmp = p - k + 1; for (octave_idx_type i (0); i<=r-k; i++) { pk (k, i) = tmp * (pk(k-1,i+1)-pk(k-1,i)) / (U(r1+i+p+1)-U(r1+i+k)); } } return 0; } int surfderivcpts (octave_idx_type n, octave_idx_type p, const RowVector& U, octave_idx_type m, octave_idx_type q, const RowVector& V, const Matrix& P, octave_idx_type d, octave_idx_type r1, octave_idx_type r2, octave_idx_type s1, octave_idx_type s2, NDArray &pkl) { octave_idx_type r = r2-r1, s = s2-s1; octave_idx_type du = d <= p ? d : p; octave_idx_type dv = d <= q ? d : q; dim_vector idxa (4, 1); Array idxta (idxa, 0); Array idxva (idxa, idx_vector (':')); idxa.resize (4); idxa(0) = (du+1); idxa(1) = (dv+1); idxa(2) = (r+1); idxa(3) = (s+1); pkl.resize (idxa, 0.0); for (octave_idx_type j(s1); j<=s2; j++) { Matrix temp (du <= n ? (du+1) : (n+1), n+1, 0.0); curvederivcpts (n, p, U, P.extract (0, j, P.rows()-1, P.cols ()-1), du, r1, r2, temp); for (octave_idx_type k(0); k<=du; k++) { for ( octave_idx_type i(0); i<=r-k; i++) { assert (k idx(dim_vector (4, 1), 0); octave_idx_type du = d <= p ? d: p; octave_idx_type dv = d <= q ? d: q; skl.resize (d+1, d+1, 0.0); octave_idx_type uspan = findspan (n, p, u, U); Matrix Nu (p+1, p+1, 0.0); for (octave_idx_type ip(0); ip<=p; ip++) { RowVector temp (ip+1, 0.0); basisfun (uspan, u, ip, U, temp); Nu.insert (temp.transpose (), 0,ip); } octave_idx_type vspan = findspan (m, q, v, V); Matrix Nv (q+1, q+1, 0.0); for (octave_idx_type iq(0); iq<=q; iq++) { RowVector temp (iq+1, 0.0); basisfun (vspan, v, iq, V, temp); Nv.insert (temp.transpose (), 0, iq); } NDArray pkl; surfderivcpts (n, p, U, m, q, V, P, d, uspan-p, uspan, vspan-q, vspan, pkl); for (octave_idx_type k(0); k<=du; k++) { octave_idx_type dd = d-k <= dv ? d-k : dv; for (octave_idx_type l(0);l <= dd; l++) { skl(k,l) = 0.0; for (octave_idx_type i(0); i<=q-l; i++) { double tmp = 0.0; for (octave_idx_type j(0); j<=p-k; j++) { idx(0) = k; idx(1)=l; idx(2)=j; idx(3) =i; tmp += Nu(j,p-k) * pkl(idx); } skl(k,l) += Nv(i,q-l) * tmp; } } } return (0); } nurbs-1.4.3/src/PaxHeaders.16248/nrb_srf_basisfun_der__.cc0000644000000000000000000000013214030323741020161 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/nrb_srf_basisfun_der__.cc0000644000175000017500000001542014030323741022262 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include #include #include "low_level_functions.h" DEFUN_DLD(nrb_srf_basisfun_der__, args, nargout,"\ NRB_SRF_BASISFUN_DER__: Undocumented private function \ ") { //function [Bu, Bv, N] = nrb_srf_basisfun_der__ (points, nrb); octave_value_list retval, newargs; if (nargout != 1 || args.length () != 2) print_usage (); const NDArray points = args(0).array_value(); const octave_scalar_map nrb = args(1).scalar_map_value(); const Cell knots = nrb.contents("knots").cell_value(); const NDArray coefs = nrb.contents("coefs").array_value(); octave_idx_type m = static_cast ((nrb.contents("number").vector_value())(0)) - 1; // m = size (nrb.coefs, 2) -1; octave_idx_type n = static_cast ((nrb.contents("number").vector_value())(1)) - 1; // n = size (nrb.coefs, 3) -1; octave_idx_type p = static_cast ((nrb.contents("order").vector_value())(0)) - 1; // p = nrb.order(1) -1; octave_idx_type q = static_cast ((nrb.contents("order").vector_value())(1)) - 1; // q = nrb.order(2) -1; Array idx(dim_vector (2, 1), idx_vector(':')); idx(0) = octave_idx_type(0); const NDArray u(points.index (idx).squeeze ()); // u = points(1,:); idx(0) = octave_idx_type(1); const NDArray v(points.index (idx).squeeze ()); // v = points(2,:); octave_idx_type npt = u.numel (); // npt = length(u); RowVector M(p+1, 0.0), N (q+1, 0.0); Matrix Nout(npt, (p+1)*(q+1), 0.0); Matrix Bu(npt, (p+1)*(q+1), 0.0); Matrix Bv(npt, (p+1)*(q+1), 0.0); RowVector Denom(npt, 0.0); RowVector Denom_du(npt, 0.0); RowVector Denom_dv(npt, 0.0); Matrix Num(npt, (p+1)*(q+1), 0.0); Matrix Num_du(npt, (p+1)*(q+1), 0.0); Matrix Num_dv(npt, (p+1)*(q+1), 0.0); const RowVector U(knots(0).row_vector_value ()); // U = nrb.knots{1}; const RowVector V(knots(1).row_vector_value ()); // V = nrb.knots{2}; Array idx2(dim_vector (3, 1), idx_vector(':')); idx2(0) = octave_idx_type(3); NDArray w (coefs.index (idx2).squeeze ()); // w = squeeze(nrb.coefs(4,:,:)); RowVector spu(u); for (octave_idx_type ii(0); ii < npt; ii++) { spu(ii) = findspan(m, p, u(ii), U); } // spu = findspan (m, p, u, U); newargs(3) = U; newargs(2) = p; newargs(1) = u; newargs(0) = spu; Matrix Ik = octave::feval (std::string("numbasisfun"), newargs, 1)(0).matrix_value (); // Ik = numbasisfun (spu, u, p, U); RowVector spv(v); for (octave_idx_type ii(0); ii < v.numel (); ii++) { spv(ii) = findspan(n, q, v(ii), V); } // spv = findspan (n, q, v, V); newargs(3) = V; newargs(2) = q; newargs(1) = v; newargs(0) = spv; Matrix Jk = octave::feval (std::string("numbasisfun"), newargs, 1)(0).matrix_value (); // Jk = numbasisfun (spv, v, q, V); Matrix NuIkuk(npt, p+1, 0.0); for (octave_idx_type ii(0); ii < npt; ii++) { basisfun (int(spu(ii)), u(ii), p, U, M); NuIkuk.insert (M, ii, 0); } // NuIkuk = basisfun (spu, u, p, U); Matrix NvJkvk(v.numel (), q+1, 0.0); for (octave_idx_type ii(0); ii < npt; ii++) { basisfun(int(spv(ii)), v(ii), q, V, N); NvJkvk.insert (N, ii, 0); } // NvJkvk = basisfun (spv, v, q, V); newargs(4) = 1; newargs(3) = U; newargs(2) = u; newargs(1) = p; newargs(0) = spu; NDArray NuIkukprime = octave::feval (std::string("basisfunder"), newargs, 1)(0).array_value (); // NuIkukprime = basisfunder (spu, p, u, U, 1); // NuIkukprime = squeeze(NuJkukprime(:,2,:)); newargs(4) = 1; newargs(3) = V; newargs(2) = v; newargs(1) = q; newargs(0) = spv; NDArray NvJkvkprime = octave::feval (std::string("basisfunder"), newargs, 1)(0).array_value (); // NvJkvkprime = basisfunder (spv, q, v, V, 1); // NvJkvkprime = squeeze(NvJkvkprime(:,2,:)); for (octave_idx_type k(0); k < npt; k++) for (octave_idx_type ii(0); ii < p+1; ii++) for (octave_idx_type jj(0); jj < q+1; jj++) { Num(k, ii+jj*(p+1)) = NuIkuk(k, ii) * NvJkvk(k, jj) * w(static_cast (Ik(k, ii)), static_cast (Jk(k, jj))); Denom(k) += Num(k, ii+jj*(p+1)); Num_du(k, ii+jj*(p+1)) = NuIkukprime(k, 1, ii) * NvJkvk(k, jj) * w(static_cast (Ik(k, ii)), static_cast (Jk(k, jj))); Denom_du(k) += Num_du(k, ii+jj*(p+1)); Num_dv(k, ii+jj*(p+1)) = NuIkuk(k, ii) * NvJkvkprime(k, 1, jj) * w(static_cast (Ik(k, ii)), static_cast (Jk(k, jj))); Denom_dv(k) += Num_dv(k, ii + jj * (p+1)); } for (octave_idx_type k(0); k < npt; k++) for (octave_idx_type ii(0); ii < p+1; ii++) for (octave_idx_type jj(0); jj < q+1; jj++) { Bu(k, octave_idx_type(ii+(p+1)*jj)) = (Num_du(k, ii+jj*(p+1))/Denom(k) - Denom_du(k)*Num(k, ii+jj*(p+1))/(Denom(k)*Denom(k))); Bv(k, octave_idx_type(ii+(p+1)*jj)) = (Num_dv(k, ii+jj*(p+1))/Denom(k) - Denom_dv(k)*Num(k, ii+jj*(p+1))/(Denom(k)*Denom(k))); Nout(k, octave_idx_type(ii+(p+1)*jj))= Ik(k, ii)+(m+1)*Jk(k, jj)+1; } // for k=1:npt // [Ika, Jkb] = meshgrid(Ik(k, :), Jk(k, :)); // N(k, :) = sub2ind([m+1, n+1], Ika(:)+1, Jkb(:)+1); // wIkaJkb(1:p+1, 1:q+1) = reshape (w(N(k, :)), p+1, q+1); // Num = (NuIkuk(k, :).' * NvJkvk(k, :)) .* wIkaJkb; // Num_du = (NuIkukprime(k, :).' * NvJkvk(k, :)) .* wIkaJkb; // Num_dv = (NuIkuk(k, :).' * NvJkvkprime(k, :)) .* wIkaJkb; // Denom = sum(sum(Num)); // Denom_du = sum(sum(Num_du)); // Denom_dv = sum(sum(Num_dv)); // Bu(k, :) = (Num_du/Denom - Denom_du.*Num/Denom.^2)(:).'; // Bv(k, :) = (Num_dv/Denom - Denom_dv.*Num/Denom.^2)(:).'; // end retval(2) = Nout; retval(1) = Bv; retval(0) = Bu; return retval; } nurbs-1.4.3/src/PaxHeaders.16248/nrb_srf_basisfun__.cc0000644000000000000000000000013214030323741017327 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/nrb_srf_basisfun__.cc0000644000175000017500000001162114030323741021427 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include #include #include "low_level_functions.h" DEFUN_DLD(nrb_srf_basisfun__, args, nargout,"\ NRB_SRF_BASISFUN__: Undocumented private function\ ") { octave_value_list retval, newargs; if (nargout != 1 || args.length () != 2) print_usage (); const NDArray points = args(0).array_value(); const octave_scalar_map nrb = args(1).scalar_map_value(); const Cell knots = nrb.contents("knots").cell_value(); const NDArray coefs = nrb.contents("coefs").array_value(); octave_idx_type m = static_cast ((nrb.contents("number").vector_value())(0)) - 1; // m = size (nrb.coefs, 2) -1; octave_idx_type n = static_cast ((nrb.contents("number").vector_value())(1)) - 1; // n = size (nrb.coefs, 3) -1; octave_idx_type p = static_cast ((nrb.contents("order").vector_value())(0)) - 1; // p = nrb.order(1) -1; octave_idx_type q = static_cast ((nrb.contents("order").vector_value())(1)) - 1; // q = nrb.order(2) -1; Array idx(dim_vector (2, 1), idx_vector(':')); idx(0) = octave_idx_type(0); const NDArray u(points.index (idx).squeeze ()); // u = points(1,:); idx(0) = octave_idx_type(1); const NDArray v(points.index (idx).squeeze ()); // v = points(2,:); octave_idx_type npt = u.numel (); // npt = length(u); RowVector M(p+1, 0.0), N (q+1, 0.0); Matrix RIkJk(npt, (p+1)*(q+1), 0.0); Matrix indIkJk(npt, (p+1)*(q+1), 0.0); RowVector denom(npt, 0.0); const RowVector U(knots(0).row_vector_value ()); // U = nrb.knots{1}; const RowVector V(knots(1).row_vector_value ()); // V = nrb.knots{2}; Array idx2(dim_vector (3, 1), idx_vector(':')); idx2(0) = octave_idx_type(3); NDArray w (coefs.index (idx2).squeeze ()); // w = squeeze(nrb.coefs(4,:,:)); RowVector spu(u); for (octave_idx_type ii(0); ii < npt; ii++) { spu(ii) = findspan(m, p, u(ii), U); } // spu = findspan (m, p, u, U); newargs(3) = U; newargs(2) = p; newargs(1) = u; newargs(0) = spu; Matrix Ik = octave::feval (std::string("numbasisfun"), newargs, 1)(0).matrix_value (); // Ik = numbasisfun (spu, u, p, U); RowVector spv(v); for (octave_idx_type ii(0); ii < v.numel (); ii++) { spv(ii) = findspan(n, q, v(ii), V); } // spv = findspan (n, q, v, V); newargs(3) = V; newargs(2) = q; newargs(1) = v; newargs(0) = spv; Matrix Jk = octave::feval (std::string("numbasisfun"), newargs, 1)(0).matrix_value (); // Jk = numbasisfun (spv, v, q, V); Matrix NuIkuk(npt, p+1, 0.0); for (octave_idx_type ii(0); ii < npt; ii++) { basisfun (int(spu(ii)), u(ii), p, U, M); NuIkuk.insert (M, ii, 0); } // NuIkuk = basisfun (spu, u, p, U); Matrix NvJkvk(v.numel (), q+1, 0.0); for (octave_idx_type ii(0); ii < npt; ii++) { basisfun(int(spv(ii)), v(ii), q, V, N); NvJkvk.insert (N, ii, 0); } // NvJkvk = basisfun (spv, v, q, V); for (octave_idx_type k(0); k < npt; k++) for (octave_idx_type ii(0); ii < p+1; ii++) for (octave_idx_type jj(0); jj < q+1; jj++) denom(k) += NuIkuk(k, ii) * NvJkvk(k, jj) * w(static_cast (Ik(k, ii)), static_cast (Jk(k, jj))); for (octave_idx_type k(0); k < npt; k++) for (octave_idx_type ii(0); ii < p+1; ii++) for (octave_idx_type jj(0); jj < q+1; jj++) { RIkJk(k, octave_idx_type(ii+(p+1)*jj)) = NuIkuk(k, ii) * NvJkvk(k, jj) * w(static_cast (Ik(k, ii)), static_cast (Jk(k, jj))) / denom(k); indIkJk(k, octave_idx_type(ii+(p+1)*jj))= Ik(k, ii) + (m+1) * Jk(k, jj) + 1; } // for k=1:npt // [Jkb, Ika] = meshgrid(Jk(k, :), Ik(k, :)); // indIkJk(k, :) = sub2ind([m+1, n+1], Ika(:)+1, Jkb(:)+1); // wIkaJkb(1:p+1, 1:q+1) = reshape (w(indIkJk(k, :)), p+1, q+1); // NuIkukaNvJkvk(1:p+1, 1:q+1) = (NuIkuk(k, :).' * NvJkvk(k, :)); // RIkJk(k, :) = (NuIkukaNvJkvk .* wIkaJkb ./ sum(sum(NuIkukaNvJkvk .* wIkaJkb)))(:).'; // end retval(0) = RIkJk; // B = RIkJk; retval(1) = indIkJk; // N = indIkJk; return retval; } nurbs-1.4.3/src/PaxHeaders.16248/nrbsurfderiveval.cc0000644000000000000000000000013214030323741017067 xustar0030 mtime=1617012705.815919627 30 atime=1617012705.815919627 30 ctime=1617012705.823919749 nurbs-1.4.3/src/nrbsurfderiveval.cc0000644000175000017500000004101014030323741021162 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include #include #include "low_level_functions.h" static double gammaln(double xx) // Compute logarithm of the gamma function // Algorithm from 'Numerical Recipes in C, 2nd Edition' pg214. { double x,y,tmp,ser; static double cof[6] = {76.18009172947146,-86.50532032291677, 24.01409824083091,-1.231739572450155, 0.12086650973866179e-2, -0.5395239384953e-5}; int j; y = x = xx; tmp = x + 5.5; tmp -= (x+0.5) * log(tmp); ser = 1.000000000190015; for (j=0; j<=5; j++) ser += cof[j]/++y; return -tmp+log(2.5066282746310005*ser/x); } static double factln(int n) // computes ln(n!) // Numerical Recipes in C // Algorithm from 'Numerical Recipes in C, 2nd Edition' pg215. { static int ntop = 0; static double a[101]; if (n <= 1) return 0.0; while (n > ntop) { ++ntop; a[ntop] = gammaln(ntop+1.0); } return a[n]; } static double bincoeff(int n, int k) // Computes the binomial coefficient. // // ( n ) n! // ( ) = -------- // ( k ) k!(n-k)! // // Algorithm from 'Numerical Recipes in C, 2nd Edition' pg215. { return floor(0.5+exp(factln(n)-factln(k)-factln(n-k))); } DEFUN_DLD(nrbsurfderiveval, args, nargout,"\ \nNRBSURFDERIVEVAL: Evaluate n-th order derivatives of a NURBS surface.\n\ \n\ \n usage: skl = nrbsurfderiveval (srf, [u; v], d) \ \n\ \n INPUT :\ \n\ \n srf : NURBS surface structure, see nrbmak\ \n\ \n u, v : parametric coordinates of the point where we compute the\ \n derivatives\ \n\ \n d : number of partial derivatives to compute\ \n\ \n OUTPUT :\ \n\ \n skl (i, j, k, l) = i-th component derived j-1,k-1 times at the\ \n l-th point.\ \n\ \n Adaptation of algorithm A4.4 from the NURBS book\n") { //function skl = nrbsurfderiveval (srf, uv, d) octave_value_list retval; if (nargout != 1 || args.length () != 3) print_usage (); octave_scalar_map srf = args(0).scalar_map_value(); Matrix uv = args(1).matrix_value (); octave_idx_type d = args(2).idx_type_value (); Array idxta (dim_vector (4, 1), 0); dim_vector idxa; idxa.resize (4); idxa(0) = 3; idxa(1) = d+1; idxa(2) = d+1; idxa(3) = uv.columns (); NDArray skl (idxa, 0.0); octave_idx_type n = octave_idx_type ((srf.contents("number").row_vector_value())(0) - 1); octave_idx_type m = octave_idx_type ((srf.contents("number").row_vector_value())(1) - 1); octave_idx_type p = octave_idx_type ((srf.contents("order").row_vector_value())(0) - 1); octave_idx_type q = octave_idx_type ((srf.contents("order").row_vector_value())(1) - 1); Cell knots = srf.contents("knots").cell_value(); RowVector knotsu = knots.elem (0).row_vector_value (); RowVector knotsv = knots.elem (1).row_vector_value (); NDArray coefs = srf.contents("coefs").array_value(); Array idx(dim_vector (3, 1), idx_vector(':')); idx (0) = idx_vector (octave_idx_type(3)); Matrix weights (NDArray (coefs.index (idx).squeeze ())); for (octave_idx_type iu(0); iu. */ #include #include "low_level_functions.h" DEFUN_DLD(curvederivcpts, args, nargout,"\ \nCURVEDERIVCPTS: Compute control points of n-th derivatives of a B-spline curve.\n \ \n \ \n usage: pk = curvederivcpts (n, p, U, P, d) \ \n pk = curvederivcpts (n, p, U, P, d, r1 r2) \ \n \ \n If r1, r2 are not given, all the control points are computed. \ \n \ \n INPUT: \ \n n+1 = number of control points \ \n p = degree of the spline \ \n d = maximum derivative order (d<=p) \ \n U = knots \ \n P = control points \ \n r1 = first control point to compute \ \n r2 = auxiliary index for the last control point to compute \ \n\ \n OUTPUT: \ \n pk(k,i) = i-th control point (k-1)-th derivative, r1 <= i <= r2-k \ \n \ \n Adaptation of algorithm A3.3 from the NURBS book\n") { octave_value_list retval; if (nargout != 1 || (args.length () != 5 && args.length() != 7)) print_usage (); octave_idx_type n = args(0).idx_type_value (); octave_idx_type p = args(1).idx_type_value (); RowVector U = args(2).row_vector_value (false, true); NDArray P = args(3).array_value (); octave_idx_type d = args(4).idx_type_value (); octave_idx_type r1(0), r2(n); if (args.length () == 7) { r1 = args (5).idx_type_value (); r2 = args (6).idx_type_value (); } else if (args.length () > 5) print_usage (); octave_idx_type r = r2 - r1; Matrix pk (d+1 <= r+1 ? d+1 : r+1, r+1, 0.0); curvederivcpts (n, p, U, P, d, r1, r2, pk); retval(0) = octave_value (pk); return retval; } /* %!test %! line = nrbmak([0.0 1.5; 0.0 3.0],[0.0 0.0 1.0 1.0]); %! pk = curvederivcpts (line.number-1, line.order-1, line.knots, %! line.coefs(1,:), 2); %! assert (pk, [0 3/2; 3/2 0], 100*eps); */ nurbs-1.4.3/src/PaxHeaders.16248/surfderiveval.cc0000644000000000000000000000013214030323741016365 xustar0030 mtime=1617012705.815919627 30 atime=1617012705.815919627 30 ctime=1617012705.823919749 nurbs-1.4.3/src/surfderiveval.cc0000644000175000017500000000602614030323741020470 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include "low_level_functions.h" #include DEFUN_DLD(surfderiveval, args, nargout,"\ \nSURFDERIVEVAL: Compute the derivatives of a B-spline surface\ \n\ \n usage: skl = surfderiveval (n, p, U, m, q, V, P, u, v, d) \ \n\ \n INPUT: \ \n\ \n n+1, m+1 = number of control points\ \n p, q = spline order\ \n U, V = knots\ \n P = control points\ \n u,v = evaluation points\ \n d = derivative order\ \n\ \n OUTPUT:\ \n\ \n skl (k+1, l+1) = surface differentiated k\ \n times in the u direction and l\ \n times in the v direction\ \n\ \n Adaptation of algorithm A3.8 from the NURBS book\n") { //function skl = surfderiveval (n, p, U, m, q, V, P, u, v, d) octave_value_list retval; if (nargout != 1 || args.length () != 10) print_usage (); octave_idx_type n = args(0).idx_type_value (); octave_idx_type p = args(1).idx_type_value (); RowVector U = args(2).row_vector_value (false, true); octave_idx_type m = args(3).idx_type_value (); octave_idx_type q = args(4).idx_type_value (); RowVector V = args(5).row_vector_value (false, true); Matrix P = args(6).matrix_value (); double u = args(7).double_value (); double v = args(8).double_value (); octave_idx_type d = args(9).idx_type_value (); Matrix skl; surfderiveval (n, p, U, m, q, V, P, u, v, d, skl); retval(0) = octave_value (skl); return retval; } /* %!shared srf %!test %! k = [0 0 0 1 1 1]; %! c = [0 1/2 1]; %! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c); %! srf = nrbmak (coef, {k, k}); %! skl = surfderiveval (srf.number(1)-1, %! srf.order(1)-1, %! srf.knots{1}, %! srf.number(2)-1, %! srf.order(2)-1, %! srf.knots{2}, %! squeeze(srf.coefs(1,:,:)), .5, .5, 1) ; %! assert (skl, [.5 0; 1 0]) %!test %! srf = nrbkntins (srf, {[], rand(1,2)}); %! skl = surfderiveval (srf.number(1)-1, %! srf.order(1)-1, %! srf.knots{1}, %! srf.number(2)-1, %! srf.order(2)-1, %! srf.knots{2}, %! squeeze(srf.coefs(1,:,:)), .5, .5, 1) ; %! assert (skl, [.5 0; 1 0], 100*eps) */ nurbs-1.4.3/src/PaxHeaders.16248/basisfun.cc0000644000000000000000000000013214030323741015316 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/basisfun.cc0000644000175000017500000000451014030323741017415 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include "low_level_functions.h" DEFUN_DLD(basisfun, args, nargout, "\n\ BASISFUN: Compute B-Spline Basis Functions \n\ \n\ Calling Sequence:\n\ \n\ N = basisfun(iv,uv,p,U)\n\ \n\ INPUT:\n\ \n\ iv - knot span ( from FindSpan() )\n\ uv - parametric point\n\ p - spline degree\n\ U - knot sequence\n\ \n\ OUTPUT:\n\ \n\ N - Basis functions vector(numel(uv)*(p+1))\n\ \n\ Algorithm A2.2 from 'The NURBS BOOK' pg70.\n\ \n\ ") { octave_value_list retval; if (nargout != 1 || args.length () != 4) print_usage (); const NDArray i = args(0).array_value(); const NDArray u = args(1).array_value(); int p = args(2).idx_type_value(); const RowVector U = args(3).row_vector_value(); RowVector N(p+1, 0.0); Matrix B(u.numel (), p+1, 0.0); for (octave_idx_type ii(0); ii < u.numel (); ii++) { basisfun(int(i(ii)), u(ii), p, U, N); B.insert(N, ii, 0); } retval(0) = octave_value(B); return retval; } /* %!shared n, U, p, u, s %!test %! n = 3; %! U = [0 0 0 1/2 1 1 1]; %! p = 2; %! u = linspace(0, 1, 10); %! s = findspan(n, p, u, U); %! assert (s, [2*ones(1, 5) 3*ones(1, 5)]); %!test %! Bref = [1.00000 0.00000 0.00000 %! 0.60494 0.37037 0.02469 %! 0.30864 0.59259 0.09877 %! 0.11111 0.66667 0.22222 %! 0.01235 0.59259 0.39506 %! 0.39506 0.59259 0.01235 %! 0.22222 0.66667 0.11111 %! 0.09877 0.59259 0.30864 %! 0.02469 0.37037 0.60494 %! 0.00000 0.00000 1.00000]; %! B = basisfun(s, u, p, U); %! assert (B, Bref, 1e-5); */ nurbs-1.4.3/src/PaxHeaders.16248/basisfunder.cc0000644000000000000000000000013214030323741016011 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/basisfunder.cc0000644000175000017500000000404114030323741020107 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include "low_level_functions.h" DEFUN_DLD(basisfunder, args, nargout,"\n\ BASISFUNDER: B-Spline Basis function derivatives\n\ \n\ Calling Sequence:\n\ \n\ ders = basisfunder (ii, pl, uu, k, nd)\n\ \n\ INPUT:\n\ \n\ ii - knot span\n\ pl - degree of curve\n\ uu - parametric points\n\ k - knot vector\n\ nd - number of derivatives to compute\n\ \n\ OUTPUT:\n\ \n\ ders - ders(n, i, :) (i-1)-th derivative at n-th point\n\ \n\ Adapted from Algorithm A2.3 from 'The NURBS BOOK' pg72. \n\ \n\ ") { octave_value_list retval; if (nargout != 1 || args.length () != 5) print_usage (); const NDArray i = args(0).array_value (); int pl = args(1).int_value (); const NDArray u = args(2).array_value (); const RowVector U = args(3).row_vector_value (); int nd = args(4).int_value (); if (i.numel () != u.numel ()) print_usage (); NDArray dersv (dim_vector (i.numel (), nd+1, pl+1), 0.0); NDArray ders(dim_vector(nd+1, pl+1), 0.0); for ( octave_idx_type jj(0); jj < i.numel (); jj++) { basisfunder (int (i(jj)), pl, u(jj), U, nd, ders); for (octave_idx_type kk(0); kk < nd+1; kk++) for (octave_idx_type ll(0); ll < pl+1; ll++) { dersv(jj, kk, ll) = ders(kk, ll); } } retval(0) = dersv; return retval; } nurbs-1.4.3/src/PaxHeaders.16248/Makefile0000644000000000000000000000013214030323741014635 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/Makefile0000644000175000017500000000217414030323741016740 0ustar00rvazquezrvazquez00000000000000## Copyright (C) 2009-2017 Carlo de Falco ## ## This program is free software: you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program. If not, see . OCTFILES=basisfun.oct \ basisfunder.oct \ bspderiv.oct \ bspeval.oct \ curvederivcpts.oct \ nrb_srf_basisfun__.oct \ nrb_srf_basisfun_der__.oct \ nrbsurfderiveval.oct \ surfderivcpts.oct \ surfderiveval.oct \ tbasisfun.oct MKOCTFILE ?= mkoctfile all: $(OCTFILES) low_level_functions.o: low_level_functions.cc $(MKOCTFILE) -c $< %.oct: %.cc low_level_functions.o $(MKOCTFILE) $< low_level_functions.o clean: -rm -f *.o core octave-core *.oct *~ nurbs-1.4.3/src/PaxHeaders.16248/bspderiv.cc0000644000000000000000000000013214030323741015322 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/src/bspderiv.cc0000644000175000017500000000415714030323741017430 0ustar00rvazquezrvazquez00000000000000/* Copyright (C) 2009, 2020 Carlo de Falco, Rafael Vazquez This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see . */ #include #include "low_level_functions.h" DEFUN_DLD(bspderiv, args, nargout,"\n\ BSPDERIV: B-Spline derivative\n\ \n\ \n\ Calling Sequence:\n\ \n\ [dc,dk] = bspderiv(d,c,k)\n\ \n\ INPUT:\n\ \n\ d - degree of the B-Spline\n\ c - control points double matrix(mc,nc)\n\ k - knot sequence double vector(nk)\n\ \n\ OUTPUT:\n\ \n\ dc - control points of the derivative double matrix(mc,nc)\n\ dk - knot sequence of the derivative double vector(nk)\n\ \n\ Modified version of Algorithm A3.3 from 'The NURBS BOOK' pg98.\n\ ") { //if (bspderiv_bad_arguments(args, nargout)) // return octave_value_list(); if ((nargout != 1 && nargout != 2) || args.length () != 3) print_usage (); int d = args(0).int_value(); const Matrix c = args(1).matrix_value(); const RowVector k = args(2).row_vector_value(); octave_value_list retval; octave_idx_type mc = c.rows(), nc = c.cols(), nk = k.numel(); Matrix dc (mc, nc-1, 0.0); RowVector dk(nk-2, 0.0); double tmp; for (octave_idx_type i(0); i<=nc-2; i++) { tmp = (double)d / (k(i+d+1) - k(i+1)); for ( octave_idx_type j(0); j<=mc-1; j++) dc(j,i) = tmp*(c(j,i+1) - c(j,i)); } for ( octave_idx_type i(1); i <= nk-2; i++) dk(i-1) = k(i); if (nargout>1) retval(1) = octave_value(dk); retval(0) = octave_value(dc); return(retval); } nurbs-1.4.3/PaxHeaders.16248/DESCRIPTION0000644000000000000000000000013214030323741014114 xustar0030 mtime=1617012705.775919014 30 atime=1617012705.775919014 30 ctime=1617012705.823919749 nurbs-1.4.3/DESCRIPTION0000644000175000017500000000063614030323741016220 0ustar00rvazquezrvazquez00000000000000Name: nurbs Version: 1.4.3 Date: 2021-03-29 Author: Mark Spink, Daniel Claxton, Carlo de Falco, Rafael Vazquez Maintainer: Carlo de Falco and Rafael Vazquez Title: Nurbs. Description: Collection of routines for the creation, and manipulation of Non-Uniform Rational B-Splines (NURBS), based on the NURBS toolbox by Mark Spink. Categories: splines Depends: octave (>= 5.1) License: GPLv3+ Url: http://octave.sf.net nurbs-1.4.3/PaxHeaders.16248/INDEX0000644000000000000000000000013214030323741013200 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/INDEX0000644000175000017500000000230314030323741015275 0ustar00rvazquezrvazquez00000000000000nurbs >> Nurbs Basic operations for NURBS curves, surfaces and volumes nrbmak nrbkntins nrbdegelev nrbderiv nrbdeval nrbeval nrbeval_der_w nrbeval_der_p nrbinverse crvkntremove Operations for constructing NURBS curves and surfaces nrbtform nrbreverse nrbtransp nrbpermute nrbline nrbcirc nrbrect nrbsquare nrb4surf nrbspheretiling nrbspheretile nrbcylind nrbextract nrbextrude nrbrevolve nrbruled nrbcoons nrbtestcrv nrbtestsrf nrbclamp nrbunclamp nrbmultipatch nrbglue nrbmodp nrbmodw Plot and export nrbplot nrbctrlplot nrbkntplot nrbexport nrb2iges B-Spline functions bspeval bspderiv bspkntins bspdegelev bspinterpcrv bspinterpsurf basisfun basisfunder basiskntins findspan numbasisfun tbasisfun B-splines geometric entities curvederivcpts curvederiveval surfderivcpts surfderiveval NURBS geometric entities and functions nrbbasisfun nrbmeasure nrbbasisfunder nrbnumbasisfun nrbcrvderiveval nrbsurfderiveval Knots construction and refinement aveknt kntuniform kntrefine kntbrkdegreg kntbrkdegmult Vector and Transformation Utilities vecnormalize vecmag vecmag2 vecangle vecdot veccross vecrot vecrotx vecroty vecrotz vecscale vectrans nurbs-1.4.3/PaxHeaders.16248/COPYING0000644000000000000000000000013214030323741013441 xustar0030 mtime=1617012705.775919014 30 atime=1617012705.775919014 30 ctime=1617012705.823919749 nurbs-1.4.3/COPYING0000644000175000017500000010451314030323741015544 0ustar00rvazquezrvazquez00000000000000 GNU GENERAL PUBLIC LICENSE Version 3, 29 June 2007 Copyright (C) 2007 Free Software Foundation, Inc. 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But first, please read . nurbs-1.4.3/PaxHeaders.16248/inst0000644000000000000000000000013214030323741013306 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.823919749 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/0000755000175000017500000000000014030323741015462 5ustar00rvazquezrvazquez00000000000000nurbs-1.4.3/inst/PaxHeaders.16248/findspan.m0000644000000000000000000000013214030323741015343 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/findspan.m0000644000175000017500000000347614030323741017454 0ustar00rvazquezrvazquez00000000000000function s = findspan(n,p,u,U) % FINDSPAN Find the span of a B-Spline knot vector at a parametric point % % Calling Sequence: % % s = findspan(n,p,u,U) % % INPUT: % % n - number of control points - 1 % p - spline degree % u - parametric point % U - knot sequence % % OUTPUT: % % s - knot span index % % Modification of Algorithm A2.1 from 'The NURBS BOOK' pg68 % % Copyright (C) 2010 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (max(u(:))>U(end) || min(u(:))= U,1,'last')-1; end end %!test %! n = 3; %! U = [0 0 0 1/2 1 1 1]; %! p = 2; %! u = linspace(0, 1, 10); %! s = findspan (n, p, u, U); %! assert (s, [2*ones(1, 5) 3*ones(1, 5)]); %!test %! p = 2; m = 7; n = m - p - 1; %! U = [zeros(1,p) linspace(0,1,m+1-2*p) ones(1,p)]; %! u = [ 0 0.11880 0.55118 0.93141 0.40068 0.35492 0.44392 0.88360 0.35414 0.92186 0.83085 1]; %! s = [2 2 3 4 3 3 3 4 3 4 4 4]; %! assert (findspan (n, p, u, U), s, 1e-10); nurbs-1.4.3/inst/PaxHeaders.16248/nrbsurfderiveval.m0000644000000000000000000000013214030323741017124 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbsurfderiveval.m0000644000175000017500000002631414030323741021231 0ustar00rvazquezrvazquez00000000000000function skl = nrbsurfderiveval (srf, uv, d) % % NRBSURFDERIVEVAL: Evaluate n-th order derivatives of a NURBS surface % % usage: skl = nrbsurfderiveval (srf, [u; v], d) % % INPUT: % % srf : NURBS surface structure, see nrbmak % % u, v : parametric coordinates of the point where we compute the % derivatives % % d : number of partial derivatives to compute % % % OUTPUT: % % skl (i, j, k, l) = i-th component derived j-1,k-1 times at the % l-th point. % % Adaptation of algorithm A4.4 from the NURBS book, pg137 % % Copyright (C) 2009 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . skl = zeros (3, d+1, d+1, size (uv, 2)); for iu = 1:size(uv, 2); wders = squeeze (surfderiveval (srf.number(1)-1, srf.order(1)-1, ... srf.knots{1}, srf.number(2)-1, ... srf.order(2)-1, srf.knots{2}, ... squeeze (srf.coefs(4, :, :)), uv(1,iu), ... uv(2,iu), d)); for idim = 1:3 Aders = squeeze (surfderiveval (srf.number(1)-1, srf.order(1)-1, ... srf.knots{1}, srf.number(2)-1, ... srf.order(2)-1, srf.knots{2}, ... squeeze (srf.coefs(idim, :, :)), uv(1,iu),... uv(2,iu), d)); for k=0:d for l=0:d-k v = Aders(k+1, l+1); for j=1:l v = v - nchoosek(l,j)*wders(1,j+1)*skl(idim, k+1, l-j+1,iu); end for i=1:k v = v - nchoosek(k,i)*wders(i+1,1)*skl(idim, k-i+1, l+1, iu); v2 =0; for j=1:l v2 = v2 + nchoosek(l,j)*wders(i+1,j+1)*skl(idim, k-i+1, l-j+1, iu); end v = v - nchoosek(k,i)*v2; end skl(idim, k+1, l+1, iu) = v/wders(1,1); end end end end end %!test %! k = [0 0 1 1]; %! c = [0 1]; %! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c); %! coef(3,:,:) = coef(1,:,:); %! srf = nrbmak (coef, {k, k}); %! [u, v] = meshgrid (linspace(0,1,11)); %! uv = [u(:)';v(:)']; %! skl = nrbsurfderiveval (srf, uv, 0); %! aux = nrbeval(srf,uv); %! assert (squeeze (skl (1:2,1,1,:)), aux(1:2,:), 1e3*eps) %!test %! k = [0 0 1 1]; %! c = [0 1]; %! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c); %! coef(3,:,:) = coef(1,:,:); %! srf = nrbmak (coef, {k, k}); %! srf = nrbkntins (srf, {[], rand(2,1)}); %! [u, v] = meshgrid (linspace(0,1,11)); %! uv = [u(:)';v(:)']; %! skl = nrbsurfderiveval (srf, uv, 0); %! aux = nrbeval(srf,uv); %! assert (squeeze (skl (1:2,1,1,:)), aux(1:2,:), 1e3*eps) %!shared srf, uv %!test %! k = [0 0 0 1 1 1]; %! c = [0 1/2 1]; %! [coef(1,:,:), coef(2,:,:)] = meshgrid (c, c); %! coef(3,:,:) = coef(1,:,:); %! srf = nrbmak (coef, {k, k}); %! ders= nrbderiv (srf); %! [u, v] = meshgrid (linspace(0,1,11)); %! uv = [u(:)';v(:)']; %! skl = nrbsurfderiveval (srf, uv, 1); %! [fun, der] = nrbdeval (srf, ders, uv); %! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps) %! %!test %! srf = nrbdegelev (srf, [3, 1]); %! ders= nrbderiv (srf); %! [fun, der] = nrbdeval (srf, ders, uv); %! skl = nrbsurfderiveval (srf, uv, 1); %! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps) %!shared uv %!test %! k = [0 0 0 1 1 1]; %! c = [0 1/2 1]; %! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c); %! coef(3,:,:) = coef(1,:,:); %! srf = nrbmak (coef, {k, k}); %! ders= nrbderiv (srf); %! [u, v] = meshgrid (linspace(0,1,11)); %! uv = [u(:)';v(:)']; %! skl = nrbsurfderiveval (srf, uv, 1); %! [fun, der] = nrbdeval (srf, ders, uv); %! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps) %! %!test %! p = 3; q = 3; %! mcp = 5; ncp = 5; %! Lx = 10*rand(1); Ly = Lx; %! srf = nrbdegelev (nrb4surf ([0 0], [Lx, 0], [0 Ly], [Lx Ly]), [p-1, q-1]); %! %%srf = nrbkntins (srf, {linspace(0,1,mcp-p+2)(2:end-1), linspace(0,1,ncp-q+2)(2:end-1)}); %! %%srf.coefs = permute (srf.coefs, [1 3 2]); %! ders= nrbderiv (srf); %! [fun, der] = nrbdeval (srf, ders, uv); %! skl = nrbsurfderiveval (srf, uv, 1); %! assert (squeeze (skl (1:2,1,1,:)), fun(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,2,1,:)), der{1}(1:2,:), 1e3*eps) %! assert (squeeze (skl (1:2,1,2,:)), der{2}(1:2,:), 1e3*eps) %!shared srf, uv, P, dPdx, d2Pdx2, c1, c2 %!test %! [u, v] = meshgrid (linspace(0,1,10)); %! uv = [u(:)';v(:)']; %! c1 = nrbmak([0 1/2 1; 0 1 0],[0 0 0 1 1 1]); %! c1 = nrbtform (c1, vecrotx (pi/2)); %! c2 = nrbtform(c1, vectrans([0 1 0])); %! srf = nrbdegelev (nrbruled (c1, c2), [3, 1]); %! skl = nrbsurfderiveval (srf, uv, 2); %! P = squeeze(skl(:,1,1,:)); %! dPdx = squeeze(skl(:,2,1,:)); %! d2Pdx2 = squeeze(skl(:,3,1,:)); %!assert(P(3,:), 2*(P(1,:)-P(1,:).^2),100*eps) %!assert(dPdx(3,:), 2-4*P(1,:), 100*eps) %!assert(d2Pdx2(3,:), -4+0*P(1,:), 100*eps) %! srf = nrbdegelev (nrbruled (c1, c2), [5, 6]); %! skl = nrbsurfderiveval (srf, uv, 2); %! P = squeeze(skl(:,1,1,:)); %! dPdx = squeeze(skl(:,2,1,:)); %! d2Pdx2 = squeeze(skl(:,3,1,:)); %! aux = nrbeval(srf,uv); %! assert (squeeze (skl (1:2,1,1,:)), aux(1:2,:), 1e3*eps) %!assert(P(3,:), 2*(P(1,:)-P(1,:).^2),100*eps) %!assert(dPdx(3,:), 2-4*P(1,:), 100*eps) %!assert(d2Pdx2(3,:), -4+0*P(1,:), 100*eps) %! %!test %! skl = nrbsurfderiveval (srf, uv, 0); %! aux = nrbeval (srf, uv); %! assert (squeeze (skl (1:2,1,1,:)), aux(1:2,:), 1e3*eps) %!shared dPdu, d2Pdu2, P, srf, uv %!test %! [u, v] = meshgrid (linspace(0,1,10)); %! uv = [u(:)';v(:)']; %! c1 = nrbmak([0 1/2 1; 0.1 1.6 1.1; 0 0 0],[0 0 0 1 1 1]); %! c2 = nrbmak([0 1/2 1; 0.1 1.6 1.1; 1 1 1],[0 0 0 1 1 1]); %! srf = nrbdegelev (nrbruled (c1, c2), [0, 1]); %! skl = nrbsurfderiveval (srf, uv, 2); %! P = squeeze(skl(:,1,1,:)); %! dPdu = squeeze(skl(:,2,1,:)); %! dPdv = squeeze(skl(:,1,2,:)); %! d2Pdu2 = squeeze(skl(:,3,1,:)); %! aux = nrbeval(srf,uv); %! assert (squeeze (skl (1:2,1,1,:)), aux(1:2,:), 1e3*eps) %!assert(dPdu(2,:), 3-4*P(1,:),100*eps) %!assert(d2Pdu2(2,:), -4+0*P(1,:),100*eps) %! %!test %! skl = nrbsurfderiveval (srf, uv, 0); %! aux = nrbeval(srf,uv); %! assert (squeeze (skl (1:2,1,1,:)), aux(1:2,:), 1e3*eps) %!test %! srf = nrb4surf([0 0], [1 0], [0 1], [1 1]); %! geo = nrbdegelev (srf, [3 3]); %1 geo.coefs (4, 2:end-1, 2:end-1) = geo.coefs (4, 2:end-1, 2:end-1) + .1 * rand (1, geo.number(1)-2, geo.number(2)-2); %! geo = nrbkntins (geo, {[.1:.1:.9], [.2:.2:.8]}); %! [u, v] = meshgrid (linspace(0,1,10)); %! uv = [u(:)';v(:)']; %! skl = nrbsurfderiveval (geo, uv, 2); %! dgeo = nrbderiv (geo); %! [pnts, ders] = nrbdeval (geo, dgeo, uv); %! assert (ders{1}, squeeze(skl(:,2,1,:)), 1e-9) %! assert (ders{2}, squeeze(skl(:,1,2,:)), 1e-9) %!test %! crv = nrbline ([1 0], [2 0]); %! srf = nrbrevolve (crv, [0 0 0], [0 0 1], pi/2); %! srf = nrbtransp (srf); %! [v, u] = meshgrid (linspace (0, 1, 11)); %! uv = [u(:)'; v(:)']; %! skl = nrbsurfderiveval (srf, uv, 2); %! c = sqrt(2); %! w = @(x, y) (2 - c)*y.^2 + (c-2)*y + 1; %! dwdy = @(x, y) 2*(2-c)*y + c - 2; %! d2wdy2 = @(x, y) 2*(2-c); %! F1 = @(x, y) (x+1) .* ((1-y).^2 + c*y.*(1-y)) ./ w(x,y); %! F2 = @(x, y) (x+1) .* (y.^2 + c*y.*(1-y)) ./ w(x,y); %! dF1dx = @(x, y) ((1-y).^2 + c*y.*(1-y)) ./ w(x,y); %! dF2dx = @(x, y) (y.^2 + c*y.*(1-y)) ./ w(x,y); %! dF1dy = @(x, y) (x+1) .* ((2 - 2*c)*y + c - 2) ./ w(x,y) - (x+1) .* ((1-y).^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2; %! dF2dy = @(x, y) (x+1) .* ((2 - 2*c)*y + c) ./ w(x,y) - (x+1) .* (y.^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2; %! d2F1dx2 = @(x, y) zeros (size (x)); %! d2F2dx2 = @(x, y) zeros (size (x)); %! d2F1dxdy = @(x, y) ((2 - 2*c)*y + c - 2) ./ w(x,y) - ((1-y).^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2; %! d2F2dxdy = @(x, y) ((2 - 2*c)*y + c) ./ w(x,y) - (y.^2 + c*y.*(1-y)) .* dwdy(x,y) ./ w(x,y).^2; %! d2F1dy2 = @(x, y) (x+1)*(2 - 2*c) ./ w(x,y) - 2*(x+1) .* ((2 - 2*c)*y + c - 2) .* dwdy(x,y) ./ w(x,y).^2 - ... %! (x+1) .* ((1-y).^2 + c*y.*(1-y)) * d2wdy2(x,y) ./ w(x,y).^2 + ... %! 2 * (x+1) .* ((1-y).^2 + c*y.*(1-y)) .* w(x,y) .*dwdy(x,y).^2 ./ w(x,y).^4; %! d2F2dy2 = @(x, y) (x+1)*(2 - 2*c) ./ w(x,y) - 2*(x+1) .* ((2 - 2*c)*y + c) .* dwdy(x,y) ./ w(x,y).^2 - ... %! (x+1) .* (y.^2 + c*y.*(1-y)) * d2wdy2(x,y) ./ w(x,y).^2 + ... %! 2 * (x+1) .* (y.^2 + c*y.*(1-y)) .* w(x,y) .*dwdy(x,y).^2 ./ w(x,y).^4; %! assert ([F1(u(:),v(:)), F2(u(:),v(:))], squeeze(skl(1:2,1,1,:))', 1e2*eps); %! assert ([dF1dx(u(:),v(:)), dF2dx(u(:),v(:))], squeeze(skl(1:2,2,1,:))', 1e2*eps); %! assert ([dF1dy(u(:),v(:)), dF2dy(u(:),v(:))], squeeze(skl(1:2,1,2,:))', 1e2*eps); %! assert ([d2F1dx2(u(:),v(:)), d2F2dx2(u(:),v(:))], squeeze(skl(1:2,3,1,:))', 1e2*eps); %! assert ([d2F1dxdy(u(:),v(:)), d2F2dxdy(u(:),v(:))], squeeze(skl(1:2,2,2,:))', 1e2*eps); %! assert ([d2F1dy2(u(:),v(:)), d2F2dy2(u(:),v(:))], squeeze(skl(1:2,1,3,:))', 1e2*eps); %!test %! knots = {[0 0 1 1] [0 0 1 1]}; %! coefs(:,1,1) = [0;0;0;1]; %! coefs(:,2,1) = [1;0;0;1]; %! coefs(:,1,2) = [0;1;0;1]; %! coefs(:,2,2) = [1;1;1;2]; %! srf = nrbmak (coefs, knots); %! [v, u] = meshgrid (linspace (0, 1, 3)); %! uv = [u(:)'; v(:)']; %! skl = nrbsurfderiveval (srf, uv, 2); %! w = @(x, y) x.*y + 1; %! F1 = @(x, y) x ./ w(x,y); %! F2 = @(x, y) y ./ w(x,y); %! F3 = @(x, y) x .* y ./ w(x,y); %! dF1dx = @(x, y) 1./w(x,y) - x.*y./w(x,y).^2; %! dF1dy = @(x, y) - x.^2./w(x,y).^2; %! dF2dx = @(x, y) - y.^2./w(x,y).^2; %! dF2dy = @(x, y) 1./w(x,y) - x.*y./w(x,y).^2; %! dF3dx = @(x, y) y./w(x,y) - x.*(y./w(x,y)).^2; %! dF3dy = @(x, y) x./w(x,y) - y.*(x./w(x,y)).^2; %! d2F1dx2 = @(x, y) -2*y./w(x,y).^2 + 2*x.*y.^2./w(x,y).^3; %! d2F1dy2 = @(x, y) 2*x.^3./w(x,y).^3; %! d2F1dxdy = @(x, y) -x./w(x,y).^2 - x./w(x,y).^2 + 2*x.^2.*y./w(x,y).^3; %! d2F2dx2 = @(x, y) 2*y.^3./w(x,y).^3; %! d2F2dy2 = @(x, y) -2*x./w(x,y).^2 + 2*y.*x.^2./w(x,y).^3; %! d2F2dxdy = @(x, y) -y./w(x,y).^2 - y./w(x,y).^2 + 2*y.^2.*x./w(x,y).^3; %! d2F3dx2 = @(x, y) -2*y.^2./w(x,y).^2 + 2*x.*y.^3./w(x,y).^3; %! d2F3dy2 = @(x, y) -2*x.^2./w(x,y).^2 + 2*y.*x.^3./w(x,y).^3; %! d2F3dxdy = @(x, y) 1./w(x,y) - 3*x.*y./w(x,y).^2 + 2*(x.*y).^2./w(x,y).^3; %! assert ([F1(u(:),v(:)), F2(u(:),v(:)), F3(u(:),v(:))], squeeze(skl(1:3,1,1,:))', 1e2*eps); %! assert ([dF1dx(u(:),v(:)), dF2dx(u(:),v(:)), dF3dx(u(:),v(:))], squeeze(skl(1:3,2,1,:))', 1e2*eps); %! assert ([dF1dy(u(:),v(:)), dF2dy(u(:),v(:)), dF3dy(u(:),v(:))], squeeze(skl(1:3,1,2,:))', 1e2*eps); %! assert ([d2F1dx2(u(:),v(:)), d2F2dx2(u(:),v(:)), d2F3dx2(u(:),v(:))], squeeze(skl(1:3,3,1,:))', 1e2*eps); %! assert ([d2F1dy2(u(:),v(:)), d2F2dy2(u(:),v(:)), d2F3dy2(u(:),v(:))], squeeze(skl(1:3,1,3,:))', 1e2*eps); %! assert ([d2F1dxdy(u(:),v(:)), d2F2dxdy(u(:),v(:)), d2F3dxdy(u(:),v(:))], squeeze(skl(1:3,2,2,:))', 1e2*eps); nurbs-1.4.3/inst/PaxHeaders.16248/vecangle.m0000644000000000000000000000013214030323741015325 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecangle.m0000644000175000017500000000270214030323741017425 0ustar00rvazquezrvazquez00000000000000function ang = vecangle(num,den) % % VECANGLE: An alternative to atan, returning an arctangent in the % range 0 to 2*pi. % % Calling Sequence: % % ang = vecmag2(num,dum) % % INPUT: % % num : Numerator, vector of size (1,nv). % dem : Denominator, vector of size (1,nv). % % OUTPUT: % ang : Arctangents, row vector of angles. % % Description: % % The components of the vector ang are the arctangent of the corresponding % enties of num./dem. This function is an alternative for % atan, returning an angle in the range 0 to 2*pi. % % Examples: % % Find the atan(1.2,2.0) and atan(1.5,3.4) using vecangle % % ang = vecangle([1.2 1.5], [2.0 3.4]); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . ang = atan2(num,den); index = find(ang < 0.0); ang(index) = 2*pi+ang(index); end nurbs-1.4.3/inst/PaxHeaders.16248/nrbmultipatch.m0000644000000000000000000000013014030323741016413 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbmultipatch.m0000644000175000017500000002315514030323741020522 0ustar00rvazquezrvazquez00000000000000function [interfaces, boundary] = nrbmultipatch (nurbs, tol) % % NRBMULTIPATCH: construct the information for gluing conforming NURBS patches, using the same format as in GeoPDEs. % % Calling Sequence: % % [interfaces, boundary] = nrbmultipatch (nurbs, [tol]); % % INPUT: % % nurbs : an array of NURBS surfaces or volumes (not both), see nrbmak. % tol : relative tolerance to compare knots and control points at the interfaces. % % OUTPUT: % % interfaces: array with the information for each interface, that is: % - number of the first patch (patch1), and the local side number (side1) % - number of the second patch (patch2), and the local side number (side2) % - flag (faces and volumes), ornt1, ornt2 (only volumes): information % on how the two patches match, see below. % boundary: array with the boundary faces that do not belong to any interface % - nsides: total number of sides on the boundary array (numel(boundary)) % - patches: number of the patch to which the boundary belongs % - sides: number of the local side on the patch % % The faces of two patches must match conformingly: the control points must be the same, % with the knot vectors (in each direction) related by an affine transformation. % % The boundary faces are stored separately, that is, nsides=1 for each boundary. % To join several faces under the same condition, the user should do it by hand. % % Copyright (C) 2014, 2015, 2016, 2017 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 2) tol = 1e-13; end npatch = numel (nurbs); if (~iscell (nurbs(1).knots)) ndim = 1; compare_sides = @(nrb1, nrb2) max(abs(nrb1.coefs - nrb2.coefs)) < tol; elseif (size(nurbs(1).knots,2) == 2) ndim = 2; compare_sides = @(nrb1, nrb2) compare_sides_univariate (nrb1, nrb2, tol); elseif (size(nurbs(1).knots,2) == 3) ndim = 3; compare_sides = @(nrb1, nrb2) compare_sides_bivariate (nrb1, nrb2, tol); end non_set_faces = cell (npatch, 1); for ii = 1:npatch if (~iscell (nurbs(ii).knots)) if (ndim ~= 1) error ('All the patches must have the same dimension (at least for now)') end elseif (ndim ~= size(nurbs(ii).knots,2)) error ('All the patches must have the same dimension (at least for now)') end non_set_faces{ii} = 1:2*ndim; end num_interfaces = 0; num_boundaries = 0; boundary = struct ('nsides', 0, 'patches', [], 'faces', []); for i1 = 1:npatch nrb_faces1 = nrbextract (nurbs(i1)); for j1 = non_set_faces{i1} % This is to fix a bug when two faces of the same patch form an interface % (for instance, in a ring or a torus) if (isempty (intersect (non_set_faces{i1}, j1))); continue; end nrb1 = nrb_faces1(j1); % corners1 = face_corners (nrb1); non_set_faces{i1} = setdiff (non_set_faces{i1}, j1); flag = 0; i2 = i1 - 1; while (~flag && i2 < npatch) i2 = i2 + 1; nrb_faces2 = nrbextract (nurbs(i2)); j2 = 0; while (~flag && j2 < numel (non_set_faces{i2})) j2 = j2 + 1; nrb2 = nrb_faces2(non_set_faces{i2}(j2)); if (ndim == 1) flag = compare_sides (nrb1, nrb2); MsgFlag = false; elseif (ndim == 2) [flag, MsgFlag] = compare_sides (nrb1, nrb2); elseif (ndim == 3) [flag, ornt1, ornt2, MsgFlag] = compare_sides (nrb1, nrb2); end if (MsgFlag) display_warning (MsgFlag, i1, j1, i2, j2); end end end if (flag) intrfc.patch1 = i1; intrfc.side1 = j1; intrfc.patch2 = i2; intrfc.side2 = non_set_faces{i2}(j2); if (ndim ==3) intrfc.flag = flag; intrfc.ornt1 = ornt1; intrfc.ornt2 = ornt2; elseif (ndim == 2) intrfc.ornt = flag; end non_set_faces{i2} = setdiff (non_set_faces{i2}, non_set_faces{i2}(j2)); num_interfaces = num_interfaces + 1; interfaces(num_interfaces) = intrfc; else bndry.nsides = 1; bndry.patches = i1; bndry.faces = j1; num_boundaries = num_boundaries + 1; boundary(num_boundaries) = bndry; end end end if (num_interfaces == 0) interfaces = []; end if (num_boundaries == 0) boundary = []; end end % Compare the sides of two volumes function [flag, ornt1, ornt2, MsgFlag] = compare_sides_bivariate (nrb1, nrb2, tol) MsgFlag = 0; face_corners = @(x) reshape (x.coefs(:, [1 end], [1 end]), 4, []); coefs1 = face_corners (nrb1); coefs2 = face_corners (nrb2); % Sort of relative error tolcp = tol * max(abs(coefs1(1:3,1) - coefs1(1:3,end))); tolknt = tol; % Should use some sort of relative error if (max (max (abs (coefs1 - coefs2))) < tolcp) flag = 1; ornt1 = 1; ornt2 = 1; elseif (max (max (abs (coefs1 - coefs2(:,[1 3 2 4])))) < tolcp) flag = -1; ornt1 = 1; ornt2 = 1; elseif (max (max (abs (coefs1 - coefs2(:,[3 1 4 2])))) < tolcp) flag = -1; ornt1 = -1; ornt2 = 1; elseif (max (max (abs (coefs1 - coefs2(:,[2 1 4 3])))) < tolcp) flag = 1; ornt1 = -1; ornt2 = 1; elseif (max (max (abs (coefs1 - coefs2(:,[4 3 2 1])))) < tolcp) flag = 1; ornt1 = -1; ornt2 = -1; elseif (max (max (abs (coefs1 - coefs2(:,[4 2 3 1])))) < tolcp) flag = -1; ornt1 = -1; ornt2 = -1; elseif (max (max (abs (coefs1 - coefs2(:,[2 4 1 3])))) < tolcp) flag = -1; ornt1 = 1; ornt2 = -1; elseif (max (max (abs (coefs1 - coefs2(:,[3 4 1 2])))) < tolcp) flag = 1; ornt1 = 1; ornt2 = -1; else flag = 0; ornt1 = 0; ornt2 = 0; end % Reorder control points and knot vectors, to make comparisons easier if (flag) if (flag == -1) nrb2 = nrbtransp (nrb2); end if (ornt1 == -1) nrb2 = nrbreverse (nrb2, 1); end if (ornt2 == -1) nrb2 = nrbreverse (nrb2, 2); end if (nrb1.order ~= nrb2.order) flag = 0; MsgFlag = -3; elseif (nrb1.number ~= nrb2.number) flag = 0; MsgFlag = -1; elseif (any (cellfun (@numel, nrb1.knots) ~= cellfun (@numel, nrb2.knots))) % This is redundant flag = 0; MsgFlag = -4; else % Pass the knots to the [0 1] interval to compare pass_to_01 = @(x) (x - x(1)) / (x(end) - x(1)); knt1 = cellfun (pass_to_01, nrb1.knots, 'UniformOutput', false); knt2 = cellfun (pass_to_01, nrb2.knots, 'UniformOutput', false); if (max (abs (nrb1.coefs(:) - nrb2.coefs(:))) > tolcp) flag = 0; MsgFlag = -2; elseif ((max (abs (knt1{1} - knt2{1})) > tolknt) || (max (abs (knt1{2} - knt2{2})) > tolknt)) flag = 0; MsgFlag = -5; end end end end % Compare the sides of two surfaces function [flag, MsgFlag] = compare_sides_univariate (nrb1, nrb2, tol) MsgFlag = 0; face_corners = @(x) reshape (x.coefs(:, [1 end]), 4, []); coefs1 = face_corners (nrb1); coefs2 = face_corners (nrb2); % Sort of relative error tolcp = tol * max(abs(coefs1(1:3,1) - coefs1(1:3,end))); tolknt = tol; if (max (max (abs (coefs1 - coefs2))) < tolcp) flag = 1; elseif (max (max (abs (coefs1 - coefs2(:,[end 1])))) < tolcp) flag = -1; else flag = 0; end if (flag) % Reorder control points and knot vectors, to make comparisons easier if (flag == -1) nrb2 = nrbreverse (nrb2); end if (nrb1.order ~= nrb2.order) flag = 0; MsgFlag = -3; elseif (nrb1.number ~= nrb2.number) flag = 0; MsgFlag = -1; elseif (numel(nrb1.knots) ~= numel(nrb2.knots)) % This is redundant flag = 0; MsgFlag = -4; else % Pass the knots to the [0 1] interval to compare knt1 = (nrb1.knots - nrb1.knots(1)) / (nrb1.knots(end) - nrb1.knots(1)); knt2 = (nrb2.knots - nrb2.knots(1)) / (nrb2.knots(end) - nrb2.knots(1)); if (max (abs (nrb1.coefs(:) - nrb2.coefs(:))) > tolcp) flag = 0; MsgFlag = -2; elseif (max (abs (knt1 - knt2)) > tolknt) flag = 0; MsgFlag = -5; end end end end function display_warning (MsgFlag, patch1, face1, patch2, face2) switch MsgFlag case {-1} warning (['The corners of PATCH %d FACE %d, and PATCH %d FACE %d coincide, but the number ' ... 'of control points is different. No information is saved in this case'], patch1, face1, patch2, face2) case {-2} warning (['The corners of PATCH %d FACE %d, and PATCH %d FACE %d coincide, but the internal ' ... 'control points do not. No information is saved in this case'], patch1, face1, patch2, face2) case {-3} warning (['The corners of PATCH %d FACE %d, and PATCH %d FACE %d coincide, but the degree ' ... 'is different. No information is saved in this case'], patch1, face1, patch2, face2) case {-4} warning (['The corners of PATCH %d FACE %d, and PATCH %d FACE %d coincide, but the number ' ... 'of knots is different. No information is saved in this case'], patch1, face1, patch2, face2) case {-5} warning (['The corners of PATCH %d FACE %d, and PATCH %d FACE %d coincide, but the ' ... 'knot vectors are different. No information is saved in this case'], patch1, face1, patch2, face2) end endnurbs-1.4.3/inst/PaxHeaders.16248/nrbkntplot.m0000644000000000000000000000013014030323741015734 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbkntplot.m0000644000175000017500000002041714030323741020041 0ustar00rvazquezrvazquez00000000000000function nrbkntplot (nurbs, nsub) % NRBKNTPLOT: Plot a NURBS entity with the knots subdivision. % % Calling Sequence: % % nrbkntplot(nurbs) % nrbkntplot(nurbs, npnts) % % INPUT: % % nurbs: NURBS curve, surface or volume, see nrbmak. % npnts: Number of evaluation points, for a surface or volume, a row % vector with the number of points along each direction. % % Example: % % Plot the test surface with its knot vector % % nrbkntplot(nrbtestsrf) % % See also: % % nrbctrlplot % % Copyright (C) 2011, 2012, 2021 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 1) error ('nrbkntplot: Need a NURBS to plot!'); end % Default values light='on'; cmap='summer'; colormap (cmap); hold_flag = ishold; if (iscell (nurbs.knots)) if (size (nurbs.knots,2) == 2) % plot a NURBS surface if (nargin < 2) nsub = [50 50]; elseif (numel(nsub) == 1) nsub = [nsub nsub]; end nrbplot (nurbs, nsub, 'light', light, 'colormap', cmap); hold on % And plot the knots knt1 = unique (nurbs.knots{1}(nurbs.order(1):end-nurbs.order(1)+1)); knt2 = unique (nurbs.knots{2}(nurbs.order(2):end-nurbs.order(2)+1)); p1 = nrbeval (nurbs, {knt1, linspace(knt2(1),knt2(end),nsub(2)+1)}); p2 = nrbeval (nurbs, {linspace(knt1(1),knt1(end),nsub(1)+1), knt2}); if (any (nurbs.coefs(3,:))) % surface in a 3D space for ii = 1:numel(knt1) plot3 (squeeze(p1(1,ii,:)), squeeze(p1(2,ii,:)), squeeze(p1(3,ii,:)),'k'); end for ii = 1:numel(knt2) plot3 (squeeze(p2(1,:,ii)), squeeze(p2(2,:,ii)), squeeze(p2(3,:,ii)),'k'); end else % plain surface for ii = 1:numel(knt1) plot (squeeze(p1(1,ii,:)), squeeze (p1(2,ii,:)),'k'); end for ii = 1:numel(knt2) plot (p2(1,:,ii),p2(2,:,ii),'k'); end end elseif (size (nurbs.knots,2) == 3) % plot a NURBS volume if (nargin < 2) nsub = [25 25 25]; elseif (numel(nsub) == 1) nsub = [nsub nsub nsub]; end % Plot the boundaries bnd = nrbextract (nurbs); nrbkntplot (bnd(1), nsub(2:3)); hold on for iface = 2:6 inds = setdiff(1:3, ceil(iface/2)); nrbkntplot (bnd(iface), nsub(inds)); end end else % plot a NURBS curve if (nargin < 2) nsub = 1000; end nrbplot (nurbs, nsub); hold on % And plot the knots order = nurbs.order; p = nrbeval (nurbs, unique (nurbs.knots(order:end-order+1))); if (any (nurbs.coefs(3,:))) % plot a 3D curve plot3 (p(1,:), p(2,:), p(3,:), 'rx'); else % plot a 2D curve plot (p(1,:), p(2,:), 'rx'); end end if (~hold_flag) hold off end end %!demo %! crv = nrbtestcrv; %! nrbkntplot(crv) %! title('Test curve') %! hold off %!demo %! sphere = nrbrevolve(nrbcirc(1,[],0.0,pi),[0.0 0.0 0.0],[1.0 0.0 0.0]); %! nrbkntplot(sphere); %! title('Ball and torus - surface construction by revolution'); %! hold on; %! torus = nrbrevolve(nrbcirc(0.2,[0.9 1.0]),[0.0 0.0 0.0],[1.0 0.0 0.0]); %! nrbkntplot(torus); %! hold off %!demo %! knots = {[0 0 0 1/2 1 1 1] [0 0 0 1 1 1]... %! [0 0 0 1/6 2/6 1/2 1/2 4/6 5/6 1 1 1]}; %! %! coefs = [-1.0000 -0.9734 -0.7071 1.4290 1.0000 3.4172 %! 0 2.4172 0 0.0148 -2.0000 -1.9734 %! 0 2.0000 4.9623 9.4508 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -0.8536 0 -0.6036 1.9571 1.2071 3.5000 %! 0.3536 2.5000 0.2500 0.5429 -1.7071 -1.0000 %! 0 2.0000 4.4900 8.5444 3.4142 2.0000 %! 0.8536 1.0000 0.6036 1.0000 0.8536 1.0000 %! -0.3536 -4.0000 -0.2500 -1.2929 1.7071 1.0000 %! 0.8536 0 0.6036 -2.7071 -1.2071 -5.0000 %! 0 2.0000 4.4900 10.0711 3.4142 2.0000 %! 0.8536 1.0000 0.6036 1.0000 0.8536 1.0000 %! 0 -4.0000 0 0.7071 2.0000 5.0000 %! 1.0000 4.0000 0.7071 -0.7071 -1.0000 -5.0000 %! 0 2.0000 4.9623 14.4142 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -2.5000 -4.0000 -1.7678 0.7071 1.0000 5.0000 %! 0 4.0000 0 -0.7071 -3.5000 -5.0000 %! 0 2.0000 6.0418 14.4142 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -2.4379 0 -1.7238 2.7071 1.9527 5.0000 %! 0.9527 4.0000 0.6737 1.2929 -3.4379 -1.0000 %! 0 2.0000 6.6827 10.0711 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -0.9734 -1.0000 -0.6883 0.7071 3.4172 1.0000 %! 2.4172 0 1.7092 -1.4142 -1.9734 -2.0000 %! 0 4.0000 6.6827 4.9623 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! 0 -0.8536 0 0.8536 3.5000 1.2071 %! 2.5000 0.3536 1.7678 -1.2071 -1.0000 -1.7071 %! 0 3.4142 6.0418 4.4900 4.0000 0 %! 1.0000 0.8536 0.7071 0.6036 1.0000 0.8536 %! -4.0000 -0.3536 -2.8284 1.2071 1.0000 1.7071 %! 0 0.8536 0 -0.8536 -5.0000 -1.2071 %! 0 3.4142 7.1213 4.4900 4.0000 0 %! 1.0000 0.8536 0.7071 0.6036 1.0000 0.8536 %! -4.0000 0 -2.8284 1.4142 5.0000 2.0000 %! 4.0000 1.0000 2.8284 -0.7071 -5.0000 -1.0000 %! 0 4.0000 10.1924 4.9623 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! -4.0000 -2.5000 -2.8284 0.7071 5.0000 1.0000 %! 4.0000 0 2.8284 -2.4749 -5.0000 -3.5000 %! 0 4.0000 10.1924 6.0418 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! 0 -2.4379 0 1.3808 5.0000 1.9527 %! 4.0000 0.9527 2.8284 -2.4309 -1.0000 -3.4379 %! 0 4.0000 7.1213 6.6827 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! -1.0000 -0.9734 0.2071 2.4163 1.0000 3.4172 %! 0 2.4172 -1.2071 -1.3954 -2.0000 -1.9734 %! 2.0000 4.0000 7.0178 6.6827 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000 %! -0.8536 0 0.3536 2.4749 1.2071 3.5000 %! 0.3536 2.5000 -0.8536 -0.7071 -1.7071 -1.0000 %! 1.7071 4.0000 6.3498 6.0418 1.7071 0 %! 0.8536 1.0000 0.8536 0.7071 0.8536 1.0000 %! -0.3536 -4.0000 0.8536 0.7071 1.7071 1.0000 %! 0.8536 0 -0.3536 -3.5355 -1.2071 -5.0000 %! 1.7071 4.0000 6.3498 7.1213 1.7071 0 %! 0.8536 1.0000 0.8536 0.7071 0.8536 1.0000 %! 0 -4.0000 1.2071 3.5355 2.0000 5.0000 %! 1.0000 4.0000 -0.2071 -3.5355 -1.0000 -5.0000 %! 2.0000 4.0000 7.0178 10.1924 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000 %! -2.5000 -4.0000 -0.5429 3.5355 1.0000 5.0000 %! 0 4.0000 -1.9571 -3.5355 -3.5000 -5.0000 %! 2.0000 4.0000 8.5444 10.1924 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000 %! -2.4379 0 -0.0355 3.5355 1.9527 5.0000 %! 0.9527 4.0000 -1.4497 -0.7071 -3.4379 -1.0000 %! 2.0000 4.0000 9.4508 7.1213 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000]; %! coefs = reshape (coefs, 4, 4, 3, 9); %! horseshoe = nrbmak (coefs, knots); %! nrbkntplot (horseshoe); nurbs-1.4.3/inst/PaxHeaders.16248/nrbreverse.m0000644000000000000000000000013214030323741015716 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbreverse.m0000644000175000017500000000567114030323741020026 0ustar00rvazquezrvazquez00000000000000function nrb = nrbreverse(nrb, idir) % % NRBREVERSE: Reverse the evaluation directions of a NURBS geometry. % % Calling Sequence: % % rnrb = nrbreverse(nrb); % rnrb = nrbreverse(nrb, idir); % % INPUT: % % nrb : NURBS data structure, see nrbmak. % idir : vector of directions to reverse. % % OUTPUT: % % rnrb : Reversed NURBS. % % Description: % % Utility function to reverse the evaluation direction of a NURBS % curve or surface. % % Copyright (C) 2000 Mark Spink % Copyright (C) 2013 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin > 2) error('Incorrect number of input arguments'); end if (iscell(nrb.knots)) % reverse a NURBS surface or volume ndim = numel (nrb.knots); if (nargin == 1 || isempty (idir)) idir = 1:ndim; end for ii = idir nrb.knots{ii} = sort (nrb.knots{ii}(end) - nrb.knots{ii}); nrb.coefs = flip (nrb.coefs, ii+1); end else % reverse a NURBS curve nrb.knots = sort (nrb.knots(end) - nrb.knots); nrb.coefs = fliplr (nrb.coefs); end end %!demo %! pnts = [0.5 1.5 3.0 7.5 8.5; %! 3.0 5.5 1.5 4.0 4.5; %! 0.0 0.0 0.0 0.0 0.0]; %! crv1 = nrbmak(pnts,[0 0 0 1/2 3/4 1 1 1]); %! crv2 = nrbreverse(crv1); %! fprintf('Knots of the original curve\n') %! disp(crv1.knots) %! fprintf('Knots of the reversed curve\n') %! disp(crv2.knots) %! fprintf('Control points of the original curve\n') %! disp(crv1.coefs(1:2,:)) %! fprintf('Control points of the reversed curve\n') %! disp(crv2.coefs(1:2,:)) %! nrbplot(crv1,100) %! hold on %! nrbplot(crv2,100) %! title('The curve and its reverse are the same') %! hold off %!test %! srf = nrbrevolve(nrbline([1 0],[2 0]), [0 0 0], [0 0 1], pi/2); %! srf = nrbkntins (srf, {0.3, 0.6}); %! srf2 = nrbreverse (srf); %! assert (srf.knots, cellfun(@(x) sort(1-x), srf2.knots, 'UniformOutput', false), 1e-15) %! assert (srf.coefs, srf2.coefs(:,end:-1:1,end:-1:1)) %!test %! srf = nrbrevolve(nrbline([1 0],[2 0]), [0 0 0], [0 0 1], pi/2); %! srf = nrbkntins (srf, {0.3, 0.6}); %! srf2 = nrbreverse (srf, 1); %! knt{1} = sort(1-srf2.knots{1}); knt{2} = srf2.knots{2}; %! assert (srf.knots, knt, 1e-15) %! assert (srf.coefs, srf2.coefs(:,end:-1:1,:)) %! srf2 = nrbreverse (srf, 2); %! knt{1} = srf2.knots{1}; knt{2} = sort(1-srf2.knots{2}); %! assert (srf.knots, knt, 1e-15) %! assert (srf.coefs, srf2.coefs(:,:,end:-1:1)) nurbs-1.4.3/inst/PaxHeaders.16248/nrbspheretiling.m0000644000000000000000000000013214030323741016740 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbspheretiling.m0000644000175000017500000002674214030323741021052 0ustar00rvazquezrvazquez00000000000000function [tiling, tile, ver3d] = nrbspheretiling (topology, radius, center) % % NRBSPHERETILING: Makes an array of NURBS patches representing a % full or partial tiling of the sphere. % % % Calling Sequences: % % [tiling, tile, ver3d] = nrbspheretiling % [tiling, tile, ver3d] = nrbspheretiling(topology) % [tiling, tile, ver3d] = nrbspheretiling(topology, [radius], [center]) % % INPUT: % % topology: String specifying the desired topology for the tiling. % Options are: - 'cube' (default) % - 'ico' for paired icosahedron (nonconforming) % - 'rdode' for rhombic dodecahedron % - 'rtria' for rhombic triacontahedron % - 'dico' for deltoidal icositetrahedron % - 'dhexe' for deltoidal hexecontahedron % - 'octant' for a tiling of the first octant % % radius: Radius of the sphere, default 1.0 % % center: Center of the sphere, default (0,0,0) % % OUTPUT: % % tiling: Structure array of NURBS objects representing the tiling of the % sphere. % % tile: NURBS object representing one tile of the unit sphere. The tile % will be a fourth-order rational Bezier patch. % % ver3d: 3-by-4 matrix with the coordinates of the 4 (ordered) vertices % on the unit sphere, ordered clockwise when viewed from outside the % sphere. % % See also: nrbspheretile % % For more details, see: % Sander Dedoncker, Laurens Coox, Florian Maurin, Francesco Greco, Wim Desmet % Bézier tilings of the sphere and their applications in benchmarking multipatch isogeometric methods % Computer Meth. Appl. Mech. Engrg., 2018 % % Copyright (C) 2017 Sander Dedoncker % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . %% Complete input if nargin < 1 topology = 'cube'; end if (nargin < 2 || isempty (radius)) radius = 1; elseif (radius <= 0) warning ('The radius should be positive. Taking radius equal to one') radius = 1; end if (nargin < 3 || isempty (center)) center = [0 0 0]; end %% Catch invalid input if (nargin > 3) error('Too many input arguments'); end %% Load tilings switch lower (topology) case 'cube' [tile,ver3d] = CubeLoad; [tiling] = Multi_SphereFromCube(tile); case 'ico' [tile,ver3d] = IcoLoad; [tiling] = Multi_SphereFromIco(tile); disp ('Note that the meshes of the icosahedral tiling are non-conforming') case 'rdode' [tile,ver3d] = RDodeLoad; [tiling] = Multi_SphereFromRDode(tile); case 'rtria' [tile,ver3d] = RTriaLoad; [tiling] = Multi_SphereFromRTria(tile); case 'dico' [tile,ver3d] = DIcoLoad; [tiling] = Multi_SphereFromDIco(tile); case 'dhexe' [tile,ver3d] = DHexeLoad; [tiling] = Multi_SphereFromDHexe(tile); case 'octant' [tile,ver3d] = DIcoLoad; [tiling] = OctantFromDIco(tile); otherwise error ('Invalid topology') end for iptc = 1:numel(tiling) tiling(iptc) = nrbtform (nrbtform (tiling(iptc), vecscale(radius*[1 1 1])), vectrans(center)); end end %% Auxiliary functions % Cube function [tile,ver3d] = CubeLoad ver3d = [1 -1 -1 1;1 1 -1 -1;-1 -1 -1 -1]/sqrt(3); tile = nrbspheretile (ver3d); end function [cubesphere] = Multi_SphereFromCube(NB) %multipatch structure cubesphere(1) = NB; %rotate the different patches rot1 = vecroty(-pi/2); cubesphere(2) = nrbtform(cubesphere(1),rot1); rot2 = vecrotz(-pi/2); for ii = 2:4 cubesphere(ii+1) = nrbtform(cubesphere(ii),rot2); end rot3 = vecroty(pi); cubesphere(6) = nrbtform(cubesphere(1),rot3); end % Paired Icosahedron function [tile,ver3d] = IcoLoad %% Define vertex points gr = (1+sqrt(5))/2; % golden ratio Q1 = [1/2;0;-gr/2]; % see en.wikipedia.org/wiki/Regular_icosahedron#Cartesian_coordinates Q2 = [0;gr/2;-1/2]; Q3 = [-1/2;0;-gr/2]; Q4 = [0;-gr/2;-1/2]; R = norm(Q1); % sphere radius ver3d = [Q1,Q2,Q3,Q4]/R; % rescale to unit sphere %% Generate tile tile = nrbspheretile(ver3d); % make tile end function [icosphere] = Multi_SphereFromIco(NB) icosphere(1) = NB; %rotate the different patches rot1 = vecrot(2*pi/5,[0;(1+sqrt(5))/2/2;-1/2]); rot2 = vecrotx(pi); icosphere(6) = nrbtform(icosphere(1),rot2); for ii = 1:4 icosphere(ii+1) = nrbtform(icosphere(ii),rot1); icosphere(ii+6) = nrbtform(icosphere(ii+5),rot1); end end % Rhombic dodecahedron function [tile,ver3d] = RDodeLoad %% Define vertex points s2 = sqrt(2); s3 = sqrt(3); Q1 = [s2/2;0;-s2/2]; % see en.wikipedia.org/wiki/Regular_icosahedron#Cartesian_coordinates Q2 = [0;s3/3;-s2/s3]; Q3 = [-s2/2;0;-s2/2]; Q4 = [0;-s3/3;-s2/s3]; R = norm(Q1); % sphere radius ver3d = [Q1,Q2,Q3,Q4]/R; % rescale to unit sphere %% Generate tile tile = nrbspheretile(ver3d); % make tile end function [rdodesphere] = Multi_SphereFromRDode(NB) %multipatch structure rdodesphere(1) = NB; %rotate the different patches rot1 = vecrot(pi/2,[sqrt(2)/2;0;-sqrt(2)/2]); rdodesphere(2) = nrbtform(rdodesphere(1),rot1); rdodesphere(3) = nrbtform(rdodesphere(2),rot1); rdodesphere(4) = nrbtform(rdodesphere(3),rot1); rot2 = vecrot(pi/2,[-sqrt(2)/2;0;-sqrt(2)/2]); rdodesphere(5) = nrbtform(rdodesphere(1),rot2); rdodesphere(6) = nrbtform(rdodesphere(5),rot2); rdodesphere(7) = nrbtform(rdodesphere(6),rot2); rot3 = vecrotx(pi); rdodesphere(8) = nrbtform(rdodesphere(1),rot3); rdodesphere(9) = nrbtform(rdodesphere(2),rot3); rdodesphere(10) = nrbtform(rdodesphere(4),rot3); rdodesphere(11) = nrbtform(rdodesphere(5),rot3); rdodesphere(12) = nrbtform(rdodesphere(7),rot3); rdodesphere = rdodesphere([1 2 7 10 11 6 5 4 3 8 12 9]); end % Rhombic triacontahedron function [tile,ver3d] = RTriaLoad %% Define vertex points gr = (1+sqrt(5))/2; % golden ratio Q1 = [gr^2;0;-gr^3]; % see http://www.rwgrayprojects.com/rbfnotes/polyhed/PolyhedraData/RhombicTriaconta/RhombicTriaconta.pdf Q2 = [0;gr;-gr^3]; Q3 = [-gr^2;0;-gr^3]; Q4 = [0;-gr;-gr^3]; R1 = norm(Q1); R2 = norm(Q2); R3 = norm(Q3); R4 = norm(Q4);% sphere radius ver3d = [Q1/R1,Q2/R2,Q3/R3,Q4/R4]; % rescale to unit sphere %% Generate tile tile = nrbspheretile(ver3d); % make tile end function [rtriasphere] = Multi_SphereFromRTria(NB) %multipatch structure rtriasphere(1) = NB; %rotate the different patches gr = (1+sqrt(5))/2; rot1 = vecrot(2*pi/5,[gr^2;0;-gr^3]); for ii = 1:3 rtriasphere(ii+1) = nrbtform(rtriasphere(ii),rot1); end rot2 = vecrot(2*pi/5,[-gr^2;0;-gr^3]); for ii = 1:4 rtriasphere(ii+4) = nrbtform(rtriasphere(ii),rot2); end for ii = 1:12 rtriasphere(ii+8) = nrbtform(rtriasphere(ii+4),rot2); end rot3 = vecroty(pi); for ii = 1:5 rtriasphere(20+ii) = nrbtform(rtriasphere(4*(ii-1)+1),rot3); rtriasphere(25+ii) = nrbtform(rtriasphere(4*(ii-1)+2),rot3); end end % Deltoidal icositetrahedron function [tile,ver3d] = DIcoLoad %% Define vertex points s3 = sqrt(3); Q1 = [0;0;-1]; % quadrisected cube tile Q2 = [s3/3;0;-s3/3]; Q3 = [s3/3;s3/3;-s3/3]; Q4 = [0;s3/3;-s3/3]; R1 = norm(Q1); R2 = norm(Q2); R3 = norm(Q3); R4 = norm(Q4);% sphere radius ver3d = [Q1/R1,Q2/R2,Q3/R3,Q4/R4]; % rescale to unit sphere %% Generate tile tile = nrbspheretile(ver3d); % make tile end function [dicosphere] = Multi_SphereFromDIco(NB) %multipatch structure dicosphere(1) = NB; %rotate the different patches rot1 = vecrotz(pi/2); for ii = 1:3 dicosphere(ii+1) = nrbtform(dicosphere(ii),rot1); end rot2 = vecrotx(pi/2); for ii = 1:12 dicosphere(ii+4) = nrbtform(dicosphere(ii),rot2); end rot3 = vecroty(pi/2); rot4 = vecroty(-pi/2); for ii = 1:4 dicosphere(16+ii) = nrbtform(dicosphere(ii),rot3); dicosphere(20+ii) = nrbtform(dicosphere(ii),rot4); end end % Deltoidal hexecontahedron function [tile,ver3d] = DHexeLoad %% Define vertex points gr = (1+sqrt(5))/2; % golden ratio % Icosahedron A1 = [1;0;-gr]; % see en.wikipedia.org/wiki/Regular_icosahedron#Cartesian_coordinates A2 = [0;gr;-1]; A3 = [-1;0;-gr]; % Trisection Q1 = A1; Q2 = (A1+A2)/2; Q3 = (A1+A2+A3)/3; Q4 = (A3+A1)/2; R1 = norm(Q1); R2 = norm(Q2); R3 = norm(Q3); R4 = norm(Q4);% sphere radius ver3d = [Q1/R1,Q2/R2,Q3/R3,Q4/R4]; % rescale to unit sphere % Rotation % rotm = axang2rotm([ver3d(:,1)',-pi/5]); rotm = vecrot(-pi/5,ver3d(:,1)'); rotm = rotm(1:3,1:3); ver3d = rotm*ver3d; %% Generate tile tile = nrbspheretile(ver3d); % make tile end function [dhexesphere] = Multi_SphereFromDHexe(NB) %multipatch structure dhexesphere(1) = NB; %make a paired ico patch gr = (1+sqrt(5))/2; A1 = [1;0;-gr]; % see en.wikipedia.org/wiki/Regular_icosahedron#Cartesian_coordinates A2 = [0;gr;-1]; A3 = [-1;0;-gr]; A4 = (A1+A2+A3)/3; rot0 = vecrot(pi/5,A1); %rotate first tile dhexesphere(1) = nrbtform(dhexesphere(1),rot0); rot1 = vecrot(2*pi/3,A4); dhexesphere(2) = nrbtform(dhexesphere(1),rot1); dhexesphere(3) = nrbtform(dhexesphere(2),rot1); rot2 = vecrot(-2*pi/5,A1); for ii = 1:3 dhexesphere(ii+3) = nrbtform(dhexesphere(ii),rot2); end %rotate the 10 different patches rot3 = vecrot(2*pi/5,A2); for ii = 1:24 dhexesphere(ii+6) = nrbtform(dhexesphere(ii),rot3); end rot4 = vecrotx(pi); for ii = 1:30 dhexesphere(30+ii) = nrbtform(dhexesphere(ii),rot4); end end % Octant function [octant] = OctantFromDIco(NB) %rotate the different patches rot1 = vecroty(-pi/2); octant(1) = nrbtform(NB,rot1); rot2 = vecrotx(pi/2); octant(2) = nrbtform(NB,rot2); rot3 = vecrotz(-pi/2); temp = nrbtform(NB,rot3); temp = nrbtform(temp,rot2); octant(3) = nrbtform(temp,rot2); end %!demo %! tiling = nrbspheretiling('cube'); %! figure %! hold on %! for ii = 1:length(tiling) %! nrbkntplot(tiling(ii), 10) %! end %! title('Spherical cubic tiling') %!demo %! tiling = nrbspheretiling('ico'); %! figure %! hold on %! for ii = 1:length(tiling) %! nrbkntplot(tiling(ii), 10) %! end %! title('Spherical icosahedral tiling') %!demo %! tiling = nrbspheretiling('rdode'); %! figure %! hold on %! for ii = 1:length(tiling) %! nrbkntplot(tiling(ii), 10) %! end %! title('Spherical rhombic dodecahedral tiling') %!demo %! tiling = nrbspheretiling('rtria'); %! figure %! hold on %! for ii = 1:length(tiling) %! nrbkntplot(tiling(ii), 10) %! end %! title('Spherical rhombic triacontahedral tiling') %!demo %! tiling = nrbspheretiling('dico'); %! figure %! hold on %! for ii = 1:length(tiling) %! nrbkntplot(tiling(ii), 10) %! end %! title('Spherical deltoidal icositetrahedral tiling') %!demo %! tiling = nrbspheretiling('dico'); %! figure %! hold on %! for ii = 1:length(tiling) %! nrbkntplot(tiling(ii), 10) %! end %! title('Spherical deltoidal hexecontahedral tiling') nurbs-1.4.3/inst/PaxHeaders.16248/nrbextrude.m0000644000000000000000000000013214030323741015723 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbextrude.m0000644000175000017500000000546314030323741020032 0ustar00rvazquezrvazquez00000000000000function srf = nrbextrude(curve,vector) % % NRBEXTRUDE: Construct a NURBS surface by extruding a NURBS curve, or % construct a NURBS volume by extruding a NURBS surface. % % Calling Sequence: % % srf = nrbextrude(crv,vec); % % INPUT: % % crv : NURBS curve or surface to extrude, see nrbmak. % % vec : Vector along which the entity is extruded. % % OUTPUT: % % srf : NURBS surface or volume constructed. % % Description: % % Constructs either a NURBS surface by extruding a NURBS curve along a % defined vector, or a NURBS volume by extruding a NURBS surface. In the % first case, the NURBS curve forms the U direction of the surface edge, and % is extruded along the vector in the V direction. In the second case, the % original surface forms the U and V direction of the volume, and is extruded % along the W direction. % % Examples: % % Form a hollow cylinder by extruding a circle along the z-axis. % % srf = nrbextrude(nrbcirc, [0,0,1]); % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 2) error('Error too few input arguments!'); end if (iscell (curve.knots)) if (numel (curve.knots) == 3) error('Nurbs volumes cannot be extruded!'); end for ii = 1:size(curve.coefs,3) coefs(:,:,ii) = vectrans(vector) * squeeze (curve.coefs(:,:,ii)); end coefs = cat(4,curve.coefs,coefs); srf = nrbmak(coefs,{curve.knots{:}, [0 0 1 1]}); else coefs = cat(3,curve.coefs,vectrans(vector)*curve.coefs); srf = nrbmak(coefs,{curve.knots, [0 0 1 1]}); end end %!demo %! crv = nrbtestcrv; %! srf = nrbextrude(crv,[0 0 5]); %! nrbplot(srf,[40 10]); %! title('Extrusion of a test curve along the z-axis'); %! hold off % %!demo %! crv1 = nrbcirc (1, [0 0], 0, pi/2); %! crv2 = nrbcirc (2, [0 0], 0, pi/2); %! srf = nrbruled (crv1, crv2); %! vol = nrbextrude (srf, [0 0 1]); %! nrbplot (vol, [30 10 10]) %! title ('Extrusion of the quarter of a ring') % %!demo %! srf = nrbtestsrf; %! vol = nrbextrude(srf, [0 0 10]); %! nrbplot(vol,[20 20 20]); %! title('Extrusion of a test surface along the z-axis'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/nrbmodw.m0000644000000000000000000000013014030323741015207 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbmodw.m0000644000175000017500000000266714030323741017323 0ustar00rvazquezrvazquez00000000000000function mnrb = nrbmodw (nrb, new_w, index) % % NRBMODW: Modify the weights of specific control points of any NURBS map. % % Calling Sequence: % % nrb = nrbmodw (nrb, new_w, index); % % INPUT: % % nrb - NURBS map to be modified. % new_w - vector specifying the new weigths. % index - indices of the control points to be modified. % % OUTPUT: % % mnrb - the modified NURBS. % % Copyright (C) 2015 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % mnrb = nrb; [ii, jj, kk] = ind2sub (nrb.number, index); for count = 1:numel (ii) mnrb.coefs(:,ii(count),jj(count),kk(count)) = ... [nrb.coefs(1:3,ii(count),jj(count),kk(count))./repmat(nrb.coefs(4,ii(count),jj(count),kk(count)),3,1).*new_w(count); new_w(count)]; end end nurbs-1.4.3/inst/PaxHeaders.16248/vecnormalize.m0000644000000000000000000000013214030323741016237 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecnormalize.m0000644000175000017500000000262414030323741020342 0ustar00rvazquezrvazquez00000000000000function nvec = vecnormalize(vec) % % VECNORMALIZE: Normalize the vectors. % % Calling Sequence: % % nvec = vecnormalize(vec); % % INPUT: % % vec : An array of column vectors represented by a matrix of % size (dim,nv), where is the dimension of the vector and % nv the number of vectors. % % OUTPUT: % % nvec : Normalized vectors, matrix the same size as vec. % % Description: % % Normalizes the array of vectors, returning the unit vectors. % % Examples: % % Normalize the two vectors (0.0,2.0,1.3) and (1.5,3.4,2.3) % % nvec = vecnormalize([0.0 1.5; 2.0 3.4; 1.3 2.3]); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . nvec = vec./repmat(sqrt(sum(vec.^2)),[size(vec,1) ones(1,ndims(vec)-1)]); end nurbs-1.4.3/inst/PaxHeaders.16248/vecscale.m0000644000000000000000000000013214030323741015326 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecscale.m0000644000175000017500000000322414030323741017426 0ustar00rvazquezrvazquez00000000000000function ss = vecscale(vector) % % VECSCALE: Transformation matrix for a scaling. % % Calling Sequence: % % ss = vecscale(svec) % % INPUT: % % svec : A vectors defining the scaling along the x,y and z axes. % i.e. [sx, sy, sy] % % OUTPUT: % % ss : Scaling Transformation Matrix % % Description: % % Returns a (4x4) Transformation matrix for scaling. % % The matrix is: % % [ sx 0 0 0] % [ 0 sy 0 0] % [ 0 0 sz 0] % [ 0 0 0 1] % % Example: % % Scale up the NURBS line (0.0,0.0,0.0) - (1.0,1.0,1.0) by 3 along % the x-axis, 2 along the y-axis and 4 along the z-axis. % % line = nrbline([0.0 0.0 0.0],[1.0 1.0 1.0]); % trans = vecscale([3.0 2.0 4.0]); % sline = nrbtform(line, trans); % % See also: % % nrbtform % % Copyright (C) 2000 Mark Spink, 2007 Daniel Claxton % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 1 error('Scaling vector not specified'); end s = [vector(:);0;0]; ss = [s(1) 0 0 0; 0 s(2) 0 0; 0 0 s(3) 0; 0 0 0 1]; end nurbs-1.4.3/inst/PaxHeaders.16248/vecmag2.m0000644000000000000000000000013214030323741015065 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecmag2.m0000644000175000017500000000250114030323741017162 0ustar00rvazquezrvazquez00000000000000function mag = vecmag2(vec) % % VECMAG2: Squared magnitude of a set of vectors. % % Calling Sequence: % % mvec = vecmag2(vec) % % INPUT: % % vec : An array of column vectors represented by a matrix of % size (dim,nv), where dim is the dimension of the vector and % nv the number of vectors. % % OUTPUT: % % mvec : Squared magnitude of the vectors, vector of size (1,nv). % % Description: % % Determines the squared magnitude of the vectors. % % Examples: % % Find the squared magnitude of the two vectors (0.0,2.0,1.3) % and (1.5,3.4,2.3) % % mvec = vecmag2([0.0 1.5; 2.0 3.4; 1.3 2.3]); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . mag = sum(vec.^2); end nurbs-1.4.3/inst/PaxHeaders.16248/surfderivcpts.m0000644000000000000000000000013214030323741016444 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/surfderivcpts.m0000644000175000017500000000454014030323741020546 0ustar00rvazquezrvazquez00000000000000function pkl = surfderivcpts (n, p, U, m, q, V, P, d, r1, r2, s1, s2) % % SURFDERIVCPTS: Compute control points of n-th derivatives of a NURBS surface. % % usage: pkl = surfderivcpts (n, p, U, m, q, V, P, d) % % INPUT: % % n+1, m+1 = number of control points % p, q = spline order % U, V = knots % P = control points % d = derivative order % % OUTPUT: % % pkl (k+1, l+1, i+1, j+1) = i,jth control point % of the surface differentiated k % times in the u direction and l % times in the v direction % % Adaptation of algorithm A3.7 from the NURBS book, pg114 % % Copyright (C) 2009 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin <= 8) r1 = 0; r2 = n; s1 = 0; s2 = m; end r = r2-r1; s = s2-s1; du = min (d, p); dv = min (d, q); for j=s1:s2 temp = curvederivcpts (n, p, U, P(:,j+1:end), du, r1, r2); for k=0:du for i=0:r-k pkl (k+1, 1, i+1, j-s1+1) = temp (k+1, i+1); end end end for k=0:du for i=0:r-k dd = min (d-k, dv); temp = curvederivcpts (m, q, V(s1+1:end), pkl(k+1, 1, i+1, :), ... dd, 0, s); for l=1:dd for j=0:s-l pkl (k+1, l+1, i+1, j+1) = temp (l+1, j+1); end end end end end %!test %! coefs = cat(3,[0 0; 0 1],[1 1; 0 1]); %! knots = {[0 0 1 1] [0 0 1 1]}; %! plane = nrbmak(coefs,knots); %! pkl = surfderivcpts (plane.number(1)-1, plane.order(1)-1,... %! plane.knots{1}, plane.number(2)-1,... %! plane.order(2)-1, plane.knots{2}, ... %! squeeze (plane.coefs(1,:,:)), 1); nurbs-1.4.3/inst/PaxHeaders.16248/nrbnumbasisfun.m0000644000000000000000000000013014030323741016573 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbnumbasisfun.m0000644000175000017500000001066614030323741020705 0ustar00rvazquezrvazquez00000000000000function idx = nrbnumbasisfun (points, nrb) % % NRBNUMBASISFUN: Numbering of basis functions for NURBS % % Calling Sequence: % % N = nrbnumbasisfun (u, crv) % N = nrbnumbasisfun ({u, v}, srf) % N = nrbnumbasisfun (p, srf) % N = nrbnumbasisfun ({u, v, w}, vol) % N = nrbnumbasisfun (p, vol) % % INPUT: % % u or p(1,:,:) - parametric points along u direction % v or p(2,:,:) - parametric points along v direction % w or p(3,:,:) - parametric points along w direction % crv - NURBS curve % srf - NURBS surface % vol - NURBS volume % % OUTPUT: % % N - Indices of the basis functions that are nonvanishing at each % point. size(N) == [npts, prod(nrb.order)] % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2016 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if ( (nargin<2) ... || (nargout>1) ... || (~isstruct(nrb)) ... || (iscell(points) && ~iscell(nrb.knots)) ... || (~iscell(points) && iscell(nrb.knots) && (size(points,1)~=numel(nrb.number))) ... ) error('Incorrect input arguments in nrbnumbasisfun'); end if (~iscell(nrb.knots)) %% NURBS curve iv = findspan (nrb.number-1, nrb.order-1, points, nrb.knots); idx = numbasisfun (iv, points, nrb.order-1, nrb.knots); else ndim = numel (nrb.number); if (iscell (points)) for idim = 1:ndim pts_dim = points{idim}; sp{idim} = findspan (nrb.number(idim)-1, nrb.order(idim)-1, pts_dim, nrb.knots{idim}); % N{idim} = basisfun(sp{idim}, pts_dim, nrb.order(idim)-1, nrb.knots{idim}); num{idim} = numbasisfun (sp{idim}, pts_dim, nrb.order(idim)-1, nrb.knots{idim}) + 1; end npts_dim = cellfun (@numel, points); cumnpts = cumprod([1 npts_dim]); npts = prod (npts_dim); numaux = 1; cumorder = cumprod ([1 nrb.order]); cumnumber = cumprod ([1 nrb.number]); for idim = 1:ndim num_dim = reshape (num{idim}, 1, npts_dim(idim), 1, nrb.order(idim)); num_dim = repmat (num_dim, cumnpts(idim), 1, cumorder(idim), 1); num_prev = reshape (numaux, cumnpts(idim), 1, cumorder(idim), 1); num_prev = repmat (num_prev, 1, npts_dim(idim), 1, nrb.order(idim)); numaux = sub2ind ([cumnumber(idim) nrb.number(idim)], num_prev, num_dim); numaux = reshape (numaux, cumnpts(idim+1), cumorder(idim+1)); end idx = reshape (numaux, npts, prod (nrb.order)); else for idim = 1:ndim pts_dim = points(idim,:); sp{idim} = findspan (nrb.number(idim)-1, nrb.order(idim)-1, pts_dim, nrb.knots{idim}); % N{idim} = basisfun(sp{idim}, pts_dim, nrb.order(idim)-1, nrb.knots{idim}); num{idim} = numbasisfun (sp{idim}, pts_dim, nrb.order(idim)-1, nrb.knots{idim}) + 1; end npts = numel (points(1,:)); idx = zeros (npts, prod(nrb.order)); local_num = cell (ndim, 1); for ipt = 1:npts for idim = 1:ndim local_num{idim} = num{idim}(ipt,:); end [local_num{:}] = ndgrid (local_num{:}); idx(ipt,:) = reshape (sub2ind (nrb.number, local_num{:}), 1, size(idx, 2)); end end end end %!test %! p = 2; q = 3; m = 4; n = 5; %! Lx = 1; Ly = 1; %! nrb = nrb4surf ([0 0], [1 0], [0 1], [1 1]); %! nrb = nrbdegelev (nrb, [p-1, q-1]); %! ikx = linspace(0,1,m); iky = linspace(0,1,n); %! nrb = nrbkntins (nrb, {ikx(2:end-1), iky(2:end-1)}); %! nrb.coefs (4,:,:) = nrb.coefs (4,:,:) + rand (size (nrb.coefs (4,:,:))); %! u = rand (1, 30); v = rand (1, 10); %! u = (u-min (u))/max (u-min (u)); %! v = (v-min (v))/max (v-min (v)); %! N = nrbnumbasisfun ({u, v}, nrb); %! assert (all (all (N>0)), true) %! assert (all (all (N <= prod (nrb.number))), true) %! assert (max (max (N)), prod (nrb.number)) %! assert (min (min (N)), 1)nurbs-1.4.3/inst/PaxHeaders.16248/nrbdeval.m0000644000000000000000000000013214030323741015336 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbdeval.m0000644000175000017500000003436214030323741017445 0ustar00rvazquezrvazquez00000000000000function varargout = nrbdeval (nurbs, dnurbs, varargin) % NRBDEVAL: Evaluation of the derivative and second derivatives of NURBS curve, surface or volume. % % [pnt, jac] = nrbdeval (crv, dcrv, tt) % [pnt, jac] = nrbdeval (srf, dsrf, {tu tv}) % [pnt, jac] = nrbdeval (vol, dvol, {tu tv tw}) % [pnt, jac, hess] = nrbdeval (crv, dcrv, dcrv2, tt) % [pnt, jac, hess] = nrbdeval (srf, dsrf, dsrf2, {tu tv}) % [pnt, jac, hess] = nrbdeval (vol, dvol, dvol2, {tu tv tw}) % [pnt, jac, hess, third_der] = nrbdeval (crv, dcrv, dcrv2, dcrv3, tt) % [pnt, jac, hess, third_der] = nrbdeval (srf, dsrf, dsrf2, dsrf3, {tu tv}) % [pnt, jac, hess, third_der, fourth_der] = nrbdeval (crv, dcrv, dcrv2, dcrv3, dcrv4, tt) % [pnt, jac, hess, third_der, fourth_der] = nrbdeval (srf, dsrf, dsrf2, dsrf3, dsrf4, {tu tv}) % % INPUTS: % % crv, srf, vol - original NURBS curve, surface or volume. % dcrv, dsrf, dvol - NURBS derivative representation of crv, srf % or vol (see nrbderiv2) % dcrv2, dsrf2, dvol2 - NURBS second derivative representation of crv, % srf or vol (see nrbderiv2) % dcrv3, dsrf3, dvol3 - NURBS third derivative representation of crv, % srf or vol (see nrbderiv) % dcrv4, dsrf4, dvol4 - NURBS fourth derivative representation of crv, % srf or vol (see nrbderiv) % % tt - parametric evaluation points % If the nurbs is a surface or a volume then tt is a cell % {tu, tv} or {tu, tv, tw} are the parametric coordinates % % OUTPUT: % % pnt - evaluated points. % jac - evaluated first derivatives (Jacobian). % hess - evaluated second derivatives (Hessian). % third_der - evaluated third derivatives % fourth_der - evaluated fourth derivatives % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Carlo de Falco % Copyright (C) 2010, 2011 Rafael Vazquez % Copyright (C) 2018 Luca Coradello % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin == 3) tt = varargin{1}; elseif (nargin == 4) dnurbs2 = varargin{1}; tt = varargin{2}; elseif (nargin == 5) dnurbs2 = varargin{1}; dnurbs3 = varargin{2}; tt = varargin{3}; elseif (nargin == 6) dnurbs2 = varargin{1}; dnurbs3 = varargin{2}; dnurbs4 = varargin{3}; tt = varargin{4}; else error ('nrbrdeval: wrong number of input parameters') end if (~isstruct(nurbs)) error('NURBS representation is not structure!'); end if (~strcmp(nurbs.form,'B-NURBS')) error('Not a recognised NURBS representation'); end [cp,cw] = nrbeval(nurbs, tt); if (iscell(nurbs.knots)) if (size(nurbs.knots,2) == 3) % NURBS structure represents a volume temp = cw(ones(3,1),:,:,:); pnt = cp./temp; [cup,cuw] = nrbeval (dnurbs{1}, tt); tempu = cuw(ones(3,1),:,:,:); jac{1} = (cup-tempu.*pnt)./temp; [cvp,cvw] = nrbeval (dnurbs{2}, tt); tempv = cvw(ones(3,1),:,:,:); jac{2} = (cvp-tempv.*pnt)./temp; [cwp,cww] = nrbeval (dnurbs{3}, tt); tempw = cww(ones(3,1),:,:,:); jac{3} = (cwp-tempw.*pnt)./temp; % second derivatives if (nargout >= 3) if (exist ('dnurbs2')) [cuup, cuuw] = nrbeval (dnurbs2{1,1}, tt); tempuu = cuuw(ones(3,1),:,:,:); hess{1,1} = (cuup - (2*cup.*tempu + cp.*tempuu)./temp + 2*cp.*tempu.^2./temp.^2)./temp; clear cuup cuuw tempuu [cvvp, cvvw] = nrbeval (dnurbs2{2,2}, tt); tempvv = cvvw(ones(3,1),:,:,:); hess{2,2} = (cvvp - (2*cvp.*tempv + cp.*tempvv)./temp + 2*cp.*tempv.^2./temp.^2)./temp; clear cvvp cvvw tempvv [cwwp, cwww] = nrbeval (dnurbs2{3,3}, tt); tempww = cwww(ones(3,1),:,:,:); hess{3,3} = (cwwp - (2*cwp.*tempw + cp.*tempww)./temp + 2*cp.*tempw.^2./temp.^2)./temp; clear cwwp cwww tempww [cuvp, cuvw] = nrbeval (dnurbs2{1,2}, tt); tempuv = cuvw(ones(3,1),:,:,:); hess{1,2} = (cuvp - (cup.*tempv + cvp.*tempu + cp.*tempuv)./temp + 2*cp.*tempu.*tempv./temp.^2)./temp; hess{2,1} = hess{1,2}; clear cuvp cuvw tempuv [cuwp, cuww] = nrbeval (dnurbs2{1,3}, tt); tempuw = cuww(ones(3,1),:,:,:); hess{1,3} = (cuwp - (cup.*tempw + cwp.*tempu + cp.*tempuw)./temp + 2*cp.*tempu.*tempw./temp.^2)./temp; hess{3,1} = hess{1,3}; clear cuwp cuww tempuw [cvwp, cvww] = nrbeval (dnurbs2{2,3}, tt); tempvw = cvww(ones(3,1),:,:,:); hess{2,3} = (cvwp - (cvp.*tempw + cwp.*tempv + cp.*tempvw)./temp + 2*cp.*tempv.*tempw./temp.^2)./temp; hess{3,2} = hess{2,3}; clear cvwp cvww tempvw else warning ('nrbdeval: dnurbs2 missing. The second derivative is not computed'); hess = []; end if (nargout > 3) warning ('nrbdeval: 3rd and 4th order derivatives not implemented for volumes'); third_der = []; fourth_der = []; end end elseif (size(nurbs.knots,2) == 2) % NURBS structure represents a surface temp = cw(ones(3,1),:,:); pnt = cp./temp; [cup,cuw] = nrbeval (dnurbs{1}, tt); tempu = cuw(ones(3,1),:,:); jac{1} = (cup-tempu.*pnt)./temp; [cvp,cvw] = nrbeval (dnurbs{2}, tt); tempv = cvw(ones(3,1),:,:); jac{2} = (cvp-tempv.*pnt)./temp; % second derivatives if (nargout >= 3) if (exist ('dnurbs2')) [cuup, cuuw] = nrbeval (dnurbs2{1,1}, tt); tempuu = cuuw(ones(3,1),:,:); hess{1,1} = (cuup - (2*cup.*tempu + cp.*tempuu)./temp + 2*cp.*tempu.^2./temp.^2)./temp; [cvvp, cvvw] = nrbeval (dnurbs2{2,2}, tt); tempvv = cvvw(ones(3,1),:,:); hess{2,2} = (cvvp - (2*cvp.*tempv + cp.*tempvv)./temp + 2*cp.*tempv.^2./temp.^2)./temp; [cuvp, cuvw] = nrbeval (dnurbs2{1,2}, tt); tempuv = cuvw(ones(3,1),:,:); hess{1,2} = (cuvp - (cup.*tempv + cvp.*tempu + cp.*tempuv)./temp + 2*cp.*tempu.*tempv./temp.^2)./temp; hess{2,1} = hess{1,2}; else warning ('nrbdeval: dnurbs2 missing. The second derivative is not computed'); hess = []; end end if (nargout >= 4) if (exist ('dnurbs3')) [cuuup, cuuuw] = nrbeval (dnurbs3{1,1,1}, tt); tempuuu = cuuuw(ones(3,1),:,:); third_der{1,1,1} = (cuuup - 3*jac{1}.*tempuu - cp.*tempuuu - 3*hess{1,1}.*tempu)./temp; [cvvvp, cvvvw] = nrbeval (dnurbs3{2,2,2}, tt); tempvvv = cvvvw(ones(3,1),:,:); third_der{2,2,2} = (cvvvp - 3*jac{2}.*tempvv - cp.*tempvvv - 3*hess{2,2}.*tempv)./temp; [cuuvp, cuuvw] = nrbeval (dnurbs3{1,1,2}, tt); tempuuv = cuuvw(ones(3,1),:,:); third_der{1,1,2} = (cuuvp - 2*jac{1}.*tempuv - jac{2}.*tempuu - cp.*tempuuv - 2*hess{1,2}.*tempu - hess{1,1}.*tempv)./temp; third_der{1,2,1} = third_der{1,1,2}; third_der{2,1,1} = third_der{1,1,2}; [cuvvp, cuvvw] = nrbeval (dnurbs3{1,2,2}, tt); tempuvv = cuvvw(ones(3,1),:,:); third_der{1,2,2} = (cuvvp - 2*jac{2}.*tempuv - jac{1}.*tempvv - cp.*tempuvv - 2*hess{1,2}.*tempv - hess{2,2}.*tempu)./temp; third_der{2,2,1} = third_der{1,2,2}; third_der{2,1,2} = third_der{1,2,2}; else warning ('nrbdeval: dnurbs3 missing. The third derivative is not computed'); third_der = []; end end if (nargout == 5) if (exist ('dnurbs4')) [cuuuup, cuuuuw] = nrbeval (dnurbs4{1,1,1,1}, tt); tempuuuu = cuuuuw(ones(3,1),:,:); fourth_der{1,1,1,1} = (cuuuup - cp.*tempuuuu - 4*jac{1}.*tempuuu -6*hess{1,1}.*tempuu - 4*third_der{1,1,1}.*tempu)./temp; [cvvvvp, cvvvvw] = nrbeval (dnurbs4{2,2,2,2}, tt); tempvvvv = cvvvvw(ones(3,1),:,:); fourth_der{2,2,2,2} = (cvvvvp - cp.*tempvvvv - 4*jac{2}.*tempvvv -6*hess{2,2}.*tempvv - 4*third_der{2,2,2}.*tempv)./temp; [cuuuvp, cuuuvw] = nrbeval (dnurbs4{1,1,1,2}, tt); tempuuuv = cuuuvw(ones(3,1),:,:); fourth_der{1,1,1,2} = (cuuuvp - cp.*tempuuuv - 3*jac{1}.*tempuuv - jac{2}.*tempuuu -3*hess{1,2}.*tempuu -3*hess{1,1}.*tempuv ... - third_der{1,1,1}.*tempv - 3*third_der{1,1,2}.*tempu)./temp; fourth_der{1,1,2,1} = fourth_der{1,1,1,2}; fourth_der{1,2,1,1} = fourth_der{1,1,1,2}; fourth_der{2,1,1,1} = fourth_der{1,1,1,2}; [cuvvvp, cuvvvw] = nrbeval (dnurbs4{1,2,2,2}, tt); tempuvvv = cuvvvw(ones(3,1),:,:); fourth_der{1,2,2,2} = (cuvvvp - cp.*tempuvvv - 3*jac{2}.*tempuvv - jac{1}.*tempvvv -3*hess{1,2}.*tempvv -3*hess{2,2}.*tempuv ... - third_der{2,2,2}.*tempu - 3*third_der{1,2,2}.*tempv)./temp; fourth_der{2,2,1,2} = fourth_der{1,2,2,2}; fourth_der{2,1,2,2} = fourth_der{1,2,2,2}; fourth_der{2,2,2,1} = fourth_der{1,2,2,2}; [cuuvvp, cuuvvw] = nrbeval (dnurbs4{1,1,2,2}, tt); tempuuvv = cuuvvw(ones(3,1),:,:); fourth_der{1,1,2,2} = (cuuvvp - cp.*tempuuvv - 2*jac{1}.*tempuvv - 2*jac{2}.*tempuuv -hess{1,1}.*tempvv -hess{2,2}.*tempuu ... -4*hess{1,2}.*tempuv - 2*third_der{1,1,2}.*tempv - 2*third_der{1,2,2}.*tempu)./temp; fourth_der{1,2,2,1} = fourth_der{1,1,2,2}; fourth_der{2,2,1,1} = fourth_der{1,1,2,2}; fourth_der{1,2,1,2} = fourth_der{1,1,2,2}; fourth_der{2,1,2,1} = fourth_der{1,1,2,2}; fourth_der{2,1,1,2} = fourth_der{1,1,2,2}; clear cuup cuuw tempuu cvvp cvvw tempvv cuvp cuvw tempuv clear cuuup cuuuw tempuuu cvvvp cvvvw tempvvv cuuvp cuuvw tempuuv cuvvp cuvvw tempuvv clear cuuuup cuuuuw tempuuuu cvvvvp cvvvvw tempvvvv cuuuvp cuuuvw tempuuuv cuuvvp cuuvvw tempuuvv cvvvup cvvvuw tempvvvu else warning ('nrbdeval: dnurbs4 missing. The fourth derivative is not computed'); fourth_der = []; end end end else % NURBS is a curve temp = cw(ones(3,1),:); pnt = cp./temp; % first derivative [cup,cuw] = nrbeval (dnurbs,tt); temp1 = cuw(ones(3,1),:); jac = (cup-temp1.*pnt)./temp; % second derivative if (nargout >= 3 && exist ('dnurbs2')) [cuup,cuuw] = nrbeval (dnurbs2, tt); temp2 = cuuw(ones(3,1),:); hess = (cuup - (2*cup.*temp1 + cp.*temp2)./temp + 2*cp.*temp1.^2./temp.^2)./temp; if (nargout >= 4 && exist ('dnurbs3')) [cuuup, cuuuw] = nrbeval (dnurbs3, tt); temp3 = cuuuw(ones(3,1),:); third_der = (cuuup - 3*jac.*temp2 - cp.*temp3 - 3*hess.*temp1)./temp; if (nargout >= 5 && exist ('dnurbs4')) [cuuuup, cuuuuw] = nrbeval (dnurbs4, tt); temp4 = cuuuuw(ones(3,1),:); fourth_der = (cuuuup - cp.*temp4 - 4*jac.*temp3 -6*hess.*temp2 - 4*third_der.*temp1)./temp; if (iscell (tt)) fourth_der = {fourth_der}; end end if (iscell (tt)) third_der = {third_der}; end end if (iscell (tt)) hess = {hess}; end end if (iscell (tt)) jac = {jac}; end end varargout{1} = pnt; varargout{2} = jac; if (nargout >= 3) varargout{3} = hess; end if (nargout >= 4) varargout{4} = third_der; end if (nargout == 5) varargout{5} = fourth_der; end end %!demo %! crv = nrbtestcrv; %! nrbplot(crv,48); %! title('First derivatives along a test curve.'); %! %! tt = linspace(0.0,1.0,9); %! %! dcrv = nrbderiv(crv); %! %! [p1, dp] = nrbdeval(crv,dcrv,tt); %! %! p2 = vecnormalize(dp); %! %! hold on; %! plot(p1(1,:),p1(2,:),'ro'); %! h = quiver(p1(1,:),p1(2,:),p2(1,:),p2(2,:),0); %! set(h,'Color','black'); %! hold off; %!demo %! srf = nrbtestsrf; %! p = nrbeval(srf,{linspace(0.0,1.0,20) linspace(0.0,1.0,20)}); %! h = surf(squeeze(p(1,:,:)),squeeze(p(2,:,:)),squeeze(p(3,:,:))); %! set(h,'FaceColor','blue','EdgeColor','blue'); %! title('First derivatives over a test surface.'); %! %! npts = 5; %! tt = linspace(0.0,1.0,npts); %! dsrf = nrbderiv(srf); %! %! [p1, dp] = nrbdeval(srf, dsrf, {tt, tt}); %! %! up2 = vecnormalize(dp{1}); %! vp2 = vecnormalize(dp{2}); %! %! hold on; %! plot3(p1(1,:),p1(2,:),p1(3,:),'ro'); %! h1 = quiver3(p1(1,:),p1(2,:),p1(3,:),up2(1,:),up2(2,:),up2(3,:)); %! h2 = quiver3(p1(1,:),p1(2,:),p1(3,:),vp2(1,:),vp2(2,:),vp2(3,:)); %! set(h1,'Color','black'); %! set(h2,'Color','black'); %! %! hold off; %!test %! knots{1} = [0 0 0 1 1 1]; %! knots{2} = [0 0 0 .5 1 1 1]; %! knots{3} = [0 0 0 0 1 1 1 1]; %! cx = [0 0.5 1]; nx = length(cx); %! cy = [0 0.25 0.75 1]; ny = length(cy); %! cz = [0 1/3 2/3 1]; nz = length(cz); %! coefs(1,:,:,:) = repmat(reshape(cx,nx,1,1),[1 ny nz]); %! coefs(2,:,:,:) = repmat(reshape(cy,1,ny,1),[nx 1 nz]); %! coefs(3,:,:,:) = repmat(reshape(cz,1,1,nz),[nx ny 1]); %! coefs(4,:,:,:) = 1; %! nurbs = nrbmak(coefs, knots); %! x = rand(5,1); y = rand(5,1); z = rand(5,1); %! tt = [x y z]'; %! ders = nrbderiv(nurbs); %! [points,jac] = nrbdeval(nurbs,ders,tt); %! assert(points,tt,1e-10) %! assert(jac{1}(1,:,:),ones(size(jac{1}(1,:,:))),1e-12) %! assert(jac{2}(2,:,:),ones(size(jac{2}(2,:,:))),1e-12) %! assert(jac{3}(3,:,:),ones(size(jac{3}(3,:,:))),1e-12) %! %!test %! knots{1} = [0 0 0 1 1 1]; %! knots{2} = [0 0 0 0 1 1 1 1]; %! knots{3} = [0 0 0 1 1 1]; %! cx = [0 0 1]; nx = length(cx); %! cy = [0 0 0 1]; ny = length(cy); %! cz = [0 0.5 1]; nz = length(cz); %! coefs(1,:,:,:) = repmat(reshape(cx,nx,1,1),[1 ny nz]); %! coefs(2,:,:,:) = repmat(reshape(cy,1,ny,1),[nx 1 nz]); %! coefs(3,:,:,:) = repmat(reshape(cz,1,1,nz),[nx ny 1]); %! coefs(4,:,:,:) = 1; %! coefs = coefs([2 1 3 4],:,:,:); %! nurbs = nrbmak(coefs, knots); %! x = rand(5,1); y = rand(5,1); z = rand(5,1); %! tt = [x y z]'; %! dnurbs = nrbderiv(nurbs); %! [points, jac] = nrbdeval(nurbs,dnurbs,tt); %! assert(points,[y.^3 x.^2 z]',1e-10); %! assert(jac{2}(1,:,:),3*y'.^2,1e-12) %! assert(jac{1}(2,:,:),2*x',1e-12) %! assert(jac{3}(3,:,:),ones(size(z')),1e-12) nurbs-1.4.3/inst/PaxHeaders.16248/nrbexport.m0000644000000000000000000000013214030323741015564 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbexport.m0000644000175000017500000001234014030323741017663 0ustar00rvazquezrvazquez00000000000000function nrbexport (varargin) % % NRBEXPORT: export NURBS geometries to a format compatible with the one used in GeoPDEs. % % Calling Sequence: % % nrbexport (nurbs, filename); % nrbexport (nurbs, interfaces, boundaries, filename); % nrbexport (nurbs, interfaces, boundaries, subdomains, filename); % nrbexport (nurbs, filename, version); % nrbexport (nurbs, interfaces, boundaries, filename, version); % nrbexport (nurbs, interfaces, boundaries, subdomains, filename, version); % % INPUT: % % nurbs : NURBS curve, surface or volume, see nrbmak. % interfaces: interface information for GeoPDEs (see nrbmultipatch) % boundaries: boundary information for GeoPDEs (see nrbmultipatch) % filename : name of the output file. % version : either '-V0.7' or '-V2.1', to select the file format % % % Description: % % The data of the nurbs structure is written in the file, in a format % that can be read by GeoPDEs. By default, the file is saved in the % format used by GeoPDEs 2.1. For the format of GeoPDEs 2.0 use the % option '-v0.7'. Earlier versions of GeoPDEs are not supported. % % Copyright (C) 2011, 2014, 2015 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (strcmpi (varargin{end}, '-v0.7')) version = '0.7'; else version = '2.1'; end if (nargin == 2 || nargin == 3) nurbs = varargin{1}; filename = varargin{2}; if (numel (nurbs) > 1) warning ('Automatically creating the interface information with nrbmultipatch') [interfaces, boundaries] = nrbmultipatch (nurbs); subdomains = []; else interfaces = []; boundaries = []; subdomains = []; end elseif (nargin == 4 || (nargin == 5 && ischar(varargin{4}))) nurbs = varargin{1}; interfaces = varargin{2}; boundaries = varargin{3}; filename = varargin{4}; subdomains = []; elseif (nargin == 6 || (nargin == 5 && ~ischar(varargin{4}))) nurbs = varargin{1}; interfaces = varargin{2}; boundaries = varargin{3}; subdomains = varargin{4}; filename = varargin{5}; else error ('nrbexport: wrong number of input arguments') end fid = fopen (filename, 'w'); if (fid < 0) error ('nrbexport: cannot open file %s', filename); end ndim = numel (nurbs(1).order); npatch = numel (nurbs); rdim = 1; if (strcmp (version, '0.7')) rdim = ndim; else for iptc = 1:npatch if (any (abs(nurbs(iptc).coefs(3,:)) > 1e-12)) rdim = 3; break elseif (any (abs(nurbs(iptc).coefs(2,:)) > 1e-12)) rdim = 2; end end end if (strcmp (version, '0.7')) fprintf (fid, '%s\n', '# nurbs mesh v.0.7'); else fprintf (fid, '%s\n', '# nurbs mesh v.2.1'); end fprintf (fid, '%s\n', '#'); fprintf (fid, '%s\n', ['# ' date]); fprintf (fid, '%s\n', '#'); if (strcmp (version, '0.7')) fprintf (fid, '%i ', ndim, npatch, numel(interfaces), numel(subdomains)); else fprintf (fid, '%i ', ndim, rdim, npatch, numel(interfaces), numel(subdomains)); end fprintf (fid, '\n'); for iptc = 1:npatch fprintf (fid, '%s %i \n', 'PATCH', iptc); fprintf (fid, '%i ', nurbs(iptc).order-1); fprintf (fid, '\n'); fprintf (fid, '%i ', nurbs(iptc).number); fprintf (fid, '\n'); if (iscell (nurbs(iptc).knots)) for ii = 1:ndim fprintf (fid, '%1.15f ', nurbs(iptc).knots{ii}); fprintf (fid, '\n'); end else fprintf (fid, '%1.15f ', nurbs(iptc).knots); fprintf (fid, '\n'); end for ii = 1:rdim fprintf (fid, '%1.15f ', nurbs(iptc).coefs(ii,:,:)); fprintf (fid, '\n'); end fprintf (fid, '%1.15f ', nurbs(iptc).coefs(4,:,:)); fprintf (fid, '\n'); end for intrfc = 1:numel(interfaces) if (isfield (interfaces, 'ref')) fprintf (fid, '%s \n', interfaces(intrfc).ref); else fprintf (fid, '%s %i \n', 'INTERFACE', intrfc); end fprintf (fid, '%i %i \n', interfaces(intrfc).patch1, interfaces(intrfc).side1); fprintf (fid, '%i %i \n', interfaces(intrfc).patch2, interfaces(intrfc).side2); if (ndim == 2) fprintf (fid, '%i \n', interfaces(intrfc).ornt); elseif (ndim == 3) fprintf (fid, '%i %i %i \n', interfaces(intrfc).flag, interfaces(intrfc).ornt1, interfaces(intrfc).ornt2); end end for isubd = 1:numel(subdomains) % The subdomain part should be fixed fprintf (fid, '%s \n', subdomains(isubd).name); fprintf (fid, '%i ', subdomains(isubd).patches); fprintf (fid, '\n'); end for ibnd = 1:numel (boundaries) if (isfield (boundaries, 'name')) fprintf (fid, '%s \n', boundaries(ibnd).name); else fprintf (fid, '%s %i \n', 'BOUNDARY', ibnd); end fprintf (fid, '%i \n', boundaries(ibnd).nsides); for ii = 1:boundaries(ibnd).nsides fprintf (fid, '%i %i \n', boundaries(ibnd).patches(ii), boundaries(ibnd).faces(ii)); end end fclose (fid); end nurbs-1.4.3/inst/PaxHeaders.16248/private0000644000000000000000000000013214030323741014760 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.823919749 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/0000755000175000017500000000000014030323741017134 5ustar00rvazquezrvazquez00000000000000nurbs-1.4.3/inst/private/PaxHeaders.16248/nrb_srf_basisfun__.m0000644000000000000000000000013214030323741021036 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/nrb_srf_basisfun__.m0000644000175000017500000000230314030323741023133 0ustar00rvazquezrvazquez00000000000000function [B, N] = nrb_srf_basisfun__ (points, nrb); % __NRB_SRF_BASISFUN__: Undocumented internal function % % Copyright (C) 2009 Carlo de Falco % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. warning ('nrb_srf_basisfun__ is deprecated. Use nrbbasisfun, instead') m = size (nrb.coefs, 2) -1; n = size (nrb.coefs, 3) -1; p = nrb.order(1) -1; q = nrb.order(2) -1; u = points(1,:); v = points(2,:); npt = length(u); U = nrb.knots{1}; V = nrb.knots{2}; w = squeeze(nrb.coefs(4,:,:)); spu = findspan (m, p, u, U); spv = findspan (n, q, v, V); NuIkuk = basisfun (spu, u, p, U); NvJkvk = basisfun (spv, v, q, V); indIkJk = nrbnumbasisfun (points, nrb); for k=1:npt wIkaJkb(1:p+1, 1:q+1) = reshape (w(indIkJk(k, :)), p+1, q+1); NuIkukaNvJkvk(1:p+1, 1:q+1) = (NuIkuk(k, :).' * NvJkvk(k, :)); RIkJk(k, :) = reshape((NuIkukaNvJkvk .* wIkaJkb ./ sum(sum(NuIkukaNvJkvk .* wIkaJkb))),1,[]); end B = RIkJk; N = indIkJk; end nurbs-1.4.3/inst/private/PaxHeaders.16248/nrb_crv_basisfun__.m0000644000000000000000000000013214030323741021036 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/nrb_crv_basisfun__.m0000644000175000017500000000135314030323741023137 0ustar00rvazquezrvazquez00000000000000 function [B, nbfu] = nrb_crv_basisfun__ (points, nrb); % __NRB_CRV_BASISFUN__: Undocumented internal function % % Copyright (C) 2009 Carlo de Falco % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. warning ('nrb_crv_basisfun__ is deprecated. Use nrbbasisfun, instead') n = size (nrb.coefs, 2) -1; p = nrb.order -1; u = points; U = nrb.knots; w = nrb.coefs(4,:); spu = findspan (n, p, u, U); nbfu = numbasisfun (spu, u, p, U); N = w(nbfu+1) .* basisfun (spu, u, p, U); B = bsxfun (@(x,y) x./y, N, sum (N,2)); end nurbs-1.4.3/inst/private/PaxHeaders.16248/nrb_crv_basisfun_der__.m0000644000000000000000000000013214030323741021670 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/nrb_crv_basisfun_der__.m0000644000175000017500000000214314030323741023767 0ustar00rvazquezrvazquez00000000000000 function [Bu, nbfu] = nrb_crv_basisfun_der__ (points, nrb) % __NRB_CRV_BASISFUN_DER__: Undocumented internal function % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2013 Rafael Vazquez % % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. warning ('nrb_crv_basisfunder__ is deprecated. Use nrbbasisfunder, instead') n = size (nrb.coefs, 2) -1; p = nrb.order -1; u = points; U = nrb.knots; w = nrb.coefs(4,:); spu = findspan (n, p, u, U); nbfu = numbasisfun (spu, u, p, U); Nprime = basisfunder (spu, p, u, U, 1); N = reshape (Nprime(:,1,:), numel(u), p+1); Nprime = reshape (Nprime(:,2,:), numel(u), p+1); [Dpc, Dpk] = bspderiv (p, w, U); D = bspeval (p, w, U, u); Dprime = bspeval (p-1, Dpc, Dpk, u); Bu1 = bsxfun (@(np, d) np/d , Nprime.', D); Bu2 = bsxfun (@(n, dp) n*dp, N.', Dprime./D.^2); Bu = w(nbfu+1) .* (Bu1 - Bu2).'; end nurbs-1.4.3/inst/private/PaxHeaders.16248/nrb_srf_numbasisfun__.m0000644000000000000000000000013214030323741021556 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/nrb_srf_numbasisfun__.m0000644000175000017500000000162414030323741023660 0ustar00rvazquezrvazquez00000000000000function idx = nrb_srf_numbasisfun__ (points, nrb) % __NRB_SRF_NUMBASISFUN__: Undocumented internal function % % Copyright (C) 2009 Carlo de Falco % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. warning ('nrb_srf_numbasisfun__ is deprecated. Use nrbnumbasisfun, instead') m = nrb.number(1)-1; n = nrb.number(2)-1; npt = size(points,2); u = points(1,:); v = points(2,:); U = nrb.knots{1}; V = nrb.knots{2}; p = nrb.order(1)-1; q = nrb.order(2)-1; spu = findspan (m, p, u, U); Ik = numbasisfun (spu, u, p, U); spv = findspan (n, q, v, V); Jk = numbasisfun (spv, v, q, V); for k=1:npt [Jkb, Ika] = meshgrid(Jk(k, :), Ik(k, :)); idx(k, :) = sub2ind([m+1, n+1], Ika(:)+1, Jkb(:)+1); end end nurbs-1.4.3/inst/private/PaxHeaders.16248/onebasisfun__.m0000644000000000000000000000013214030323741020025 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/onebasisfun__.m0000644000175000017500000000253314030323741022127 0ustar00rvazquezrvazquez00000000000000function Nip = onebasisfun__ (u, p, U) % __ONEBASISFUN__: Undocumented internal function % % Adapted from Algorithm A2.4 from 'The NURBS BOOK' pg74. % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2012 Rafael Vazquez % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. Nip = zeros (size (u)); N = zeros (p+1, 1); for ii = 1:numel(u) if ((u(ii) == U(1)) && (U(1) == U(end-1)) || ... (u(ii) == U(end)) && (U(end) == U(2))) Nip(ii) = 1; continue end if (~ any (U <= u(ii))) || (~ any (U > u(ii))) continue; end for jj = 1:p+1 % Initialize zero-th degree functions if (u(ii) >= U(jj) && u(ii) < U(jj+1)) N(jj) = 1; else N(jj) = 0; end end for k = 1:p if (N(1) == 0) saved = 0; else saved = (u(ii) - U(1))*N(1) / (U(k+1)-U(1)); end for jj = 1:p-k+1 Uleft = U(1+jj); Uright = U(1+jj+k); if (N(jj+1) == 0) N(jj) = saved; saved = 0; else temp = N(jj+1)/(Uright-Uleft); N(jj) = saved + (Uright - u(ii))*temp; saved = (u(ii) - Uleft)*temp; end end end Nip(ii) = N(1); end end nurbs-1.4.3/inst/private/PaxHeaders.16248/nrb_srf_basisfun_der__.m0000644000000000000000000000013214030323741021670 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/nrb_srf_basisfun_der__.m0000644000175000017500000000312214030323741023765 0ustar00rvazquezrvazquez00000000000000function [Bu, Bv, N] = nrb_srf_basisfun_der__ (points, nrb); % __NRB_SRF_BASISFUN_DER__: Undocumented internal function % % Copyright (C) 2009 Carlo de Falco % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. warning ('nrb_srf_basisfunder__ is deprecated. Use nrbbasisfunder, instead') m = size (nrb.coefs, 2) -1; n = size (nrb.coefs, 3) -1; p = nrb.order(1) -1; q = nrb.order(2) -1; u = points(1,:); v = points(2,:); npt = length(u); U = nrb.knots{1}; V = nrb.knots{2}; w = squeeze(nrb.coefs(4,:,:)); spu = findspan (m, p, u, U); spv = findspan (n, q, v, V); N = nrbnumbasisfun (points, nrb); NuIkuk = basisfun (spu, u, p, U); NvJkvk = basisfun (spv, v, q, V); NuIkukprime = basisfunder (spu, p, u, U, 1); NuIkukprime = reshape (NuIkukprime(:,2,:), npt, []); NvJkvkprime = basisfunder (spv, q, v, V, 1); NvJkvkprime = reshape (NvJkvkprime(:,2,:), npt, []); for k=1:npt wIkaJkb(1:p+1, 1:q+1) = reshape (w(N(k, :)), p+1, q+1); Num = (NuIkuk(k, :).' * NvJkvk(k, :)) .* wIkaJkb; Num_du = (NuIkukprime(k, :).' * NvJkvk(k, :)) .* wIkaJkb; Num_dv = (NuIkuk(k, :).' * NvJkvkprime(k, :)) .* wIkaJkb; Denom = sum(sum(Num)); Denom_du = sum(sum(Num_du)); Denom_dv = sum(sum(Num_dv)); Bu(k, :) = reshape((Num_du/Denom - Denom_du.*Num/Denom.^2),1,[]); Bv(k, :) = reshape((Num_dv/Denom - Denom_dv.*Num/Denom.^2),1,[]); end endnurbs-1.4.3/inst/private/PaxHeaders.16248/onebasisfunder__.m0000644000000000000000000000013214030323741020520 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/private/onebasisfunder__.m0000644000175000017500000000176414030323741022627 0ustar00rvazquezrvazquez00000000000000function [N, Nder] = onebasisfunder__ (u, p, U) % __ONEBASISFUNDER__: Undocumented internal function % % Copyright (C) 2012 Rafael Vazquez % This software comes with ABSOLUTELY NO WARRANTY; see the file % COPYING for details. This is free software, and you are welcome % to distribute it under the conditions laid out in COPYING. N = zeros (size (u)); Nder = zeros (size (u)); for ii = 1:numel (u) if (~ any (U <= u(ii))) || (~ any (U > u(ii))) continue; elseif (p == 0) N(ii) = 1; Nder(ii) = 0; continue; else ln = u(ii) - U(1); ld = U(end-1) - U(1); if (ld ~= 0) aux = onebasisfun__ (u(ii), p-1, U(1:end-1))/ ld; N(ii) = N(ii) + ln * aux; Nder(ii) = Nder(ii) + p * aux; end dn = U(end) - u(ii); dd = U(end) - U(2); if (dd ~= 0) aux = onebasisfun__ (u(ii), p-1, U(2:end))/ dd; N(ii) = N(ii) + dn * aux; Nder(ii) = Nder(ii) - p * aux; end end end end nurbs-1.4.3/inst/PaxHeaders.16248/nrbextract.m0000644000000000000000000000013214030323741015715 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbextract.m0000644000175000017500000000704614030323741020023 0ustar00rvazquezrvazquez00000000000000function crvs = nrbextract(srf, sides) % % NRBEXTRACT: construct NURBS curves by extracting the boundaries of a NURBS surface, or NURBS surfaces by extracting the boundary of a NURBS volume. % It only works for geometries constructed with open knot vectors. For a NURBS curve, % it returns two structures with the the boundary knots and control points. % % Calling Sequence: % % crvs = nrbextract(surf, [sides]); % % INPUT: % % surf : NURBS surface or volume, see nrbmak. % sides : the list of boundary sides to be extracted % % OUTPUT: % % crvs : array of NURBS curves or NURBS surfaces extracted. % % Description: % % Constructs either an array of four NURBS curves, by extracting the boundaries % of a NURBS surface, or an array of six surfaces, by extracting the boundaries % of a NURBS volume. The new entities are ordered in the following way % % 1: U = 0 % 2: U = 1 % 3: V = 0 % 4: V = 1 % 5: W = 0 (only for volumes) % 6: W = 1 (only for volumes) % % Copyright (C) 2010,2014,2015 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 2) if (~iscell (srf.knots)) ndim = 1; else ndim = numel(srf.knots); end sides = 1:2*ndim; end if (~iscell (srf.knots)) crvs(1).knots = srf.knots(1); crvs(1).coefs = srf.coefs(:,1); crvs(2).knots = srf.knots(end); crvs(2).coefs = srf.coefs(:,end); crvs = crvs(sides); return end for idim = 1:numel(srf.knots) ord = srf.order(idim); if (srf.knots{idim}(1) ~= srf.knots{idim}(ord) || ... srf.knots{idim}(end) ~= srf.knots{idim}(end-ord+1)) error ('nrbextract: only working for open knot vectors') end end if (numel (srf.knots) == 2) for ind = 1:2 ind2 = mod (ind, 2) + 1; %ind2 = [2 1]; bnd1 = (ind - 1) * 2 + 1; bnd2 = (ind - 1) * 2 + 2; if (ind == 1) coefs1 = squeeze (srf.coefs(:,1,:)); coefs2 = squeeze (srf.coefs(:,end,:)); elseif (ind == 2) coefs1 = squeeze (srf.coefs(:,:,1)); coefs2 = squeeze (srf.coefs(:,:,end)); end crvs(bnd1) = nrbmak (coefs1, srf.knots{ind2}); crvs(bnd2) = nrbmak (coefs2, srf.knots{ind2}); end elseif (numel (srf.knots) == 3) for ind = 1:3 inds = setdiff (1:3, ind); bnd1 = (ind - 1) * 2 + 1; bnd2 = (ind - 1) * 2 + 2; if (ind == 1) coefs1 = squeeze (srf.coefs(:,1,:,:)); coefs2 = squeeze (srf.coefs(:,end,:,:)); elseif (ind == 2) coefs1 = squeeze (srf.coefs(:,:,1,:)); coefs2 = squeeze (srf.coefs(:,:,end,:)); elseif (ind == 3) coefs1 = squeeze (srf.coefs(:,:,:,1)); coefs2 = squeeze (srf.coefs(:,:,:,end)); end crvs(bnd1) = nrbmak (coefs1, {srf.knots{inds(1)} srf.knots{inds(2)}}); crvs(bnd2) = nrbmak (coefs2, {srf.knots{inds(1)} srf.knots{inds(2)}}); end else error ('The entity is not a surface nor a volume') end crvs = crvs(sides); end nurbs-1.4.3/inst/PaxHeaders.16248/basisfunder.m0000644000000000000000000000013214030323741016046 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/basisfunder.m0000644000175000017500000000770114030323741020152 0ustar00rvazquezrvazquez00000000000000function dersv = basisfunder (ii, pl, uu, u_knotl, nders) % BASISFUNDER: B-Spline Basis function derivatives. % % Calling Sequence: % % ders = basisfunder (ii, pl, uu, k, nd) % % INPUT: % % ii - knot span index (see findspan) % pl - degree of curve % uu - parametric points % k - knot vector % nd - number of derivatives to compute % % OUTPUT: % % ders - ders(n, i, :) (i-1)-th derivative at n-th point % % Adapted from Algorithm A2.3 from 'The NURBS BOOK' pg72. % % See also: % % numbasisfun, basisfun, findspan % % Copyright (C) 2009,2011 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . dersv = zeros(numel(uu), nders+1, pl+1); for jj = 1:numel(uu) i = ii(jj)+1; %% convert to base-1 numbering of knot spans u = uu(jj); ders = zeros(nders+1,pl+1); ndu = zeros(pl+1,pl+1); left = zeros(pl+1); right = zeros(pl+1); a = zeros(2,pl+1); ndu(1,1) = 1; for j = 1:pl left(j+1) = u - u_knotl(i+1-j); right(j+1) = u_knotl(i+j) - u; saved = 0; for r = 0:j-1 ndu(j+1,r+1) = right(r+2) + left(j-r+1); temp = ndu(r+1,j)/ndu(j+1,r+1); ndu(r+1,j+1) = saved + right(r+2)*temp; saved = left(j-r+1)*temp; end ndu(j+1,j+1) = saved; end for j = 0:pl ders(1,j+1) = ndu(j+1,pl+1); end for r = 0:pl s1 = 0; s2 = 1; a(1,1) = 1; for k = 1:nders %compute kth derivative d = 0; rk = r-k; pk = pl-k; if (r >= k) a(s2+1,1) = a(s1+1,1)/ndu(pk+2,rk+1); d = a(s2+1,1)*ndu(rk+1,pk+1); end if (rk >= -1) j1 = 1; else j1 = -rk; end if ((r-1) <= pk) j2 = k-1; else j2 = pl-r; end for j = j1:j2 a(s2+1,j+1) = (a(s1+1,j+1) - a(s1+1,j))/ndu(pk+2,rk+j+1); d = d + a(s2+1,j+1)*ndu(rk+j+1,pk+1); end if (r <= pk) a(s2+1,k+1) = -a(s1+1,k)/ndu(pk+2,r+1); d = d + a(s2+1,k+1)*ndu(r+1,pk+1); end ders(k+1,r+1) = d; j = s1; s1 = s2; s2 = j; end end r = pl; for k = 1:nders for j = 0:pl ders(k+1,j+1) = ders(k+1,j+1)*r; end r = r*(pl-k); end dersv(jj, :, :) = ders; end end %!test %! k = [0 0 0 0 1 1 1 1]; %! p = 3; %! u = rand (1); %! i = findspan (numel(k)-p-2, p, u, k); %! ders = basisfunder (i, p, u, k, 1); %! sumders = sum (squeeze(ders), 2); %! assert (sumders(1), 1, 1e-15); %! assert (sumders(2:end), 0, 1e-15); %!test %! k = [0 0 0 0 1/3 2/3 1 1 1 1]; %! p = 3; %! u = rand (1); %! i = findspan (numel(k)-p-2, p, u, k); %! ders = basisfunder (i, p, u, k, 7); %! sumders = sum (squeeze(ders), 2); %! assert (sumders(1), 1, 1e-15); %! assert (sumders(2:end), zeros(rows(squeeze(ders))-1, 1), 1e-13); %!test %! k = [0 0 0 0 1/3 2/3 1 1 1 1]; %! p = 3; %! u = rand (100, 1); %! i = findspan (numel(k)-p-2, p, u, k); %! ders = basisfunder (i, p, u, k, 7); %! for ii=1:10 %! sumders = sum (squeeze(ders(ii,:,:)), 2); %! assert (sumders(1), 1, 1e-15); %! assert (sumders(2:end), zeros(rows(squeeze(ders(ii,:,:)))-1, 1), 1e-13); %! end %! assert (ders(:, (p+2):end, :), zeros(numel(u), 8-p-1, p+1), 1e-13) %! assert (all(all(ders(:, 1, :) <= 1)), true) nurbs-1.4.3/inst/PaxHeaders.16248/nrbtestcrv.m0000644000000000000000000000013214030323741015735 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbtestcrv.m0000644000175000017500000000200414030323741020030 0ustar00rvazquezrvazquez00000000000000function crv = nrbtestcrv % NRBTESTCRV: Constructs a simple test curve. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . pnts = [0.5 1.5 4.5 3.0 7.5 6.0 8.5; 3.0 5.5 5.5 1.5 1.5 4.0 4.5; 0.0 0.0 0.0 0.0 0.0 0.0 0.0]; crv = nrbmak(pnts,[0 0 0 1/4 1/2 3/4 3/4 1 1 1]); end %!demo %! crv = nrbtestcrv; %! nrbplot(crv,100) %! title('Test curve') %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/vecmag.m0000644000000000000000000000013214030323741015003 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecmag.m0000644000175000017500000000242514030323741017105 0ustar00rvazquezrvazquez00000000000000function mag = vecmag(vec) % % VECMAG: Magnitude of the vectors. % % Calling Sequence: % % mvec = vecmag(vec) % % INPUT: % % vec : An array of column vectors represented by a matrix of % size (dim,nv), where is the dimension of the vector and % nv the number of vectors. % % OUTPUT: % % mvec : Magnitude of the vectors, vector of size (1,nv). % % Description: % % Determines the magnitude of the vectors. % % Examples: % % Find the magnitude of the two vectors (0.0,2.0,1.3) and (1.5,3.4,2.3) % % mvec = vecmag([0.0 1.5; 2.0 3.4; 1.3 2.3]); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . mag = sqrt(sum(vec.^2)); end nurbs-1.4.3/inst/PaxHeaders.16248/nrbsquare.m0000644000000000000000000000013214030323741015543 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbsquare.m0000644000175000017500000000516114030323741017645 0ustar00rvazquezrvazquez00000000000000function srf = nrbsquare (corner, lengthx, lengthy, varargin) % % NRBSQUARE: create the NURBS surface for a square. % % Calling Sequence: % % srf = nrbsquare (corner, lengthx, lengthy); % srf = nrbsquare (corner, lengthx, lengthy, degree); % srf = nrbsquare (corner, lengthx, lengthy, degree, nsub); % % INPUT: % corner : the coordinates of the bottom left corner of the square. % lenghtx : the length along the x direction. % lenghty : the length along the y direction. % degree : the degree of the NURBS surface, in each direction. % nsub : the number of subdivision of the NURBS surface, in each direction. % % OUTPUT: % srf : the NURBS surface. % % Copyright (C) 2016 Jacopo Corno, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (isempty (corner)) corner = [0 0]; end nsub = [1 1]; degree = [1 1]; if (numel (varargin) >= 1) if (numel (varargin{1}) == 1) degree = [varargin{1} varargin{1}]; elseif (numel (varargin{1}) == 2) degree = varargin{1}; else error ('The degree vector should provide the degree in each direction (two values).'); end if (numel (varargin) == 2) if (numel (varargin{2}) == 1) nsub = [varargin{2} varargin{2}]; elseif (numel (varargin{2}) == 2) nsub = varargin{2}; else error ('The nsub vector should provide the number of intervals in each direction (two values).'); end end end srf = nrb4surf (corner, corner+[lengthx 0], corner+[0 lengthy], corner+[lengthx lengthy]); srf = nrbdegelev (srf, degree-[1 1]); [~,~,new_knots] = kntrefine (srf.knots, nsub-1, degree, degree-[1 1]); srf = nrbkntins (srf, new_knots); end %!test %! srf = nrbsquare ([], 1, 2, 2, 4); %! assert (srf.order, [3 3]); %! knt = [0 0 0 1/4 1/2 3/4 1 1 1]; %! assert (srf.knots, {knt knt}) %! x = linspace (0, 1, 100); %! [X,Y] = ndgrid (x, x); %! vals = nrbeval (srf, {x x}); %! assert (squeeze(vals(1,:,:)), X, 1e-15); %! assert (squeeze(vals(2,:,:)), 2*Y, 1e-15); nurbs-1.4.3/inst/PaxHeaders.16248/bspinterpcrv.m0000644000000000000000000000013214030323741016262 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/bspinterpcrv.m0000644000175000017500000000571714030323741020373 0ustar00rvazquezrvazquez00000000000000function [crv, u] = bspinterpcrv (Q, p, method) % % BSPINTERPCRV: B-Spline interpolation of a 3d curve. % % Calling Sequence: % % crv = bspinterpcrv (Q, p); % crv = bspinterpcrv (Q, p, method); % [crv, u] = bspinterpcrv (Q, p); % [crv, u] = bspinterpcrv (Q, p, method); % % INPUT: % % Q - points to be interpolated in the form [x_coord; y_coord; z_coord]. % p - degree of the interpolating curve. % method - parametrization method. The available choices are: % 'equally_spaced' % 'chord_length' % 'centripetal' (Default) % % OUTPUT: % % crv - the B-Spline curve. % u - the parametric points corresponding to the interpolation ones. % % See The NURBS book pag. 364 for more information. % % % Copyright (C) 2015 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % if (nargin < 3 || isempty (method)) method = 'centripetal'; end n = size (Q, 2); if (strcmpi (method, 'equally_spaced')) u = linspace (0, 1, n); elseif (strcmpi (method, 'chord_length')) d = sum (sqrt (sum (diff (Q')'.^2,1))); u = zeros (1, n); u(2:n) = cumsum (sqrt (sum (diff(Q, [], 2).^2, 1)))/d; % for ii = 2:n-1 % u(ii) = u(ii-1) + norm (Q(:,ii) - Q(:,ii-1)) / d; % end u(end) = 1; elseif (strcmpi (method, 'centripetal')) d = sum (sqrt (sqrt (sum (diff (Q')'.^2,1)))); u = zeros (1, n); u(2:n) = cumsum (sqrt (sqrt (sum (diff(Q, [], 2).^2, 1))))/d; % for ii = 2:n-1 % u(ii) = u(ii-1) + sqrt (norm (Q(:,ii) - Q(:,ii-1))) / d; % end u(end) = 1; else error ('BSPINTERPCRV: unrecognized parametrization method.') end knts = zeros (1, n+p+1); for jj = 2:n-p knts(jj+p) = 1/p * sum (u(jj:jj+p-1)); end knts(end-p:end) = ones(1,p+1); A = zeros (n, n); A(1,1) = 1; A(n,n) = 1; for ii=2:n-1 span = findspan (n, p, u(ii), knts); A(ii,span-p+1:span+1) = basisfun (span, u(ii), p, knts); end x = A \ Q(1,:)'; y = A \ Q(2,:)'; z = A \ Q(3,:)'; pnts = [x'; y'; z'; ones(size(x'))]; crv = nrbmak (pnts, knts); end %!demo %! Q = [1 0 -1 -1 -2 -3; %! 0 1 0 -1 -1 0; %! 0 0 0 0 0 0]; %! p = 2; %! crv = bspinterpcrv (Q, p); %! %! plot (Q(1,:), Q(2,:), 'xk'); %! hold on; grid on; %! nrbkntplot (crv); nurbs-1.4.3/inst/PaxHeaders.16248/crvkntremove.m0000644000000000000000000000013214030323741016266 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/crvkntremove.m0000644000175000017500000001036114030323741020366 0ustar00rvazquezrvazquez00000000000000 function [rcrv, t] = crvkntremove (crv, u, r, s, num, d) % % CRVKNTREMOVE: Remove one knot from the knot-vector of a NURBS curve. % % Calling Sequence: % % [rcrv, remflag] = crvkntremove (crv, u, r, s, num, d); % % INPUT: % % crv : NURBS curve, see nrbmak. % % u : knot to be removed. % % r : index of the knot to be removed. % % s : multiplicity of the knot to be removed. % % num : number of knot removals requested. % % d : curve deviation tolerance. % % OUTPUT: % % rcrv : new NURBS structure for the curve with knot u remuved. % % t : actual number of knot removals performed. % % % % DESCRIPTION: % % Remove knot u from the NURBS curve crv at most num times. % Check that the maximum deviation of the curve be less than d. % Based on algorithm A5.8 NURBS Book (pag183) % % SEE ALSO: % % nrbkntins % % Copyright (C) 2013 Jacopo Corno % Copyright (C) 2013 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . [U, Pw, t] = RemoveCurveKnot (crv.number, crv.order - 1, crv.knots, ... crv.coefs, u, r, s, num, d); rcrv = nrbmak (Pw, U); end %!test %! crv = nrbdegelev (nrbline (), 3); %! acrv = nrbkntins (crv, [.11 .11 .11]); %! [rcrv, t] = crvkntremove (acrv, .11, 8, 3, 3, 1e-10); %! assert (crv.knots, rcrv.knots, 1e-10); %! assert (t, 3); %!test %! crv = nrbcirc (); %! acrv = nrbkntins (crv, [.3 .3]); %! [rcrv, t] = crvkntremove (acrv, .3, 7, 2, 2, 1e-10); %! assert (crv.knots, rcrv.knots, 1e-10); %! assert (t, 2); function [U, Pw, t] = RemoveCurveKnot (n, p, U, Pw, u, r, s, num, d) % see algorithm A5.8 NURBS Book (pag183) w = min (Pw(4,:)); Pmax = max (sqrt (sum (Pw.^2, 1))); TOL = d*w / (1 + Pmax); m = n + p + 1; ord = p + 1; fout = (2*r - s - p) / 2; % first control point out last = r - s; first = r - p; temp = zeros (4, 2*p + 1); for t = 0:num-1 off = first - 1; % diff in index between temp and P temp(:,1) = Pw(:,off); temp(:,last+1-off+1) = Pw(:,last+1); i = first; j = last; ii = 1; jj = last - off; remflag = 0; while (j - i > t) % compute new control points for one removal step alfi = (u-U(i)) / (U(i+ord+t)-U(i)); alfj = (u-U(j-t)) / (U(j+ord)-U(j-t)); temp(:,ii+1) = (Pw(:,i)-(1.0-alfi).*temp(:,ii-1+1))./alfi; temp(:,jj+1) = (Pw(:,j)-alfj.*temp(:,jj+1+1))./(1.0-alfj); i = i + 1; ii = ii + 1; j = j - 1; jj = jj - 1; end if (j - i <= t) % check if knot removable if (norm (temp(:,ii-1+1) - temp(:,jj+1+1)) <= TOL) remflag = 1; else alfi = (u-U(i)) / (U(i+ord+t)-U(i)); if (norm (Pw(:,i) - (alfi.*temp(:,ii+t+1+1) + ... (1-alfi).*temp(:,ii-1+1))) <= TOL) remflag = 1; end%if end%if end%if if (remflag == 0) break; % cannot remove any more knots -> get out of for loop else % successful removal -> save new control points i = first; j = last; while (j - i > t) Pw(:,i) = temp(:,i-off+1); Pw(:,j) = temp(:,j-off+1); i = i + 1; j = j - 1; end end%if first = first - 1; last = last + 1; t = t + 1; end % end of for loop if (t == 0) return; end%if % shift knots for k = r+1:m U(k-t) = U(k); end U = U(1:end-t); j = floor(fout); i = j; for k = 1:t-1 if (mod (k, 2) == 1) i = i+1; else j = j-1; end%if end % shift points for k = i+1:n Pw(:,j) = Pw(:,k); j = j+1; end Pw = Pw(:,1:end-t); return; end nurbs-1.4.3/inst/PaxHeaders.16248/nrbmodp.m0000644000000000000000000000013014030323741015200 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbmodp.m0000644000175000017500000000270214030323741017302 0ustar00rvazquezrvazquez00000000000000function mnrb = nrbmodp (nrb, move, index) % % NRBMODP: Modify the coordinates of specific control points of any NURBS % map. The weight is not changed. % % Calling Sequence: % % nrb = nrbmodp (nrb, move, index); % % INPUT: % % nrb - NURBS map to be modified. % move - vector specifying the displacement of all the ctrl points. % index - indeces of the control points to be modified. % % OUTPUT: % % mnrb - the modified NURBS. % % Copyright (C) 2015 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % move = reshape (move, 3, 1); mnrb = nrb; [ii, jj, kk] = ind2sub (nrb.number, index); for count = 1:numel (ii) mnrb.coefs(1:3,ii(count),jj(count),kk(count)) = nrb.coefs(1:3,ii(count),jj(count),kk(count)) + ... move * nrb.coefs(4,ii(count),jj(count),kk(count)); end end nurbs-1.4.3/inst/PaxHeaders.16248/basiskntins.m0000644000000000000000000000013214030323741016071 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/basiskntins.m0000644000175000017500000000610414030323741020171 0ustar00rvazquezrvazquez00000000000000function C = basiskntins (deg, kc, kf) % Compute the coefficient matrix for non-uniform B-splines subdivision. % % This represents the B-spline basis given by a coarse knot vector % in terms of the B-spline basis of a finer knot vector. % % The function is implemented for the univariate case, based on % Algorithm A5.4 from 'The NURBS BOOK' pg164. % % % Calling Sequence: % % S = basiskntins (deg, kc, kf); % % INPUT: % % deg - degree of the first knot vector % kc - coarse knot vector % kf - fine knot vector % % OUTPUT: % % S - The matrix relating the two spaces, of size (deg-nu, deg-nt) % with nu = numel(u)-deg-1, nt = numel(t)-deg-1 % % Copyright (C) 2015, 2016, 2018 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . nk = numel(kc); nc = nk - (deg+1); u = new_knots(kc, kf); nu = numel(u); nf = nc + nu; if (nu == 0) C = speye (nf, nc); return else C = sparse (nf, nc); end ik = zeros(1,nk+nu); n = nc - 1; r = nu - 1; m = nc + deg; a = findspan(n, deg, u(1), kc); b = findspan(n, deg, u(end), kc); b = b+1; C(1:a-deg+1,1:a-deg+1) = speye(a-deg+1); C(b+nu:nc+nu,b:nc) = speye(nc-b+1); ik(1:a+1) = kc(1:a+1); ik(b+deg+nu+1:m+nu+1) = kc(b+deg+1:m+1); ii = b + deg - 1; ss = ii + nu; for jj=r:-1:0 ind = (a+1):ii; ind = ind(u(jj+1)<=kc(ind+1)); C(ind+ss-ii-deg,:) = 0; C(ind+ss-ii-deg,ind-deg) = speye(numel(ind)); ik(ind+(ss-ii)+1) = kc(ind+1); ii = ii - numel(ind); ss = ss - numel(ind); C(ss-deg,:) = C(ss-deg+1,:); for l=1:deg ind = ss - deg + l; alfa = ik(ss+l+1) - u(jj+1); if abs(alfa) == 0 C(ind,:) = C(ind+1,:); else alfa = alfa/(ik(ss+l+1) - kc(ii-deg+l+1)); C(ind,:) = C(ind,:)*alfa + C(ind+1,:)*(1-alfa); end end ik(ss+1) = u(jj+1); ss = ss - 1; end end function u = new_knots (kc, kf) % Find the new knots, with the correct multiplicity [valc, multc] = unique (kc, 'last'); multc = diff ([0 multc(:)']); [valf, multf] = unique (kf, 'last'); multf = diff ([0 multf(:)']); unew = setdiff (kf, kc); [~,posf] = ismember (unew, valf); mult_new = multf(posf); [urep, indc, indf] = intersect (valc, valf); mult_rep = multf(indf) - multc(indc); urep = urep(mult_rep>0); mult_rep = mult_rep(mult_rep>0); mult = [mult_new mult_rep]; u = [unew, urep]; ind = zeros (numel(kf)-numel(kc), 1); ind(cumsum([1 mult(:)'])) = 1; u = sort (u(cumsum(ind(1:end-1)))); endnurbs-1.4.3/inst/PaxHeaders.16248/nrb4surf.m0000644000000000000000000000013214030323741015306 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrb4surf.m0000644000175000017500000000404714030323741017412 0ustar00rvazquezrvazquez00000000000000function srf = nrb4surf(p11,p12,p21,p22) % % NRB4SURF: Constructs a NURBS bilinear surface. % % Calling Sequence: % % srf = nrb4surf(p11,p12,p21,p22) % % INPUT: % % p11 : Cartesian coordinate of the lhs bottom corner point. % % p12 : Cartesian coordinate of the rhs bottom corner point. % % p21 : Cartesian coordinate of the lhs top corner point. % % p22 : Cartesian coordinate of the rhs top corner point. % % OUTPUT: % % srf : NURBS bilinear surface, see nrbmak. % % Description: % % Constructs a bilinear surface defined by four coordinates. % % The position of the corner points % % ^ V direction % | % ---------------- % |p21 p22| % | | % | SRF | % | | % |p11 p12| % -------------------> U direction % % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin ~= 4 error('Four corner points must be defined'); end coefs = cat (1, zeros (3,2,2), ones (1,2,2)); coefs(1:length(p11),1,1) = p11(:); coefs(1:length(p12),2,1) = p12(:); coefs(1:length(p21),1,2) = p21(:); coefs(1:length(p22),2,2) = p22(:); knots = {[0 0 1 1] [0 0 1 1]}; srf = nrbmak(coefs, knots); end %!demo %! srf = nrb4surf([0.0 0.0 0.5],[1.0 0.0 -0.5],[0.0 1.0 -0.5],[1.0 1.0 0.5]); %! nrbplot(srf,[10,10]); %! title('Construction of a bilinear surface.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/nrbclamp.m0000644000000000000000000000013214030323741015337 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbclamp.m0000644000175000017500000000750014030323741017440 0ustar00rvazquezrvazquez00000000000000function ccrv = nrbclamp (crv, k, xdim) % NRBCLAMP: Compute the knot vector and control points of the clamped curve/surface. % % Calling Sequence: % % ccrv = nrbrclamp (crv) % ccrv = nrbrclamp (crv, k) % ccrv = nrbrclamp (crv, k, dim) % % INPUT: % % crv : unclamped NURBS curve or surface, see nrbmak. % k : continuity desired afterclamping (from -1 up to p-1, -1 by default) % dim : dimension in which to clamp (all by default). % % OUTPUT: % % ccrv: NURBS curve with clamped knot vector, see nrbmak % % Description: % % Clamps a curve or surface, using an open knot vector. Computes the new % knot vector and control points by knot insertion. % % Copyright (C) 2016 Monica Montardini, Filippo Remonato, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (iscell (crv.knots)) knt = crv.knots; curve = false; else knt = {crv.knots}; curve = true; end ndim = numel (knt); if (nargin < 2 || isempty(k)) k = (-1) * ones (1, ndim); end if (nargin < 3) xdim = 1:ndim; end %if (iscell (crv.knots)) if (numel(k) ~= ndim) k = k * ones(1, ndim); end new_knots = cell (1, ndim); for idim = xdim p = crv.order(idim) - 1; U = knt{idim}; kk = k(idim); if (kk >= p) warning ('Taking the maximum k allowed, degree - 1') kk = p - 1; end n_ins_start(idim) = max (0, p - sum(U==U(p+1)) - kk); n_ins_end(idim) = max (0, p - sum(U==U(end-p)) - kk); new_knots{idim} = [U(p+1)*ones(1,n_ins_start(idim)), U(end-p)*ones(1,n_ins_end(idim))]; end % Clamp, and remove unused coefficients and knots if (curve) ccrv = nrbkntins (crv, new_knots{1}); ccrv.coefs = ccrv.coefs(:, n_ins_start+1 : end - n_ins_end); ccrv.knots = ccrv.knots(n_ins_start+1 : end - n_ins_end); else ccrv = nrbkntins (crv, new_knots); for idim = 1:ndim ccrv.knots{idim} = ccrv.knots{idim}(n_ins_start(idim)+1 : end - n_ins_end(idim)); indices{idim} = n_ins_start(idim)+1 : ccrv.number(idim)-n_ins_end(idim); end ccrv.coefs = ccrv.coefs(1:4,indices{:}); end ccrv.number = ccrv.number - n_ins_start - n_ins_end; end %!test %! crv = nrbdegelev (nrbcirc (1, [], 0, pi/2), 2); %! crv = nrbunclamp (crv, 3); %! xx = linspace (0, 1, 20); %! crv1 = nrbclamp (crv); %! assert (crv1.knots, [0 0 0 0 0 1 1 1 1 1]) %! assert (nrbeval(crv, xx), nrbeval(crv1, xx), 1e-14) %! crv1 = nrbclamp (crv, 2); %! assert (crv1.knots, [-3 -2 -1 0 0 1 1 2 3 4]) %! assert (nrbeval(crv, xx), nrbeval(crv1, xx), 1e-14) %!test %! crv1 = nrbcirc(1,[],0,pi/4); %! crv2 = nrbcirc(2,[],0,pi/4); %! srf = nrbkntins (nrbdegelev (nrbruled(crv1, crv2), [3 2]), {0.25 []}); %! srf = nrbunclamp (srf, [4 2]); %! srf1 = nrbclamp (srf); %! xx = linspace(0,1,20); %! assert(srf1.knots, {[0 0 0 0 0 0 0.2500 1 1 1 1 1 1] [0 0 0 0 1 1 1 1]}) %! assert (nrbeval(srf, {xx xx}), nrbeval(srf1, {xx xx}), 1e-14); %! srf1 = nrbclamp (srf, [3 1]); %! assert (srf1.knots, {[-2 -1.75 -1 -0.75 0 0 0.25 1 1 1.25 2 2.25 3], [-2 -1 0 0 1 1 2 3]}) %! assert (nrbeval(srf, {xx xx}), nrbeval(srf1, {xx xx}), 1e-14); %! srf1 = nrbclamp (srf, [], 2); %! assert(srf1.knots, {[-2.75 -2 -1.75 -1 -0.75 0 0.25 1 1.25 2 2.25 3 3.25] [0 0 0 0 1 1 1 1]}) %! assert (nrbeval(srf, {xx xx}), nrbeval(srf1, {xx xx}), 1e-14); nurbs-1.4.3/inst/PaxHeaders.16248/nrbinverse.m0000644000000000000000000000013014030323741015714 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbinverse.m0000644000175000017500000001141014030323741020012 0ustar00rvazquezrvazquez00000000000000function [u, convergence] = nrbinverse (nrb, x, varargin) % % NRBINVERSE: compute parametric point starting from physical point by % inverting the NURBS map with a Newton scheme % % Calling Sequence: % % u = nrbinverse (nrb, x) % u = nrbinverse (nrb, x, options) % % INPUT: % % nrb - NURBS object % x - physical point % options - options in the FIELD/VALUE format. Possible choices: % 'u0' : starting point in the parametric domain for Newton % (Default = .5 * ones (ndim, 1)) % 'MaxIter' : maximum number of Newton iterations (Default = 10) % 'Display' : if true the some info are shown (Default = true) % 'TolX' : tolerance for the step size in Newton iterations % (Default = 1e-8) % 'TolFun' : tolerance for the residual in Newton iterations % (Default = 1e-8) % % OUTPUT: % % u - the parametric points corresponding to x % convergence - false if the method reached the maximum number % of iteration without converging, true otherwise % % Copyright (C) 2016 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % ndim = numel (nrb.number); % % Default options % persistent p; % p = inputParser (); % p.addParameter ('u0', .5*ones(ndim, 1), @(x) validateattributes (x, {'numeric'}, {'numel', ndim, '>=', 0, '<=', 1})); % p.addParameter ('MaxIter', 10, @(x) validateattributes (x, {'numeric'}, {'scalar'})); % p.addParameter ('Display', true, @(x) validateattributes (x, {'logical'}, {})); % p.addParameter ('TolX', 1e-8, @(x) validateattributes (x, {'numeric'}, {'scalar'})); % p.addParameter ('TolFun', 1e-8, @(x) validateattributes (x, {'numeric'}, {'scalar'})); % p.parse (varargin{:}); % options = p.Results; % Default options options = struct ('u0' , .5*ones (ndim, 1), ... 'MaxIter' , 10, ... 'Display' , true, ... 'TolX', 1e-8, ... 'TolFun', 1e-8); % Read the acceptable names optionNames = fieldnames (options); % Count arguments nargin = length (varargin); if (round (nargin/2) ~= nargin/2) error ('NRBINVERSE needs propertyName/propertyValue pairs'); end % Check options passed for pair = reshape (varargin, 2, []) if any (strcmp (pair{1}, optionNames)) options.(pair{1}) = pair{2}; else error('%s is not a recognized parameter name', pair{1}); end end % x as column vector x = x(:); % Define functions for Newton iteration f = @(U) nrbeval (nrb, num2cell (U)) - x; jac = @(U) nrbjacobian (nrb, num2cell (U)); % Newton cycle u_old = options.u0(:); if (iscell (nrb.knots)) first_knot = reshape (cellfun (@(x) x(1),nrb.knots), size(u_old)); last_knot = reshape (cellfun (@(x) x(end),nrb.knots), size(u_old)); else first_knot = nrb.knots(1); last_knot = nrb.knots(end); end convergence = false; for iter = 1:options.MaxIter u_new = u_old - jac (u_old) \ f (u_old); % Check if the point is outside the parametric domain u_new = max (u_new, first_knot); u_new = min (u_new, last_knot); % Error control if (norm (u_new - u_old) < options.TolX && norm (f (u_new)) < options.TolFun) if (options.Display) fprintf ('Newton scheme converged in %i iteration.\n', iter); end convergence = true; break; end u_old = u_new; end if (~convergence) fprintf ('Newton scheme reached the maximum number of iterations (%i) without converging.\n', options.MaxIter); end u = u_new; end function jac = nrbjacobian (nrb, u) ders = nrbderiv (nrb); [~, jac] = nrbdeval (nrb, ders, u); jac = [jac{:}]; end %!test %! nrb = nrb4surf ([0 0], [1 0], [2 3], [5 4]); %! p = nrbeval (nrb, {.25 .75}); %! u = nrbinverse (nrb, p, 'Display', false); %! assert (norm (u - [.25; .75]) < 1e-8); %! %!test %! nrb = nrb4surf ([0 0], [1 0], [2 3], [5 4]); %! nrb = nrbdegelev (nrbextrude (nrb, [0 2 1]), [3 3 3]); %! p = nrbeval (nrb, {.25 .75 .05}); %! u = nrbinverse (nrb, p, 'Display', false, 'TolX', 1e-12, 'TolFun', 1e-10); %! assert (norm (u - [.25; .75; .05]) < 1e-8); %! nurbs-1.4.3/inst/PaxHeaders.16248/nrbcoons.m0000644000000000000000000000013214030323741015364 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbcoons.m0000644000175000017500000001302114030323741017460 0ustar00rvazquezrvazquez00000000000000function srf = nrbcoons(u1, u2, v1, v2) % % NRBCOONS: Construction of a Coons patch. % % Calling Sequence: % % srf = nrbcoons(ucrv1, ucrv2, vcrv1, vcrv2) % % INPUT: % % ucrv1 : NURBS curve defining the bottom U direction boundary of % the constructed NURBS surface. % % ucrv2 : NURBS curve defining the top U direction boundary of % the constructed NURBS surface. % % vcrv1 : NURBS curve defining the bottom V direction boundary of % the constructed NURBS surface. % % vcrv2 : NURBS curve defining the top V direction boundary of % the constructed NURBS surface. % % OUTPUT: % % srf : Coons NURBS surface patch. % % Description: % % Construction of a bilinearly blended Coons surface patch from four NURBS % curves that define the boundary. % % The orientation of the four NURBS boundary curves. % % ^ V direction % | % | ucrv2 % ------->-------- % | | % | | % vcrv1 ^ Surface ^ vcrv2 % | | % | | % ------->-----------> U direction % ucrv1 % % % Examples: % % // Define four NURBS curves and construct a Coons surface patch. % pnts = [ 0.0 3.0 4.5 6.5 8.0 10.0; % 0.0 0.0 0.0 0.0 0.0 0.0; % 2.0 2.0 7.0 4.0 7.0 9.0]; % crv1 = nrbmak(pnts, [0 0 0 1/3 0.5 2/3 1 1 1]); % % pnts= [ 0.0 3.0 5.0 8.0 10.0; % 10.0 10.0 10.0 10.0 10.0; % 3.0 5.0 8.0 6.0 10.0]; % crv2 = nrbmak(pnts, [0 0 0 1/3 2/3 1 1 1]); % % pnts= [ 0.0 0.0 0.0 0.0; % 0.0 3.0 8.0 10.0; % 2.0 0.0 5.0 3.0]; % crv3 = nrbmak(pnts, [0 0 0 0.5 1 1 1]); % % pnts= [ 10.0 10.0 10.0 10.0 10.0; % 0.0 3.0 5.0 8.0 10.0; % 9.0 7.0 7.0 10.0 10.0]; % crv4 = nrbmak(pnts, [0 0 0 0.25 0.75 1 1 1]); % % srf = nrbcoons(crv1, crv2, crv3, crv4); % nrbplot(srf,[20 20],220,45); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin ~= 4 error('Incorrect number of input arguments'); end if (max (abs (nrbeval (u1, u1.knots(1)) - nrbeval (v1, v1.knots(1)))) > 1e-10 || ... max (abs (nrbeval (u1, u1.knots(end)) - nrbeval (v2, v2.knots(1)))) > 1e-10 || ... max (abs (nrbeval (u2, u2.knots(1)) - nrbeval (v1, v1.knots(end)))) > 1e-10 || ... max (abs (nrbeval (u2, u2.knots(end)) - nrbeval (v2, v2.knots(end)))) > 1e-10) error ('The four curves do not define a closed boundary') end r1 = nrbruled(u1, u2); r2 = nrbtransp(nrbruled(v1, v2)); t = nrb4surf(u1.coefs(:,1), u1.coefs(:,end), u2.coefs(:,1), u2.coefs(:,end)); % raise all surfaces to a common degree du = max([r1.order(1), r2.order(1), t.order(1)]); dv = max([r1.order(2), r2.order(2), t.order(2)]); r1 = nrbdegelev(r1, [du - r1.order(1), dv - r1.order(2)]); r2 = nrbdegelev(r2, [du - r2.order(1), dv - r2.order(2)]); t = nrbdegelev(t, [du - t.order(1), dv - t.order(2)]); % merge the knot vectors, to obtain a common knot vector % U knots k1 = r1.knots{1}; k2 = r2.knots{1}; k3 = t.knots{1}; k = unique([k1 k2 k3]); n = length(k); kua = []; kub = []; kuc = []; for i = 1:n i1 = length(find(k1 == k(i))); i2 = length(find(k2 == k(i))); i3 = length(find(k3 == k(i))); m = max([i1, i2, i3]); kua = [kua k(i)*ones(1,m-i1)]; kub = [kub k(i)*ones(1,m-i2)]; kuc = [kuc k(i)*ones(1,m-i3)]; end % V knots k1 = r1.knots{2}; k2 = r2.knots{2}; k3 = t.knots{2}; k = unique([k1 k2 k3]); n = length(k); kva = []; kvb = []; kvc = []; for i = 1:n i1 = length(find(k1 == k(i))); i2 = length(find(k2 == k(i))); i3 = length(find(k3 == k(i))); m = max([i1, i2, i3]); kva = [kva k(i)*ones(1,m-i1)]; kvb = [kvb k(i)*ones(1,m-i2)]; kvc = [kvc k(i)*ones(1,m-i3)]; end r1 = nrbkntins(r1, {kua, kva}); r2 = nrbkntins(r2, {kub, kvb}); t = nrbkntins(t, {kuc, kvc}); % combine coefficient to construct Coons surface coefs(1,:,:) = r1.coefs(1,:,:) + r2.coefs(1,:,:) - t.coefs(1,:,:); coefs(2,:,:) = r1.coefs(2,:,:) + r2.coefs(2,:,:) - t.coefs(2,:,:); coefs(3,:,:) = r1.coefs(3,:,:) + r2.coefs(3,:,:) - t.coefs(3,:,:); coefs(4,:,:) = r1.coefs(4,:,:) + r2.coefs(4,:,:) - t.coefs(4,:,:); srf = nrbmak(coefs, r1.knots); end %!demo %! pnts = [ 0.0 3.0 4.5 6.5 8.0 10.0; %! 0.0 0.0 0.0 0.0 0.0 0.0; %! 2.0 2.0 7.0 4.0 7.0 9.0]; %! crv1 = nrbmak(pnts, [0 0 0 1/3 0.5 2/3 1 1 1]); %! %! pnts= [ 0.0 3.0 5.0 8.0 10.0; %! 10.0 10.0 10.0 10.0 10.0; %! 3.0 5.0 8.0 6.0 10.0]; %! crv2 = nrbmak(pnts, [0 0 0 1/3 2/3 1 1 1]); %! %! pnts= [ 0.0 0.0 0.0 0.0; %! 0.0 3.0 8.0 10.0; %! 2.0 0.0 5.0 3.0]; %! crv3 = nrbmak(pnts, [0 0 0 0.5 1 1 1]); %! %! pnts= [ 10.0 10.0 10.0 10.0 10.0; %! 0.0 3.0 5.0 8.0 10.0; %! 9.0 7.0 7.0 10.0 10.0]; %! crv4 = nrbmak(pnts, [0 0 0 0.25 0.75 1 1 1]); %! %! srf = nrbcoons(crv1, crv2, crv3, crv4); %! %! nrbplot(srf,[20 20]); %! title('Construction of a bilinearly blended Coons surface.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/nrbruled.m0000644000000000000000000000013214030323741015356 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbruled.m0000644000175000017500000000657114030323741017466 0ustar00rvazquezrvazquez00000000000000function srf = nrbruled (crv1, crv2) % NRBRULED: Construct a ruled surface between two NURBS curves, or a ruled volume between two NURBS surfaces. % % Calling Sequence: % % srf = nrbruled(crv1, crv2) % % INPUT: % % crv1 : First NURBS curve (or surface), see nrbmak. % % crv2 : Second NURBS curve (or surface), see nrbmak. % % OUTPUT: % % srf : Ruled NURBS surface (or volume). % % Description: % % Constructs a ruled surface between two NURBS curves. The ruled surface is % ruled along the V (or W) direction. % % Examples: % % Construct a ruled surface between a semicircle and a straight line. % % cir = nrbcirc(1,[0 0 0],0,pi); % line = nrbline([-1 0.5 1],[1 0.5 1]); % srf = nrbruled(cir,line); % nrbplot(srf,[20 20]); % % Copyright (C) 2000 Mark Spink % Copyright (C) 2018 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . ndim = numel (crv1.order); if (ndim ~= numel (crv2.order)) error ('Both NURBS must be of the same kind (both curves or both surfaces)') elseif (ndim == 3 || numel (crv2.order) == 3) error ('The input cannot be a NURBS volume') end % ensure both curves have a common degree d = max ([crv1.order, crv2.order]); crv1 = nrbdegelev (crv1, d - crv1.order); crv2 = nrbdegelev (crv2, d - crv2.order); knt1 = crv1.knots; knt2 = crv2.knots; if (~iscell (knt1)) knt1 = {knt1}; end if (~iscell (knt2)) knt2 = {knt2}; end % merge the knot vectors, to obtain a common knot vector ka = cell (1, ndim); kb = cell (1, ndim); for idim = 1:ndim k1 = knt1{idim}; k2 = knt2{idim}; ku = unique ([k1 k2]); n = length (ku); ka{idim} = []; kb{idim} = []; for i = 1:n i1 = length (find (k1 == ku(i))); i2 = length (find (k2 == ku(i))); m = max (i1, i2); ka{idim} = [ka{idim} ku(i)*ones(1,m-i1)]; kb{idim} = [kb{idim} ku(i)*ones(1,m-i2)]; end end if (ndim == 1) crv1 = nrbkntins (crv1, ka{1}); crv2 = nrbkntins (crv2, kb{1}); knots = {crv1.knots, [0 0 1 1]}; coefs(:,:,1) = crv1.coefs; coefs(:,:,2) = crv2.coefs; else crv1 = nrbkntins (crv1, ka); crv2 = nrbkntins (crv2, kb); knots = {crv1.knots{:}, [0 0 1 1]}; coefs(:,:,:,1) = crv1.coefs; coefs(:,:,:,2) = crv2.coefs; end srf = nrbmak (coefs, knots); end %!demo %! pnts = [0.5 1.5 4.5 3.0 7.5 6.0 8.5; %! 3.0 5.5 5.5 1.5 1.5 4.0 4.5; %! 0.0 0.0 0.0 0.0 0.0 0.0 0.0]; %! crv1 = nrbmak (pnts,[0 0 0 1/4 1/2 3/4 3/4 1 1 1]); %! crv2 = nrbtform (nrbcirc (4,[4.5;0],pi,0.0),vectrans([0.0 4.0 -4.0])); %! srf = nrbruled (crv1,crv2); %! nrbplot (srf,[40 20]); %! title ('Ruled surface construction from two NURBS curves.'); %! hold off %!demo %! srf1 = nrbtestsrf; %! srf2 = nrb4surf([0 0 -1], [10 0 -1], [0 10 -1], [10 10 -1]); %! vol = nrbruled (srf1, srf2); %! nrbkntplot (vol); %! title ('Ruled volume construction from two NURBS surfaces')nurbs-1.4.3/inst/PaxHeaders.16248/nrbmak.m0000644000000000000000000000013014030323741015011 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbmak.m0000644000175000017500000002223514030323741017116 0ustar00rvazquezrvazquez00000000000000function nurbs = nrbmak(coefs,knots,normalize) % % NRBMAK: Construct the NURBS structure given the control points % and the knots. % % Calling Sequence: % % nurbs = nrbmak(cntrl,knots,[normalize]); % % INPUT: % % cntrl : Control points, these can be either Cartesian or % homogeneous coordinates. % % For a curve the control points are represented by a % matrix of size (dim,nu), for a surface a multidimensional % array of size (dim,nu,nv), for a volume a multidimensional array % of size (dim,nu,nv,nw). Where nu is number of points along % the parametric U direction, nv the number of points along % the V direction and nw the number of points along the W direction. % dim is the dimension. Valid options % are % 2 .... (x,y) 2D Cartesian coordinates % 3 .... (x,y,z) 3D Cartesian coordinates % 4 .... (wx,wy,wz,w) 4D homogeneous coordinates % % knots : Non-decreasing knot sequence spanning the interval % [0.0,1.0]. It's assumed that the geometric entities % are clamped to the start and end control points by knot % multiplicities equal to the spline order (open knot vector). % For curve knots form a vector and for surfaces (volumes) % the knots are stored by two (three) vectors for U and V (and W) % in a cell structure {uknots vknots} ({uknots vknots wknots}). % % normalize: if true, the knot vector is normalized to the interval [0, 1]. % Default value is false. % % OUTPUT: % % nurbs : Data structure for representing a NURBS entity % % NURBS Structure: % % Both curves and surfaces are represented by a structure that is % compatible with the Spline Toolbox from Mathworks % % nurbs.form .... Type name 'B-NURBS' % nurbs.dim .... Dimension of the control points % nurbs.number .... Number of Control points % nurbs.coefs .... Control Points % nurbs.order .... Order of the spline % nurbs.knots .... Knot sequence % % Note: the control points are always converted and stored within the % NURBS structure as 4D homogeneous coordinates. A curve is always stored % along the U direction, and the vknots element is an empty matrix. For % a surface the spline order is a vector [du,dv] containing the order % along the U and V directions respectively. For a volume the order is % a vector [du dv dw]. Recall that order = degree + 1. % % Description: % % This function is used as a convenient means of constructing the NURBS % data structure. Many of the other functions in the toolbox rely on the % NURBS structure been correctly defined as shown above. The nrbmak not % only constructs the proper structure, but also checks for consistency. % The user is still free to build his own structure, in fact a few % functions in the toolbox do this for convenience. % % Examples: % % Construct a 2D line from (0.0,0.0) to (1.5,3.0). % For a straight line a spline of order 2 is required. % Note that the knot sequence has a multiplicity of 2 at the % start (0.0,0.0) and end (1.0 1.0) in order to clamp the ends. % % line = nrbmak([0.0 1.5; 0.0 3.0],[0.0 0.0 1.0 1.0]); % nrbplot(line, 2); % % Construct a surface in the x-y plane i.e % % ^ (0.0,1.0) ------------ (1.0,1.0) % | | | % | V | | % | | Surface | % | | | % | | | % | (0.0,0.0) ------------ (1.0,0.0) % | % |------------------------------------> % U % % coefs = cat(3,[0 0; 0 1],[1 1; 0 1]); % knots = {[0 0 1 1] [0 0 1 1]} % plane = nrbmak(coefs,knots); % nrbplot(plane, [2 2]); % % Copyright (C) 2000 Mark Spink, 2010-2018 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 3) normalize = false; end nurbs = struct ('form', 'B-NURBS', 'dim', 4, 'number', [], 'coefs', [], ... 'knots', [], 'order', []); nurbs.form = 'B-NURBS'; nurbs.dim = 4; np = size(coefs); dim = np(1); if (iscell(knots) && (size(knots,2) > 1)) if size(knots,2) == 3 if (numel(np) == 3) np(4) = 1; elseif (numel(np)==2) np(3:4) = 1; end % constructing a volume nurbs.number = np(2:4); if (dim < 4) nurbs.coefs = repmat([0.0 0.0 0.0 1.0]',[1 np(2:4)]); nurbs.coefs(1:dim,:,:,:) = coefs; else nurbs.coefs = coefs; end uorder = size(knots{1},2)-np(2); vorder = size(knots{2},2)-np(3); worder = size(knots{3},2)-np(4); if (normalize) uknots = (uknots-uknots(uorder))/(uknots(end-uorder+1)-uknots(uorder)); vknots = (vknots-vknots(vorder))/(vknots(end-vorder+1)-vknots(vorder)); wknots = (wknots-wknots(worder))/(wknots(end-worder+1)-wknots(worder)); else uknots = sort(knots{1}); vknots = sort(knots{2}); wknots = sort(knots{3}); end nurbs.knots = {uknots vknots wknots}; nurbs.order = [uorder vorder worder]; elseif size(knots,2) == 2 if (numel(np)==2); np(3) = 1; end % constructing a surface nurbs.number = np(2:3); if (dim < 4) nurbs.coefs = repmat([0.0 0.0 0.0 1.0]',[1 np(2:3)]); nurbs.coefs(1:dim,:,:) = coefs; else nurbs.coefs = coefs; end uorder = size(knots{1},2)-np(2); vorder = size(knots{2},2)-np(3); if (normalize) uknots = (uknots-uknots(uorder))/(uknots(end-uorder+1)-uknots(uorder)); vknots = (vknots-vknots(vorder))/(vknots(end-vorder+1)-vknots(vorder)); else uknots = sort(knots{1}); vknots = sort(knots{2}); end nurbs.knots = {uknots vknots}; nurbs.order = [uorder vorder]; end elseif(~iscell(knots) || (iscell(knots)&&(size(knots,2) == 1)) ) if iscell(knots) knots = knots{1}; end % constructing a curve nurbs.number = np(2); if (dim < 4) nurbs.coefs = repmat([0.0 0.0 0.0 1.0]',[1 np(2)]); nurbs.coefs(1:dim,:) = coefs; else nurbs.coefs = coefs; end order = size (knots,2) - np(2); nurbs.order = order; knots = sort(knots); if (normalize) nurbs.knots = (knots-knots(order))/(knots(end-order+1)-knots(order)); else nurbs.knots = knots; end end if (any(nurbs.order < 1)) error ('The knot sequence is too short, as it gives order<1 (degree<0)') elseif (any (nurbs.number-nurbs.order < 0)) error ('Not enough control points, or too long not sequence (nurbs.number < nurbs.order).') end end %!demo %! pnts = [0.5 1.5 4.5 3.0 7.5 6.0 8.5; %! 3.0 5.5 5.5 1.5 1.5 4.0 4.5; %! 0.0 0.0 0.0 0.0 0.0 0.0 0.0]; %! crv = nrbmak(pnts,[0 0 0 1/4 1/2 3/4 3/4 1 1 1]); %! nrbplot(crv,100) %! title('Test curve') %! hold off %!demo %! pnts = [0.5 1.5 4.5 3.0 7.5 6.0 8.5; %! 3.0 5.5 5.5 1.5 1.5 4.0 4.5; %! 0.0 0.0 0.0 0.0 0.0 0.0 0.0]; %! crv = nrbmak(pnts,[0 0 0 0.1 1/2 3/4 3/4 1 1 1]); %! nrbplot(crv,100) %! title('Test curve with a slight variation of the knot vector') %! hold off %!demo %! pnts = zeros(3,5,5); %! pnts(:,:,1) = [ 0.0 3.0 5.0 8.0 10.0; %! 0.0 0.0 0.0 0.0 0.0; %! 2.0 2.0 7.0 7.0 8.0]; %! pnts(:,:,2) = [ 0.0 3.0 5.0 8.0 10.0; %! 3.0 3.0 3.0 3.0 3.0; %! 0.0 0.0 5.0 5.0 7.0]; %! pnts(:,:,3) = [ 0.0 3.0 5.0 8.0 10.0; %! 5.0 5.0 5.0 5.0 5.0; %! 0.0 0.0 5.0 5.0 7.0]; %! pnts(:,:,4) = [ 0.0 3.0 5.0 8.0 10.0; %! 8.0 8.0 8.0 8.0 8.0; %! 5.0 5.0 8.0 8.0 10.0]; %! pnts(:,:,5) = [ 0.0 3.0 5.0 8.0 10.0; %! 10.0 10.0 10.0 10.0 10.0; %! 5.0 5.0 8.0 8.0 10.0]; %! %! knots{1} = [0 0 0 1/3 2/3 1 1 1]; %! knots{2} = [0 0 0 1/3 2/3 1 1 1]; %! %! srf = nrbmak(pnts,knots); %! nrbplot(srf,[20 20]) %! title('Test surface') %! hold off %!demo %! coefs =[ 6.0 0.0 6.0 1; %! -5.5 0.5 5.5 1; %! -5.0 1.0 -5.0 1; %! 4.5 1.5 -4.5 1; %! 4.0 2.0 4.0 1; %! -3.5 2.5 3.5 1; %! -3.0 3.0 -3.0 1; %! 2.5 3.5 -2.5 1; %! 2.0 4.0 2.0 1; %! -1.5 4.5 1.5 1; %! -1.0 5.0 -1.0 1; %! 0.5 5.5 -0.5 1; %! 0.0 6.0 0.0 1]'; %! knots = [0 0 0 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 1 1 1]; %! %! crv = nrbmak(coefs,knots); %! nrbplot(crv,100); %! grid on; %! title('3D helical curve.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/kntrefine.m0000644000000000000000000000013214030323741015526 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/kntrefine.m0000644000175000017500000001372414030323741017634 0ustar00rvazquezrvazquez00000000000000% KNTREFINE: Refine a given knot vector by dividing each interval uniformly, % maintaining the continuity in previously existing knots. % % [rknots] = kntrefine (knots, n_sub, degree, regularity) % [rknots, zeta] = kntrefine (knots, n_sub, degree, regularity) % [rknots, zeta, new_knots] = kntrefine (knots, n_sub, degree, regularity) % % INPUT: % % knots: initial knot vector. % n_sub: number of new knots to be added in each interval. % degree: polynomial degree of the refined knot vector % regularity: maximum global regularity % % OUTPUT: % % rknots: refined knot vector % zeta: refined knot vector without repetitions % new_knots: new knots, to apply the knot insertion % % The regularity at the new inserted knots is the one given by the user. % At previously existing knots, the regularity is the minimum % between the previous regularity, and the one given by the user. % This ensures optimal convergence rates in the context of IGA. % % Copyright (C) 2010 Carlo de Falco, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . function varargout = kntrefine (knots, n_sub, degree, regularity) if (iscell(knots)) if (numel(n_sub)~=numel(degree) || numel(n_sub)~=numel(regularity) || ... numel(n_sub)~=numel(knots)) error('kntrefine: n_sub, degree and regularity must have the same length as the number of knot vectors') end aux_knots = knots; else if (numel(n_sub)~=numel(degree) || numel(n_sub)~=numel(regularity) || ... numel(n_sub)~=1) error('kntrefine: n_sub, degree and regularity must have the same length as the number of knot vectors') end aux_knots = {knots}; end if (nargout == 3) for idim = 1:numel(n_sub) if (degree(idim)+1 ~= sum (aux_knots{idim}==aux_knots{idim}(1))) error ('kntrefine: new_knots is only computed when the degree is maintained'); end end for idim = 1:numel(n_sub) min_mult = degree(idim) - regularity(idim); z = unique (aux_knots{idim}); nz = numel (z); deg = sum (aux_knots{idim} == z(1)) - 1; rknots{idim} = z(ones(1, degree(idim)+1)); new_knots{idim} = []; for ik = 2:nz insk = linspace (z(ik-1), z(ik), n_sub(idim) + 2); insk = vec (repmat (insk(2:end-1), min_mult, 1))'; old_mult = sum (aux_knots{idim} == z(ik)); mult = max (min_mult, degree(idim) - deg + old_mult); rknots{idim} = [rknots{idim}, insk, z(ik*ones(1, mult))]; new_knots{idim} = [new_knots{idim}, insk, z(ik*ones(1, mult-old_mult))]; end zeta{idim} = unique (rknots{idim}); end if (~iscell(knots)) rknots = rknots{1}; zeta = zeta{1}; new_knots = new_knots{1}; end varargout{1} = rknots; varargout{2} = zeta; varargout{3} = new_knots; else for idim = 1:numel(n_sub) min_mult = degree(idim) - regularity(idim); z = unique (aux_knots{idim}); nz = numel (z); deg = sum (aux_knots{idim} == z(1)) - 1; rknots{idim} = z(ones(1, degree(idim)+1)); for ik = 2:nz insk = linspace (z(ik-1), z(ik), n_sub(idim) + 2); insk = vec (repmat (insk(2:end-1), min_mult, 1))'; old_mult = sum (aux_knots{idim} == z(ik)); mult = max (min_mult, degree(idim) - deg + old_mult); rknots{idim} = [rknots{idim}, insk, z(ik*ones(1, mult))]; end zeta{idim} = unique (rknots{idim}); end if (~iscell(knots)) rknots = rknots{1}; zeta = zeta{1}; end varargout{1} = rknots; if (nargout == 2) varargout{2} = zeta; end end end function v = vec (in) v = in(:); end %!shared nrbs %!test %! knots = {[0 0 1 1] [0 0 0 1 1 1]}; %! coefs(1,:,:) = [1 sqrt(2)/2 0; 2 sqrt(2) 0]; %! coefs(2,:,:) = [0 sqrt(2)/2 1; 0 sqrt(2) 2]; %! coefs(4,:,:) = [1 sqrt(2)/2 1; 1 sqrt(2)/2 1]; %! nrbs = nrbmak (coefs, knots); %! nrbs = nrbkntins (nrbs, {[] [0.5 0.6 0.6]}); %! nrbs = nrbdegelev (nrbs, [0 1]); %! nrbs = nrbkntins (nrbs, {[] [0.4]}); %! rknots = kntrefine (nrbs.knots, [1 1], [1 1], [0 0]); %! assert (rknots{1} == [0 0 0.5 1 1]); %! assert (rknots{2} == [0 0 0.2 0.4 0.45 0.5 0.55 0.6 0.8 1 1]); %! %!test %! rknots = kntrefine (nrbs.knots, [1 1], [3 3], [0 0]); %! assert (rknots{1}, [0 0 0 0 0.5 0.5 0.5 1 1 1 1]); %! assert (rknots{2}, [0 0 0 0 0.2 0.2 0.2 0.4 0.4 0.4 0.45 0.45 0.45 0.5 0.5 0.5 0.55 0.55 0.55 0.6 0.6 0.6 0.8 0.8 0.8 1 1 1 1]); %! %!test %! rknots = kntrefine (nrbs.knots, [1 1], [3 3], [2 2]); %! assert (rknots{1}, [0 0 0 0 0.5 1 1 1 1]); %! assert (rknots{2}, [0 0 0 0 0.2 0.4 0.45 0.5 0.5 0.55 0.6 0.6 0.6 0.8 1 1 1 1]); %! %!test %! rknots = kntrefine (nrbs.knots, [1 1], [4 4], [0 0]); %! assert (rknots{1}, [0 0 0 0 0 0.5 0.5 0.5 0.5 1 1 1 1 1]); %! assert (rknots{2}, [0 0 0 0 0 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.45 0.45 0.45 0.45 0.5 0.5 0.5 0.5 0.55 0.55 0.55 0.55 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 1 1 1 1 1]); %! %!test %! rknots = kntrefine (nrbs.knots, [1 1], [4 4], [3 3]); %! assert (rknots{1}, [0 0 0 0 0 0.5 1 1 1 1 1]); %! assert (rknots{2}, [0 0 0 0 0 0.2 0.4 0.4 0.45 0.5 0.5 0.5 0.55 0.6 0.6 0.6 0.6 0.8 1 1 1 1 1]); %! %!test %! knots = [0 0 0 0 0.4 0.5 0.5 0.6 0.6 0.6 1 1 1 1]; %! rknots = kntrefine (knots, 1, 4, 3); %! assert (rknots, [0 0 0 0 0 0.2 0.4 0.4 0.45 0.5 0.5 0.5 0.55 0.6 0.6 0.6 0.6 0.8 1 1 1 1 1]); nurbs-1.4.3/inst/PaxHeaders.16248/nrbbasisfunder.m0000644000000000000000000000013214030323741016550 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbbasisfunder.m0000644000175000017500000001534114030323741020653 0ustar00rvazquezrvazquez00000000000000function varargout = nrbbasisfunder (points, nrb) % NRBBASISFUNDER: NURBS basis functions derivatives % % Calling Sequence: % % Bu = nrbbasisfunder (u, crv) % [Bu, N] = nrbbasisfunder (u, crv) % [Bu, Bv] = nrbbasisfunder ({u, v}, srf) % [Bu, Bv, N] = nrbbasisfunder ({u, v}, srf) % [Bu, Bv, N] = nrbbasisfunder (pts, srf) % [Bu, Bv, Bw, N] = nrbbasisfunder ({u, v, w}, vol) % [Bu, Bv, Bw, N] = nrbbasisfunder (pts, vol) % % INPUT: % % u - parametric coordinates along u direction % v - parametric coordinates along v direction % w - parametric coordinates along w direction % pts - array of scattered points in parametric domain, array size: (ndim,num_points) % crv - NURBS curve % srf - NURBS surface % vol - NURBS volume % % If the parametric coordinates are given in a cell-array, the values % are computed in a tensor product set of points % % OUTPUT: % % Bu - Basis functions derivatives WRT direction u % size(Bu)=[npts, prod(nrb.order)] % % Bv - Basis functions derivatives WRT direction v % size(Bv) == size(Bu) % % Bw - Basis functions derivatives WRT direction w % size(Bw) == size(Bu) % % N - Indices of the basis functions that are nonvanishing at each % point. size(N) == size(Bu) % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2016 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if ( (nargin<2) ... || (nargout>4) ... || (~isstruct(nrb)) ... || (iscell(points) && ~iscell(nrb.knots)) ... || (~iscell(points) && iscell(nrb.knots) && (size(points,1)~=numel(nrb.number))) ... || (~iscell(nrb.knots) && (nargout>2)) ... || iscell(points) && numel(points) ~= numel(nrb.number) ... ) error('Incorrect input arguments in nrbbasisfunder'); end if (~iscell (nrb.knots)) %% NURBS curve knt = {nrb.knots}; else %% NURBS surface or volume knt = nrb.knots; end ndim = numel (nrb.number); w = reshape (nrb.coefs(4,:), [nrb.number 1]); for idim = 1:ndim if (iscell (points)) pts_dim = points{idim}; else pts_dim = points(idim,:); end sp{idim} = findspan (nrb.number(idim)-1, nrb.order(idim)-1, pts_dim, knt{idim}); Nprime = basisfunder (sp{idim}, nrb.order(idim)-1, pts_dim, knt{idim}, 1); N{idim} = reshape (Nprime(:,1,:), numel(pts_dim), nrb.order(idim)); Nder{idim} = reshape (Nprime(:,2,:), numel(pts_dim), nrb.order(idim)); num{idim} = numbasisfun (sp{idim}, pts_dim, nrb.order(idim)-1, knt{idim}) + 1; end if (ndim == 1) B1 = reshape (w(num{1}), size(N{1})) .* N{1}; W = sum (B1, 2); B2 = reshape (w(num{1}), size(N{1})) .* Nder{1}; Wder = sum (B2, 2); B2 = bsxfun (@(x,y) x./y, B2, W); B1 = bsxfun (@(x,y) x.*y, B1, Wder./W.^2); B = B2 - B1; varargout{1} = B; varargout{2} = num{1}; else id = nrbnumbasisfun (points, nrb); if (iscell (points)) npts_dim = cellfun (@numel, points); npts = prod (npts_dim); val_aux = 1; val_ders = repmat ({1}, ndim, 1); for idim = 1:ndim val_aux = kron (N{idim}, val_aux); for jdim = 1:ndim if (idim == jdim) val_ders{idim} = kron(Nder{jdim}, val_ders{idim}); else val_ders{idim} = kron(N{jdim}, val_ders{idim}); end end end B1 = w(id) .* reshape (val_aux, npts, prod(nrb.order)); W = sum (B1, 2); for idim = 1:ndim B2 = w(id) .* reshape (val_ders{idim}, npts, prod(nrb.order)); Wder = sum (B2, 2); varargout{idim} = bsxfun (@(x,y) x./y, B2, W) - bsxfun (@(x,y) x.*y, B1, Wder ./ W.^2); end else npts = numel (points(1,:)); B = zeros (npts, prod(nrb.order)); Bder = repmat ({B}, ndim, 1); for ipt = 1:npts val_aux = 1; val_ders = repmat ({1}, ndim, 1); for idim = 1:ndim val_aux = reshape (val_aux.' * N{idim}(ipt,:), 1, []); % val_aux = kron (N{idim}(ipt,:), val_aux); for jdim = 1:ndim if (idim == jdim) val_ders{idim} = reshape (val_ders{idim}.' * Nder{jdim}(ipt,:), 1, []); else val_ders{idim} = reshape (val_ders{idim}.' * N{jdim}(ipt,:), 1, []); end end end wval = reshape (w(id(ipt,:)), size(val_aux)); val_aux = val_aux .* wval; W = sum (val_aux); for idim = 1:ndim val_ders{idim} = val_ders{idim} .* wval; Wder = sum (val_ders{idim}); Bder{idim}(ipt,:) = bsxfun (@(x,y) x./y, val_ders{idim}, W) - bsxfun (@(x,y) x.*y, val_aux, Wder ./ W.^2); end end varargout(1:ndim) = Bder(1:ndim); end if (nargout > ndim) varargout{ndim+1} = id; end end end %!demo %! U = [0 0 0 0 1 1 1 1]; %! x = [0 1/3 2/3 1] ; %! y = [0 0 0 0]; %! w = [1 1 1 1]; %! nrb = nrbmak ([x;y;y;w], U); %! u = linspace(0, 1, 30); %! [Bu, id] = nrbbasisfunder (u, nrb); %! plot(u, Bu) %! title('Derivatives of the cubic Bernstein polynomials') %! hold off %!test %! U = [0 0 0 0 1 1 1 1]; %! x = [0 1/3 2/3 1] ; %! y = [0 0 0 0]; %! w = rand(1,4); %! nrb = nrbmak ([x;y;y;w], U); %! u = linspace(0, 1, 30); %! [Bu, id] = nrbbasisfunder (u, nrb); %! #plot(u, Bu) %! assert (sum(Bu, 2), zeros(numel(u), 1), 1e-10), %!test %! U = [0 0 0 0 1/2 1 1 1 1]; %! x = [0 1/4 1/2 3/4 1] ; %! y = [0 0 0 0 0]; %! w = rand(1,5); %! nrb = nrbmak ([x;y;y;w], U); %! u = linspace(0, 1, 300); %! [Bu, id] = nrbbasisfunder (u, nrb); %! assert (sum(Bu, 2), zeros(numel(u), 1), 1e-10) %!test %! p = 2; q = 3; m = 4; n = 5; %! Lx = 1; Ly = 1; %! nrb = nrb4surf ([0 0], [1 0], [0 1], [1 1]); %! nrb = nrbdegelev (nrb, [p-1, q-1]); %! aux1 = linspace(0,1,m); aux2 = linspace(0,1,n); %! nrb = nrbkntins (nrb, {aux1(2:end-1), aux2(2:end-1)}); %! nrb.coefs (4,:,:) = nrb.coefs(4,:,:) + rand (size (nrb.coefs (4,:,:))); %! [Bu, Bv, N] = nrbbasisfunder ({rand(1, 20), rand(1, 20)}, nrb); %! #plot3(squeeze(u(1,:,:)), squeeze(u(2,:,:)), reshape(Bu(:,10), 20, 20),'o') %! assert (sum (Bu, 2), zeros(20^2, 1), 1e-10) nurbs-1.4.3/inst/PaxHeaders.16248/numbasisfun.m0000644000000000000000000000013214030323741016073 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/numbasisfun.m0000644000175000017500000000276614030323741020205 0ustar00rvazquezrvazquez00000000000000function B = numbasisfun (iv, uv, p, U) % NUMBASISFUN: List non-zero Basis functions for B-Spline in a given knot-span % % Calling Sequence: % % N = numbasisfun(i,u,p,U) % % INPUT: % % i - knot span ( from FindSpan() ) % u - parametric point % p - spline degree % U - knot sequence % % OUTPUT: % % N - Basis functions (numel(u)x(p+1)) % % See also: % % basisfun, basisfunder % % Copyright (C) 2009 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . B = bsxfun (@(a, b) a+b,iv-p, (0:p).').'; end %!test %! n = 3; %! U = [0 0 0 1/2 1 1 1]; %! p = 2; %! u = linspace (0, 1, 10); %! s = findspan (n, p, u, U); %! Bref = [0 0 0 0 0 1 1 1 1 1; ... %! 1 1 1 1 1 2 2 2 2 2; ... %! 2 2 2 2 2 3 3 3 3 3].'; %! B = numbasisfun (s, u, p, U); %! assert (B, Bref)nurbs-1.4.3/inst/PaxHeaders.16248/kntbrkdegreg.m0000644000000000000000000000013214030323741016212 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/kntbrkdegreg.m0000644000175000017500000000644314030323741020320 0ustar00rvazquezrvazquez00000000000000% KNTBRKDEGREG: Construct an open knot vector by giving the sequence of % knots, the degree and the regularity. % % knots = kntbrkdegreg (breaks, degree) % knots = kntbrkdegreg (breaks, degree, regularity) % % INPUT: % % breaks: sequence of knots. % degree: polynomial degree of the splines associated to the knot vector. % regularity: splines regularity. % % OUTPUT: % % knots: knot vector. % % If REGULARITY has as many entries as BREAKS, or as the number of interior % knots, a different regularity will be assigned to each knot. If % REGULARITY is not present, it will be taken equal to DEGREE-1. % % Copyright (C) 2010 Carlo de Falco, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . function knots = kntbrkdegreg (breaks, degree, reg) if (iscell (breaks)) if (nargin == 2) reg = degree - 1; end if (numel(breaks)~=numel(degree) || numel(breaks)~=numel(reg)) error('kntbrkdegreg: degree and regularity must have the same length as the number of knot vectors') end degree = num2cell (degree); if (~iscell (reg)) reg = num2cell (reg); end knots = cellfun (@do_kntbrkdegreg, breaks, degree, reg, 'uniformoutput', false); else if (nargin == 2) reg = degree - 1; end knots = do_kntbrkdegreg (breaks, degree, reg); end end function knots = do_kntbrkdegreg (breaks, degree, reg) if (numel (breaks) < 2) error ('kntbrkdegreg: the knots sequence should contain at least two points') end if (numel (reg) == 1) mults = [-1, (degree (ones (1, numel (breaks) - 2)) - reg), -1]; elseif (numel (reg) == numel (breaks)) mults = degree - reg; elseif (numel (reg) == numel (breaks) - 2) mults = [-1 degree-reg -1]; else error('kntbrkdegreg: the length of mult should be equal to one or the number of knots') end if (any (reg < -1)) warning ('kntbrkdegreg: for some knots the regularity is lower than -1') elseif (any (reg > degree-1)) error('kntbrkdegreg: the regularity should be lower than the degree') end knots = kntbrkdegmult (breaks, degree, mults); end %!test %! breaks = [0 1 2 3 4]; %! degree = 3; %! knots = kntbrkdegreg (breaks, degree); %! assert (knots, [0 0 0 0 1 2 3 4 4 4 4]) %!test %! breaks = [0 1 2 3 4]; %! degree = 3; %! reg = 1; %! knots = kntbrkdegreg (breaks, degree, reg); %! assert (knots, [0 0 0 0 1 1 2 2 3 3 4 4 4 4]) %!test %! breaks = [0 1 2 3 4]; %! degree = 3; %! reg = [0 1 2]; %! knots = kntbrkdegreg (breaks, degree, reg); %! assert (knots, [0 0 0 0 1 1 1 2 2 3 4 4 4 4]) %!test %! breaks = {[0 1 2 3 4] [0 1 2 3]}; %! degree = [3 2]; %! reg = {[0 1 2] 0}; %! knots = kntbrkdegreg (breaks, degree, reg); %! assert (knots, {[0 0 0 0 1 1 1 2 2 3 4 4 4 4] [0 0 0 1 1 2 2 3 3 3]}) nurbs-1.4.3/inst/PaxHeaders.16248/vectrans.m0000644000000000000000000000013214030323741015366 xustar0030 mtime=1617012705.811919565 30 atime=1617012705.811919565 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vectrans.m0000644000175000017500000000323014030323741017463 0ustar00rvazquezrvazquez00000000000000function dd = vectrans(vector) % % VECTRANS: Transformation matrix for a translation. % % Calling Sequence: % % st = vectrans(tvec) % % INPUT: % % tvec : A vectors defining the translation along the x,y and % z axes. i.e. [tx, ty, ty] % % OUTPUT: % % st : Translation Transformation Matrix % % Description: % % Returns a (4x4) Transformation matrix for translation. % % The matrix is: % % [ 1 0 0 tx ] % [ 0 1 0 ty ] % [ 0 0 1 tz ] % [ 0 0 0 1 ] % % Examples: % % Translate the NURBS line (0.0,0.0,0.0) - (1.0,1.0,1.0) by 3 along % the x-axis, 2 along the y-axis and 4 along the z-axis. % % line = nrbline([0.0 0.0 0.0],[1.0 1.0 1.0]); % trans = vectrans([3.0 2.0 4.0]); % tline = nrbtform(line, trans); % % See also: % % nrbtform % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 1 error('Translation vector required'); end v = [vector(:);0;0]; dd = [1 0 0 v(1); 0 1 0 v(2); 0 0 1 v(3); 0 0 0 1]; end nurbs-1.4.3/inst/PaxHeaders.16248/curvederiveval.m0000644000000000000000000000013214030323741016567 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/curvederiveval.m0000644000175000017500000000406114030323741020667 0ustar00rvazquezrvazquez00000000000000function ck = curvederiveval (n, p, U, P, u, d) % % CURVEDERIVEVAL: Compute the derivatives of a B-spline curve. % % usage: ck = curvederiveval (n, p, U, P, u, d) % % INPUT: % % n+1 = number of control points % p = spline order % U = knots % P = control points % u = evaluation point % d = derivative order % % OUTPUT: % % ck (k+1) = curve differentiated k times % % Adaptation of algorithm A3.4 from the NURBS book, pg99 % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2010 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . ck = zeros (d+1, 1); du = min (d, p); span = findspan (n, p, u, U); N = zeros (p+1, p+1); for ip=0:p N(1:ip+1,ip+1) = basisfun (span, u, ip, U)'; end pk = curvederivcpts (n, p, U, P, du, span-p, span); for k = 0:du for j = 0:p-k ck(k+1) = ck(k+1) + N(j+1,p-k+1)*pk(k+1,j+1); end end end %!test %! k = [0 0 0 1 1 1]; %! coefs(:,1) = [0;0;0;1]; %! coefs(:,2) = [1;0;1;1]; %! coefs(:,3) = [1;1;1;1]; %! crv = nrbmak (coefs, k); %! ck = curvederiveval (crv.number-1, crv.order-1, crv.knots, squeeze (crv.coefs(1,:,:)), 0.5, 2); %! assert(ck, [0.75; 1; -2]); %! ck = curvederiveval (crv.number-1, crv.order-1, crv.knots, squeeze (crv.coefs(2,:,:)), 0.5, 2); %! assert(ck, [0.25; 1; 2]); %! ck = curvederiveval (crv.number-1, crv.order-1, crv.knots, squeeze (crv.coefs(3,:,:)), 0.5, 2); %! assert(ck, [0.75; 1; -2]); nurbs-1.4.3/inst/PaxHeaders.16248/nrbcrvderiveval.m0000644000000000000000000000013214030323741016737 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbcrvderiveval.m0000644000175000017500000000572114030323741021043 0ustar00rvazquezrvazquez00000000000000% NRBCRVDERIVEVAL: Evaluate n-th order derivatives of a NURBS curve. % % usage: skl = nrbcrvderiveval (crv, u, d) % % INPUT: % % crv : NURBS curve structure, see nrbmak % % u : parametric coordinate of the points where we compute the derivatives % % d : number of partial derivatives to compute % % % OUTPUT: % % ck (i, j, l) = i-th component derived j-1 times at the l-th point. % % Adaptation of algorithm A4.2 from the NURBS book, pg127 % % Copyright (C) 2010 Carlo de Falco, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . function ck = nrbcrvderiveval (crv, u, d) ck = arrayfun (@(x) nrbcrvderiveval__ (crv, x, d), u, 'UniformOutput', false); ck = cat (3, ck{:}); end function ck = nrbcrvderiveval__ (crv, u, d) persistent nc; if isempty (nc) nc = [0 0 0 0 0; 1 0 0 0 0; 2 1 0 0 0; 3 3 1 0 0; 4 6 4 1 0]; end ck = zeros (3, d+1); wders = curvederiveval (crv.number-1, crv.order-1, crv.knots, squeeze (crv.coefs(4, :)), u, d); for idim = 1:3 Aders = curvederiveval (crv.number-1, crv.order-1, crv.knots, squeeze (crv.coefs(idim, :)), u, d); ck(idim, 1) = Aders(1) / wders(1); for k = 1:d ck(idim, k+1) = (Aders(k+1) - sum (nc(k+1, 1:k) .* wders(2:k+1).' .* squeeze (ck(idim, k:-1:1)))) / wders(1); end end end %!test %! knots = [0 0 0 1 1 1]; %! coefs(:,1) = [0; 0; 0; 1]; %! coefs(:,2) = [1; 0; 1; 1]; %! coefs(:,3) = [1; 1; 1; 2]; %! crv = nrbmak (coefs, knots); %! u = linspace (0, 1, 100); %! ck = nrbcrvderiveval (crv, u, 2); %! w = @(x) 1 + x.^2; %! dw = @(x) 2*x; %! F1 = @(x) (2*x - x.^2)./w(x); %! F2 = @(x) x.^2./w(x); %! F3 = @(x) (2*x - x.^2)./w(x); %! dF1 = @(x) (2 - 2*x)./w(x) - 2*(2*x - x.^2).*x./w(x).^2; %! dF2 = @(x) 2*x./w(x) - 2*x.^3./w(x).^2; %! dF3 = @(x) (2 - 2*x)./w(x) - 2*(2*x - x.^2).*x./w(x).^2; %! d2F1 = @(x) -2./w(x) - 2*x.*(2-2*x)./w(x).^2 - (8*x-6*x.^2)./w(x).^2 + 8*x.^2.*(2*x-x.^2)./w(x).^3; %! d2F2 = @(x) 2./w(x) - 4*x.^2./w(x).^2 - 6*x.^2./w(x).^2 + 8*x.^4./w(x).^3; %! d2F3 = @(x) -2./w(x) - 2*x.*(2-2*x)./w(x).^2 - (8*x-6*x.^2)./w(x).^2 + 8*x.^2.*(2*x-x.^2)./w(x).^3; %! assert ([F1(u); F2(u); F3(u)], squeeze(ck(:, 1, :)), 1e2*eps); %! assert ([dF1(u); dF2(u); dF3(u)], squeeze(ck(:, 2, :)), 1e2*eps); %! assert ([d2F1(u); d2F2(u); d2F3(u)], squeeze(ck(:, 3, :)), 1e2*eps); nurbs-1.4.3/inst/PaxHeaders.16248/kntuniform.m0000644000000000000000000000013214030323741015735 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/kntuniform.m0000644000175000017500000000342014030323741020033 0ustar00rvazquezrvazquez00000000000000% KNTUNIFORM: generate uniform open knot vectors in the reference domain. % % [csi, zeta] = kntuniform (num, degree, regularity) % % INPUT: % % num: number of breaks (in each direction) % degree: polynomial degree (in each direction) % regularity: global regularity (in each direction) % % OUTPUT: % % csi: knots % zeta: breaks = knots without repetitions % % Copyright (C) 2009, 2010 Carlo de Falco % Copyright (C) 2011 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . function [csi, zeta] = kntuniform (num, degree, regularity) if (numel(num)~=numel(degree) || numel(num)~=numel(regularity)) error('kntuniform: num, degree and regularity must have the same length') else for idim=1:numel(num) zeta{idim} = linspace (0, 1, num(idim)); rep = degree(idim) - regularity(idim); if (rep > 0) csi{idim} = [zeros(1, degree(idim)+1-rep)... reshape(repmat(zeta{idim}, rep, 1), 1, []) ones(1, degree(idim)+1-rep)]; else error ('kntuniform: regularity requested is too high') end end if (numel(num) == 1) csi = csi{1}; zeta = zeta{1}; end end end nurbs-1.4.3/inst/PaxHeaders.16248/kntbrkdegmult.m0000644000000000000000000000013214030323741016416 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/kntbrkdegmult.m0000644000175000017500000000661214030323741020522 0ustar00rvazquezrvazquez00000000000000% KNTBRKDEGMULT: Construct an open knot vector by giving the sequence of % knots, the degree and the multiplicity. % % knots = kntbrkdegreg (breaks, degree) % knots = kntbrkdegreg (breaks, degree, mult) % % INPUT: % % breaks: sequence of knots. % degree: polynomial degree of the splines associated to the knot vector. % mult: multiplicity of the knots. % % OUTPUT: % % knots: knot vector. % % If MULT has as many entries as BREAKS, or as the number of interior % knots, a different multiplicity will be assigned to each knot. If % MULT is not present, it will be taken equal to 1. % % Copyright (C) 2010 Carlo de Falco, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . function knots = kntbrkdegmult (breaks, degree, mult) if (iscell (breaks)) if (nargin == 2) mult = 1; end if (numel(breaks)~=numel(degree) || numel(breaks)~=numel(mult)) error('kntbrkdegmult: degree and multiplicity must have the same length as the number of knot vectors') end degree = num2cell (degree); if (~iscell (mult)) mult = num2cell (mult); end knots = cellfun (@do_kntbrkdegmult, breaks, degree, mult, 'uniformoutput', false); else if (nargin == 2) mult = 1; end knots = do_kntbrkdegmult (breaks, degree, mult); end end function knots = do_kntbrkdegmult (breaks, degree, mult) if (numel (breaks) < 2) error ('kntbrkdegmult: the knots sequence should contain at least two points') end if (numel (mult) == 1) mults = [degree+1, mult(ones (1, numel (breaks) - 2)), degree+1]; elseif (numel (mult) == numel (breaks)) mults = [degree+1 mult(2:end-1) degree+1]; elseif (numel (mult) == numel (breaks) - 2) mults = [degree+1 mult degree+1]; else error('kntbrkdegmult: the length of mult should be equal to one or the number of knots') end if (any (mults > degree+1)) warning ('kntbrkdegmult: some knots have higher multiplicity than the degree+1') end breaks = sort (breaks); lm = numel (mults); sm = sum (mults); mm = zeros (1,sm); mm (cumsum ([1 reshape(mults (1:end-1), 1, lm-1)])) = ones (1,lm); knots = breaks (cumsum (mm)); end %!test %! breaks = [0 1 2 3 4]; %! degree = 3; %! knots = kntbrkdegmult (breaks, degree); %! assert (knots, [0 0 0 0 1 2 3 4 4 4 4]) %!test %! breaks = [0 1 2 3 4]; %! degree = 3; %! mult = 2; %! knots = kntbrkdegmult (breaks, degree, mult); %! assert (knots, [0 0 0 0 1 1 2 2 3 3 4 4 4 4]) %!test %! breaks = [0 1 2 3 4]; %! degree = 3; %! mult = [1 2 3]; %! knots = kntbrkdegmult (breaks, degree, mult); %! assert (knots, [0 0 0 0 1 2 2 3 3 3 4 4 4 4]) %!test %! breaks = {[0 1 2 3 4] [0 1 2 3]}; %! degree = [3 2]; %! mult = {[1 2 3] 2}; %! knots = kntbrkdegmult (breaks, degree, mult); %! assert (knots, {[0 0 0 0 1 2 2 3 3 3 4 4 4 4] [0 0 0 1 1 2 2 3 3 3]}) nurbs-1.4.3/inst/PaxHeaders.16248/nrbderiv.m0000644000000000000000000000013214030323741015354 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbderiv.m0000644000175000017500000027562214030323741017471 0ustar00rvazquezrvazquez00000000000000function varargout = nrbderiv (nurbs) % % NRBDERIV: Construct up to the fourth derivative representation of a % NURBS curve, surface or volume. % % Calling Sequence: % % ders = nrbderiv (nrb); % [ders, ders2] = nrbderiv (nrb); % [ders, ders2, ders3] = nrbderiv (nrb); % [ders, ders2, ders, ders4] = nrbderiv (nrb); % % INPUT: % % nrb : NURBS data structure, see nrbmak. % % OUTPUT: % % ders: A data structure that represents the first % derivatives of a NURBS curve, surface or volume. % ders2: A data structure that represents the second % derivatives of a NURBS curve, surface or volume. % ders3: A data structure that represents the third % derivatives of a NURBS curve, surface or volume. (only surface % for now) % ders4: A data structure that represents the fourth % derivatives of a NURBS curve, surface or volume. (only surface % for now) % % % Description: % % The derivatives of a B-Spline are themselves a B-Spline of lower degree, % giving an efficient means of evaluating multiple derivatives. However, % although the same approach can be applied to NURBS, the situation for % NURBS is more complex. We have followed in this function the same idea % that was already used for the first derivative in the function nrbderiv. % The second derivative data structure can be evaluated later with the % function nrbdeval. % % See also: % % nrbdeval % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Carlo de Falco % Copyright (C) 2010, 2011 Rafael Vazquez % Copyright (C) 2018 Luca Coradello % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (~isstruct(nurbs)) error('NURBS representation is not structure!'); end if (~strcmp(nurbs.form,'B-NURBS')) error('Not a recognised NURBS representation'); end % % % We raise the degree to avoid errors in the computation of the second % % % derivative % if (iscell (nurbs.knots)) % ndim = size(nurbs.knots, 2); % else % ndim = 1; % end % % if (nargout == 2) % degelev = max (2*ones(1, ndim) - (nurbs.order-1), 0); % nurbs = nrbdegelev (nurbs, degelev); % end % % degree = nurbs.order - 1; % % if (ndim == 3) % % NURBS structure represents a volume % num1 = nurbs.number(1); % num2 = nurbs.number(2); % num3 = nurbs.number(3); % % % taking derivatives along the u direction % dknots = nurbs.knots; % dcoefs = permute (nurbs.coefs,[1 3 4 2]); % dcoefs = reshape (dcoefs,4*num2*num3,num1); % [dcoefs,dknots{1}] = bspderiv (degree(1),dcoefs,nurbs.knots{1}); % dcoefs = permute (reshape (dcoefs,[4 num2 num3 size(dcoefs,2)]),[1 4 2 3]); % dnurbs{1} = nrbmak (dcoefs, dknots); % % if (nargout == 2) % % taking second derivative along the u direction (duu) % dknots2 = dknots; % dcoefs2 = permute (dcoefs, [1 3 4 2]); % dcoefs2 = reshape (dcoefs2, 4*num2*num3, []); % [dcoefs2, dknots2{1}] = bspderiv (degree(1)-1, dcoefs2, dknots{1}); % dcoefs2 = permute (reshape (dcoefs2, 4, num2, num3, []), [1 4 2 3]); % dnurbs2{1,1} = nrbmak (dcoefs2, dknots2); % % % taking second derivative along the v direction (duv and dvu) % dknots2 = dknots; % dcoefs2 = permute (dcoefs,[1 2 4 3]); % dcoefs2 = reshape (dcoefs2, 4*(num1-1)*num3, num2); % [dcoefs2, dknots2{2}] = bspderiv (degree(2), dcoefs2, dknots{2}); % dcoefs2 = permute (reshape (dcoefs2, 4, num1-1, num3, []), [1 2 4 3]); % dnurbs2{1,2} = nrbmak (dcoefs2, dknots2); % dnurbs2{2,1} = dnurbs2{1,2}; % % % taking second derivative along the w direction (duw and dwu) % dknots2 = dknots; % dcoefs2 = reshape (dcoefs, 4*(num1-1)*num2, num3); % [dcoefs2, dknots2{3}] = bspderiv (degree(3), dcoefs2, dknots{3}); % dcoefs2 = reshape (dcoefs2, 4, num1-1, num2, []); % dnurbs2{1,3} = nrbmak (dcoefs2, dknots2); % dnurbs2{3,1} = dnurbs2{1,3}; % end % % % taking derivatives along the v direction % dknots = nurbs.knots; % dcoefs = permute (nurbs.coefs,[1 2 4 3]); % dcoefs = reshape (dcoefs,4*num1*num3,num2); % [dcoefs,dknots{2}] = bspderiv (degree(2),dcoefs,nurbs.knots{2}); % dcoefs = permute (reshape (dcoefs,[4 num1 num3 size(dcoefs,2)]),[1 2 4 3]); % dnurbs{2} = nrbmak (dcoefs, dknots); % % if (nargout == 2) % % taking second derivative along the v direction (dvv) % dknots2 = dknots; % dcoefs2 = permute (dcoefs,[1 2 4 3]); % dcoefs2 = reshape (dcoefs2, 4*num1*num3, num2-1); % [dcoefs2, dknots2{2}] = bspderiv (degree(2)-1, dcoefs2, dknots{2}); % dcoefs2 = permute (reshape (dcoefs2, 4, num1, num3, []), [1 2 4 3]); % dnurbs2{2,2} = nrbmak (dcoefs2, dknots2); % % % taking second derivative along the w direction (dvw and dwv) % dknots2 = dknots; % dcoefs2 = reshape (dcoefs, 4*num1*(num2-1), num3); % [dcoefs2, dknots2{3}] = bspderiv (degree(3), dcoefs2, dknots{3}); % dcoefs2 = reshape (dcoefs2, 4, num1, num2-1, []); % dnurbs2{2,3} = nrbmak (dcoefs2, dknots2); % dnurbs2{3,2} = dnurbs2{2,3}; % end % % % taking derivatives along the w direction % dknots = nurbs.knots; % dcoefs = reshape (nurbs.coefs,4*num1*num2,num3); % [dcoefs,dknots{3}] = bspderiv (degree(3),dcoefs,nurbs.knots{3}); % dcoefs = reshape (dcoefs,[4 num1 num2 size(dcoefs,2)]); % dnurbs{3} = nrbmak (dcoefs, dknots); % % if (nargout == 2) % % taking second derivative along the w direction (dww) % dknots2 = dknots; % dcoefs2 = reshape (dcoefs, 4*num1*num2, num3-1); % [dcoefs2, dknots2{3}] = bspderiv (degree(3)-1, dcoefs2, dknots{3}); % dcoefs2 = reshape (dcoefs2, 4, num1, num2, []); % dnurbs2{3,3} = nrbmak (dcoefs2, dknots2); % end % % elseif (ndim == 2) % % NURBS structure represents a surface % num1 = nurbs.number(1); % num2 = nurbs.number(2); % % % taking first derivative along the u direction % dknots = nurbs.knots; % dcoefs = permute (nurbs.coefs,[1 3 2]); % dcoefs = reshape (dcoefs,4*num2,num1); % [dcoefs,dknots{1}] = bspderiv (degree(1),dcoefs,nurbs.knots{1}); % dcoefs = permute (reshape (dcoefs,[4 num2 size(dcoefs,2)]),[1 3 2]); % dnurbs{1} = nrbmak (dcoefs, dknots); % % if (nargout >= 2) % % taking second derivative along the u direction (duu) % dknots2 = dknots; % dcoefs2 = permute (dcoefs, [1 3 2]); % dcoefs2 = reshape (dcoefs2, 4*num2, []); % [dcoefs2, dknots2{1}] = bspderiv (degree(1)-1, dcoefs2, dknots{1}); % dcoefs2 = permute (reshape (dcoefs2, 4, num2, []), [1 3 2]); % dnurbs2{1,1} = nrbmak (dcoefs2, dknots2); % % % taking second derivative along the v direction (duv and dvu) % dknots2 = dknots; % dcoefs2 = reshape (dcoefs, 4*(num1-1), num2); % [dcoefs2, dknots2{2}] = bspderiv (degree(2), dcoefs2, dknots{2}); % dcoefs2 = reshape (dcoefs2, 4, num1-1, []); % dnurbs2{1,2} = nrbmak (dcoefs2, dknots2); % dnurbs2{2,1} = dnurbs2{1,2}; % end % % % taking first derivative along the v direction % dknots = nurbs.knots; % dcoefs = reshape (nurbs.coefs,4*num1,num2); % [dcoefs,dknots{2}] = bspderiv (degree(2),dcoefs,nurbs.knots{2}); % dcoefs = reshape (dcoefs,[4 num1 size(dcoefs,2)]); % dnurbs{2} = nrbmak (dcoefs, dknots); % % if (nargout >= 2) % % taking second derivative along the v direction (dvv) % dknots2 = dknots; % dcoefs2 = reshape (dcoefs, 4*num1, num2-1); % [dcoefs2, dknots2{2}] = bspderiv (degree(2)-1, dcoefs2, dknots{2}); % dcoefs2 = reshape (dcoefs2, 4, num1, []); % dnurbs2{2,2} = nrbmak (dcoefs2, dknots2); % end % % else % % NURBS structure represents a curve % [dcoefs,dknots] = bspderiv (degree, nurbs.coefs, nurbs.knots); % dnurbs = nrbmak (dcoefs, dknots); % if (nargout == 2) % [dcoefs2,dknots2] = bspderiv (degree-1, dcoefs, dknots); % dnurbs2 = nrbmak (dcoefs2, dknots2); % end % end if (iscell (nurbs.knots)) ndim = size(nurbs.knots, 2); else ndim = 1; end % We raise the degree to avoid errors in the computation of the higher % order derivatives if (nargout >= 2) degelev = max (nargout*ones(1, ndim) - (nurbs.order-1), 0); nurbs = nrbdegelev (nurbs, degelev); end if (ndim == 1) % in case of a curve create a cell array to use a dimension-indipendent algorithm tmp = nurbs.knots; nurbs.knots = {}; nurbs.knots{1} = tmp; end for idim = 1:ndim num = nurbs.number; degree = nurbs.order - 1; dknots = nurbs.knots; coord = idim + 1; vec_permute = [setdiff(1:ndim+1,coord),coord]; dcoefs = permute (nurbs.coefs, vec_permute); vec_reshape = setdiff(1:ndim,idim); dcoefs = reshape (dcoefs,4*prod(num(vec_reshape)), num(idim)); [dcoefs, dknots{idim}] = bspderiv (degree(idim), dcoefs, nurbs.knots{idim}); dcoefs = reshape (dcoefs, [4 num(vec_reshape) num(idim)-1]); [~,~,ib] = intersect(1:ndim+1, vec_permute); dcoefs = permute(dcoefs, ib'); dnurbs{idim} = nrbmak (dcoefs, dknots); if (nargout >= 2) % second derivatives degree(idim) = degree(idim) - 1; num(idim) = num(idim) - 1; for jdim = 1:ndim dknots2 = dknots; coord = jdim + 1; vec_permute = [setdiff(1:ndim+1,coord),coord]; dcoefs2 = permute (dcoefs, vec_permute); vec_reshape = setdiff(1:ndim,jdim); dcoefs2 = reshape (dcoefs2, 4*prod(num(vec_reshape)), num(jdim)); [dcoefs2, dknots2{jdim}] = bspderiv (degree(jdim), dcoefs2, dknots{jdim}); dcoefs2 = reshape (dcoefs2, [4 num(vec_reshape) num(jdim)-1]); [~,~,ib] = intersect(1:ndim+1, vec_permute); dcoefs2 = permute(dcoefs2, ib'); dnurbs2{idim,jdim} = nrbmak (dcoefs2, dknots2); if(nargout >= 3) %third derivatives degree(jdim) = degree(jdim) - 1; num(jdim) = num(jdim) - 1; for kdim = 1:ndim dknots3 = dknots2; coord = kdim + 1; vec_permute = [setdiff(1:ndim+1,coord),coord]; dcoefs3 = permute (dcoefs2, vec_permute); vec_reshape = setdiff(1:ndim,kdim); dcoefs3 = reshape (dcoefs3, 4*prod(num(vec_reshape)), num(kdim)); [dcoefs3, dknots3{kdim}] = bspderiv (degree(kdim), dcoefs3, dknots2{kdim}); dcoefs3 = reshape (dcoefs3, [4 num(vec_reshape) num(kdim)-1]); [~,~,ib] = intersect(1:ndim+1, vec_permute); dcoefs3 = permute(dcoefs3, ib'); dnurbs3{idim,jdim,kdim} = nrbmak (dcoefs3, dknots3); if(nargout == 4) %fourth derivatives degree(kdim) = degree(kdim) - 1; num(kdim) = num(kdim) - 1; for ldim = 1:ndim dknots4 = dknots3; coord = ldim + 1; vec_permute = [setdiff(1:ndim+1,coord),coord]; dcoefs4 = permute (dcoefs3, vec_permute); vec_reshape = setdiff(1:ndim,ldim); dcoefs4 = reshape (dcoefs4, 4*prod(num(vec_reshape)), num(ldim)); [dcoefs4, dknots4{ldim}] = bspderiv (degree(ldim), dcoefs4, dknots3{ldim}); dcoefs4 = reshape (dcoefs4, [4 num(vec_reshape) num(ldim)-1]); [~,~,ib] = intersect(1:ndim+1, vec_permute); dcoefs4 = permute(dcoefs4, ib'); dnurbs4{idim,jdim,kdim,ldim} = nrbmak (dcoefs4, dknots4); % degree(ldim) = degree(ldim) - 1; % num(ldim) = num(ldim) - 1; end degree(kdim) = degree(kdim) + 1; num(kdim) = num(kdim) + 1; end end degree(jdim) = degree(jdim) + 1; num(jdim) = num(jdim) + 1; end end end end if (ndim == 1) % in the case of a curve transform everything back otherwise it will throw errors somewhere else in the code ! tmp = nurbs.knots{1}; nurbs.knots = tmp; tmp = dnurbs{1}; dnurbs = tmp; if (nargout >= 2) tmp = dnurbs2{1,1}; dnurbs2 = tmp; end if (nargout >= 3) tmp = dnurbs3{1,1,1}; dnurbs3 = tmp; end if (nargout == 4) tmp = dnurbs4{1,1,1,1}; dnurbs4 = tmp; end end varargout{1} = dnurbs; if (nargout >= 2) varargout{2} = dnurbs2; if (iscell (dnurbs2)) dnurbs2 = [dnurbs2{:}]; end if (any (arrayfun(@(x) any(isnan(x.coefs(:)) | isinf(x.coefs(:))), dnurbs2))) warning ('nrbderiv:SecondDerivative', ... ['The structure for the second derivative contains Inf and/or NaN coefficients, ' ... 'probably due to low continuity at repeated knots. This should not affect the ' ... 'computation of the second derivatives, except at those knots.']) end end if(nargout >= 3) varargout{3} = dnurbs3; end if(nargout == 4) varargout{4} = dnurbs4; end end %!demo %! crv = nrbtestcrv; %! nrbplot(crv,48); %! title('First derivatives along a test curve.'); %! %! tt = linspace(0.0,1.0,9); %! %! dcrv = nrbderiv(crv); %! %! [p1, dp] = nrbdeval(crv,dcrv,tt); %! %! p2 = vecnormalize(dp); %! %! hold on; %! plot(p1(1,:),p1(2,:),'ro'); %! h = quiver(p1(1,:),p1(2,:),p2(1,:),p2(2,:),0); %! set(h,'Color','black'); %! hold off; %!demo %! srf = nrbtestsrf; %! p = nrbeval(srf,{linspace(0.0,1.0,20) linspace(0.0,1.0,20)}); %! h = surf(squeeze(p(1,:,:)),squeeze(p(2,:,:)),squeeze(p(3,:,:))); %! set(h,'FaceColor','blue','EdgeColor','blue'); %! title('First derivatives over a test surface.'); %! %! npts = 5; %! tt = linspace(0.0,1.0,npts); %! dsrf = nrbderiv(srf); %! %! [p1, dp] = nrbdeval(srf, dsrf, {tt, tt}); %! %! up2 = vecnormalize(dp{1}); %! vp2 = vecnormalize(dp{2}); %! %! hold on; %! plot3(p1(1,:),p1(2,:),p1(3,:),'ro'); %! h1 = quiver3(p1(1,:),p1(2,:),p1(3,:),up2(1,:),up2(2,:),up2(3,:)); %! h2 = quiver3(p1(1,:),p1(2,:),p1(3,:),vp2(1,:),vp2(2,:),vp2(3,:)); %! set(h1,'Color','black'); %! set(h2,'Color','black'); %! %! hold off; %!test %! knots = [0 0 0 0.5 1 1 1]; %! coefs(1,:) = [0 2 4 2]; %! coefs(2,:) = [0 2 2 0]; %! coefs(3,:) = [0 4 2 0]; %! coefs(4,:) = [1 2 2 1]; %! nrb = nrbmak (coefs, knots); %! [dnrb, dnrb2] = nrbderiv (nrb); %! x = linspace (0, 1, 10); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, x); %! w = -4*x.^2 + 4*x + 1; %! F = zeros (3,numel(x)); DF = zeros (3, numel(x)); D2F = zeros (3, numel(x)); %! F(1,:) = (-4*x.*(x-2)./w) .* (x<0.5) + ((4*x - 5)./w + 3) .* (x>0.5); %! F(2,:) = (2-2./w); %! F(3,:) = (-4*x.*(5*x-4)./w) .* (x<0.5) + (-4*(x.^2 - 1)./w) .* (x>0.5); %! DF(1,:) = (8*(2*x.^2-x+1)./w.^2) .* (x<0.5) + (8*(2*x-3).*(x-1)./w.^2) .* (x>0.5); %! DF(2,:) = -8*(2*x-1)./w.^2; %! DF(3,:) = -(8*(2*x.^2+5*x-2)./w.^2) .* (x<0.5) - (8*(2*x.^2-3*x+2)./w.^2) .* (x>0.5); %! D2F(1,:) = 8*(16*x.^3-12*x.^2+24*x-9)./w.^3 .* (x<0.5) + 8*(16*x.^3-60*x.^2+72*x-29)./w.^3 .* (x>0.5); %! D2F(2,:) = -16*(12*x.^2-12*x+5)./w.^3; %! D2F(3,:) = -8*(16*x.^3+60*x.^2-48*x+21)./w.^3 .* (x<0.5) -8*(16*x.^3-36*x.^2+48*x-19)./w.^3 .* (x>0.5); %! assert (F, pnt, 1e3*eps) %! assert (DF, jac, 1e3*eps) %! assert (D2F, hess, 1e3*eps) %!test %! knots = {[0 0 0 1 1 1], [0 0 0 0.5 1 1 1]}; %! coefs = ones (4,3,4); %! coefs(1,:,:) = reshape ([0 0 0 0; 1 1 1 1; 2 2 4 2], 1, 3, 4); %! coefs(2,:,:) = reshape ([0 1 2 3; 0 1 2 3; 0 1 4 3], 1, 3, 4); %! coefs(3,:,:) = reshape ([0 1 0 0; 0 0 0 0; 0 0 0 0], 1, 3, 4); %! coefs(4,:,:) = reshape ([1 1 1 1; 1 1 1 1; 1 1 2 1], 1, 3, 4); %! nrb = nrbmak (coefs, knots); %! [dnrb, dnrb2] = nrbderiv (nrb); %! X = linspace (0, 1, 4); Y = linspace (0, 1, 4); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, {X Y}); %! [y, x] = meshgrid (X, Y); %! w = (2*x.^2.*y.^2 + 1) .* (y < 0.5) + (-6*x.^2.*y.^2 + 8*x.^2.*y - 2*x.^2 + 1) .* (y > 0.5); %! F = zeros ([3,size(x)]); %! F(1,:,:) = ((2*x - 2) ./w + 2) .* (y<0.5) + (2 + (2*x-2)./w) .* (y > 0.5); %! F(2,:,:) = (2 - (2*(y-1).^2)./w).*(y<0.5) + ... %! ((-12*x.^2.*y.^2 + 16*x.^2.*y - 4*x.^2 + 2*y.^2 + 1)./w).*(y>0.5); %! F(3,:,:) = (-2*y.*(3*y - 2).*(x - 1).^2./w) .* (y<0.5) + ... %! (2*(x - 1).^2.*(y - 1).^2./w) .* (y>0.5); %! dFdu = zeros ([3,size(x)]); %! dFdu(1,:,:) = (((8*x - 4*x.^2).*y.^2 + 2)./w.^2).*(y<0.5) + ... %! (((12*y.^2 - 16*y + 4).*x.^2 + (-24*y.^2 + 32*y - 8).*x + 2)./w.^2).*(y>0.5); %! dFdu(2,:,:) = (8*x.*y.^2.*(y - 1).^2./w.^2).*(y<0.5) + ... %! ((4*x.*(3*y - 1).*(2*y.^2 - 1).*(y - 1))./w.^2).*(y>0.5); %! dFdu(3,:,:) = (-4*y.*(2.*x.*y.^2 + 1).*(3*y - 2).*(x - 1)./w.^2).*(y<0.5) + ... %! ((-4*(x - 1).*(y - 1).^2.*(6*x.*y.^2 - 8*x.*y + 2*x - 1))./w.^2).*(y>0.5); %! dFdv = zeros ([3,size(x)]); %! dFdv(1,:,:) = (-8*x.^2.*y.*(x - 1)./w.^2).*(y<0.5) + ... %! (8*x.^2.*(3*y - 2).*(x - 1)./w.^2).*(y>0.5); %! dFdv(2,:,:) = (-4*(2*y.*x.^2 + 1).*(y - 1)./w.^2).*(y<0.5) + ... %! (((16*y.^2 - 20*y + 8).*x.^2 + 4*y)./w.^2).*(y>0.5); %! dFdv(3,:,:) = (-4*(x - 1).^2.*(2*x.^2.*y.^2 + 3*y - 1)./w.^2).*(y<0.5) + ... %! (4*(x - 1).^2.*(y - 1).*(2*x.^2 - 2*x.^2.*y + 1)./w.^2).*(y>0.5); %! d2Fduu = zeros ([3, size(x)]); %! d2Fduu(1,:,:) = (-((48*x.^2 - 16*x.^3).*y.^4 + (24*x - 8).*y.^2)./w.^3).*(y<0.5) + ... %! (((32*(3*y - 1).*(x - 1).*(y - 1))-(8*(3*y - 1).*(x - 3).*(y - 1).*w))./w.^3).*(y>0.5); %! d2Fduu(2,:,:) = (-(8*y.^2.*(6*x.^2.*y.^2 - 1).*(y - 1).^2)./w.^3).*(y<0.5) + ... %! ((4*(3*y - 1).*(2*y.^2 - 1).*(y - 1).*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 + 1))./w.^3).*(y>0.5); %! d2Fduu(3,:,:) = ((4*y.*(3*y - 2).*(8*x.^3.*y.^4 - 12*x.^2.*y.^4 + 6*x.^2.*y.^2 - 12*x.*y.^2 + 2*y.^2 - 1))./w.^3).*(y<0.5) + ... %! ((4*(y - 1).^2.*(6*y.^2 - 8*y + 3) - 4*x.^3.*(y - 1).^2.*(72*y.^4 - 192*y.^3 + 176*y.^2 - 64*y + 8) + 4*x.^2.*(y - 1).^2.*(108*y.^4 - 288*y.^3 + 282*y.^2 - 120*y + 18) - 4*x.*(y - 1).^2.*(36*y.^2 - 48*y + 12))./w.^3) .* (y>0.5); %! d2Fdvv = zeros ([3, size(x)]); %! d2Fdvv(1,:,:) = (8*x.^2.*(6*x.^2.*y.^2 - 1).*(x - 1)./w.^3) .* (y<0.5) + ... %! (8*x.^2.*(x - 1).*(54*x.^2.*y.^2 - 72*x.^2.*y + 26*x.^2 + 3)./w.^3) .* (y>0.5); %! d2Fdvv(2,:,:) = (-((48*y.^2 - 32*y.^3).*x.^4 + (- 24*y.^2 + 48*y - 8).*x.^2 + 4)./w.^3) .*(y<0.5) + ... %! (((192*y.^3 - 360*y.^2 + 288*y - 88).*x.^4 + (72*y.^2 - 28).*x.^2 + 4)./w.^3) .* (y>0.5); %! d2Fdvv(3,:,:) = (4*(x - 1).^2.*(8*x.^4.*y.^3 + 18*x.^2.*y.^2 - 12*x.^2.*y - 3))./w.^3 .* (y<0.5) + ... %! ((4*(x - 1).^2.*(24*x.^4 + 18*x.^2 + 1) + 4*y.^2.*(72*x.^4 + 18*x.^2).*(x - 1).^2 - 96*x.^4.*y.^3.*(x - 1).^2 - 4*y.*(72*x.^4 + 36*x.^2).*(x - 1).^2)./w.^3) .* (y>0.5); %! d2Fduv = zeros ([3, size(x)]); %! d2Fduv(1,:,:) = (-(y.^3.*(32*x.^3 - 16*x.^4) - y.*(16*x - 24*x.^2))./w.^3) .* (y<0.5) + ... %! (-(-8*(3*y - 2).*(6*y.^2 - 8*y + 2).*x.^4 + 8*(3*y - 2).*(12*y.^2 - 16*y + 4).*x.^3 + (48 - 72*y).*x.^2 + (48*y - 32).*x)./w.^3) .* (y>0.5); %! d2Fduv(2,:,:) = (16*x.*y.*(y - 1).*(2*x.^2.*y.^2 + 2*y - 1)./w.^3) .* (y<0.5) + ... %! (-(8*x.*(4*y.^2 - 5*y + 2))./w.^2 + (16*x.*(3*y - 2).*(2*y.^2 - 1))./w.^3) .* (y>0.5); %! d2Fduv(3,:,:) = (-(8*(x - 1).*(4*x.^3.*y.^4 - 6*x.^2.*y.^3 + 6*x.^2.*y.^2 + 12*x.*y.^3 - 6*x.*y.^2 + 3*y - 1))./w.^3) .* (y<0.5) + ... %! ((8*(x - 1).*(y - 1).*(12*x.^3.*y.^3 - 28*x.^3.*y.^2 + 20*x.^3.*y - 4*x.^3 + 6*x.^2.*y.^2 - 12*x.^2.*y + 6*x.^2 - 12*x.*y.^2 + 18*x.*y - 6*x + 1))./w.^3) .* (y>0.5); %! assert (F, pnt, 1e3*eps) %! assert (dFdu, jac{1}, 1e3*eps) %! assert (dFdv, jac{2}, 1e3*eps) %! assert (d2Fduu, hess{1,1}, 1e3*eps) %! assert (d2Fduv, hess{1,2}, 1e3*eps) %! assert (d2Fduv, hess{2,1}, 1e3*eps) %! assert (d2Fdvv, hess{2,2}, 1e3*eps) %!test %! knots = {[0 0 0 1 1 1], [0 0 0 0.5 1 1 1]}; %! coefs = ones (4,3,4); %! coefs(1,:,:) = reshape ([0 0 0 0; 1 1 1 1; 2 2 4 2], 1, 3, 4); %! coefs(2,:,:) = reshape ([0 1 2 3; 0 1 2 3; 0 1 4 3], 1, 3, 4); %! coefs(3,:,:) = reshape ([0 1 0 0; 0 0 0 0; 0 0 0 0], 1, 3, 4); %! coefs(4,:,:) = reshape ([1 1 1 1; 1 1 1 1; 1 1 2 1], 1, 3, 4); %! nrb = nrbmak (coefs, knots); %! nrb = nrbdegelev (nrbextrude (nrb, [0.4 0.6 2]), [0 0 1]); %! nrb.coefs(4,2,3,3) = 1.5; %! [dnrb, dnrb2] = nrbderiv (nrb); %! X = linspace (0, 1, 4); Y = linspace (0, 1, 4); Z = linspace (0, 1, 4); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, {X Y Z}); %! [y, x, z] = meshgrid (X, Y, Z); %! w = (-2*x.^2.*y.^2.*z.^2 + 2*x.^2.*y.^2 + 2*x.*y.^2.*z.^2 + 1) .* (y < 0.5) + ... %! (6*x.^2.*y.^2.*z.^2 - 6*x.^2.*y.^2 - 8*x.^2.*y.*z.^2 + 8*x.^2.*y + 2*x.^2.*z.^2 - 2*x.^2 - 6*x.*y.^2.*z.^2 + 8*x.*y.*z.^2 - 2*x.*z.^2 + 1) .* (y > 0.5); %! F = zeros ([3,size(x)]); %! F(1,:,:,:) = ((10*x + 20*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2))./(5*w)) .* (y<0.5) + ... %! (60*x.^2.*y.^2 - 10*x + z.*(12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2) - 80*x.^2.*y + 20*x.^2)./(-5*w) .* (y > 0.5); %! F(2,:,:,:) = ((20*y + 20*x.^2.*y.^2 + z.*(6*x.^2.*y.^2 + 3) - 10*y.^2)./(5*w)).*(y<0.5) + ... %! ((60*x.^2.*y.^2 + z.*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 3) - 80*x.^2.*y + 20*x.^2 - 10*y.^2 - 5)./(-5*w)).*(y>0.5); %! F(3,:,:,:) = ((4*y - 6*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2) - 8*x.*y + 12*x.*y.^2 + 4*x.^2.*y - 6*y.^2)./w) .* (y<0.5) + ... %! ((2*z - 4*y - 4*x + 2*x.^2.*y.^2 + 8*x.*y - 4*x.*y.^2 - 4*x.^2.*y - 4*x.^2.*z + 2*x.^2 + 2*y.^2 + 16*x.^2.*y.*z - 12*x.^2.*y.^2.*z + 2)./w) .* (y>0.5); %! dFdu = zeros ([3,size(x)]); %! dFdu(1,:,:,:) = ((x.*((8*y.^2.*z.^3)/5 + 8*y.^2) - (4*y.^2.*z.^3)/5 + x.^2.*(z.^2.*(8*y.^4 + 4*y.^2) + (8*y.^4.*z.^3)/5 - 4*y.^2) + 2)./w.^2).*(y<0.5) + ... %! ((z.^3.*(x.^2.*((72*y.^4)/5 - (192*y.^3)/5 + (176*y.^2)/5 - (64*y)/5 + 8/5) - (16*y)/5 - x.*((24*y.^2)/5 - (32*y)/5 + 8/5) + (12*y.^2)/5 + 4/5) - x.*(24*y.^2 - 32*y + 8) + x.^2.*(12*y.^2 - 16*y + 4) + x.^2.*z.^2.*(72*y.^4 - 192*y.^3 + 164*y.^2 - 48*y + 4) + 2)./w.^2).*(y>0.5); %! dFdu(2,:,:,:) = ((z.^2.*(8*x.^2.*y.^4 - y.^2.*(8*y - 4*y.^2) + (2*x.*y.^2.*(40*y - 20*y.^2))/5) + z.^3.*((12*x.^2.*y.^4)/5 + (12*x.*y.^2)/5 - (6*y.^2)/5) + (2*x.*y.^2.*(20*y.^2 - 40*y + 20))/5)./w.^2).*(y<0.5) + ... %! (((2*(3*y.^2 - 4*y + 1).*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 6*x + 3).*z.^3)/5 + (2*(3*y.^2 - 4*y + 1).*(60*x.^2.*y.^2 - 80*x.^2.*y + 20*x.^2 - 20*x.*y.^2 - 10*x + 10*y.^2 + 5).*z.^2)/5 - (2*(10*x - 20*x.*y.^2).*(3*y.^2 - 4*y + 1))/5)./w.^2).*(y>0.5); %! dFdu(3,:,:,:) = ((4*y.*(3*y - 2) + z.^3.*(8*x.^2.*y.^4 + 8*x.*y.^2 - 4*y.^2) - z.^2.*(4*y.*(2*y.^2 - 3*y.^3).*x.^2 - 4*y.*(4*y.^2 - 6*y.^3).*x + 4*y.*(2*y.^2 - 3*y.^3)) + 4*x.^2.*y.*(4*y.^2 - 6*y.^3) - 4*x.*y.*(- 6*y.^3 + 4*y.^2 + 3*y - 2)) ./w.^2).*(y<0.5) + ... %! ((z.^2.*(4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1).*x.^2 - 4*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2).*x + 4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1)) - 4*(y - 1).^2 + z.^3.*(4*(y - 1).*(18*y.^3 - 30*y.^2 + 14*y - 2).*x.^2 - 4*(6*y - 2).*(y - 1).*x + 4*(3*y - 1).*(y - 1)) + 4*x.*(y - 1).*(6*y.^3 - 14*y.^2 + 11*y - 3) - 4*x.^2.*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2))./w.^2) .* (y > 0.5); %! dFdv = zeros ([3,size(x)]); %! dFdv(1,:,:,:) = ((8*x.*y.*(x - 1).*(z.^3 + 5*x.*z.^2 - 5*x))/5./w.^2).*(y<0.5) + ... %! (-(8*x.*(3*y - 2).*(x - 1).*(z.^3 + 5*x.*z.^2 - 5*x))/5./w.^2).*(y>0.5); %! dFdv(2,:,:,:) = (-((8*x.*z.^2 - x.^2.*(8*z.^2 - 8)).*y.^2 + ((12*x.*z.^3)/5 - x.^2.*((12*z.^3)/5 + 8) + 4).*y - 4)./w.^2).*(y<0.5) + ... %! ((4*y + z.^3.*(x.*((36*y)/5 - 24/5) - x.^2.*((36*y)/5 - 24/5)) + z.^2.*(x.*(16*y.^2 + 4*y - 8) - x.^2.*(16*y.^2 + 4*y - 8)) + x.^2.*(16*y.^2 - 20*y + 8))./w.^2).*(y>0.5); %! dFdv(3,:,:,:) = ((4*(x - 1).^2 - y.*(4*(3*x - 3).*(x - 1) - 8*x.*z.^3.*(x - 1)) + y.^2.*(4*(x - 1).*(2*x.^3 - 4*x.^2 + 2*x).*z.^2 + 4*(2*x.^2 - 2*x.^3).*(x - 1)))./w.^2).*(y<0.5) + ... %! ((y.^2.*(4*(x - 1).*(2*x.^3 - 4*x.^2 + 2*x).*z.^2 + 4*(2*x.^2 - 2*x.^3).*(x - 1)) - 4*(x - 1).*(2*x.^3 - 2*x.^2 + x - 1) - y.*(24*x.*(x - 1).*z.^3 + 4*(x - 1).*(4*x.^3 - 8*x.^2 + 4*x).*z.^2 - 4*(x - 1).*(4*x.^3 - 4*x.^2 + x - 1)) + 16*x.*z.^3.*(x - 1) + 4*z.^2.*(x - 1).*(2*x.^3 - 4*x.^2 + 2*x))./w.^2).*(y>0.5); %! dFdw = zeros ([3,size(x)]); %! dFdw(1,:,:,:) = ((4*x.^2.*y.^2 + 2)./(- 10*x.^2.*y.^2.*z.^2 + 10*x.^2.*y.^2 + 10*x.*y.^2.*z.^2 + 5) - ((20*x.*y.^2.*z - 20*x.^2.*y.^2.*z).*(10*x + 20*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2)))./(5*w).^2).*(y<0.5) + ... %! ((12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2)./(- 30*x.^2.*y.^2.*z.^2 + 30*x.^2.*y.^2 + 40*x.^2.*y.*z.^2 - 40*x.^2.*y - 10*x.^2.*z.^2 + 10*x.^2 + 30*x.*y.^2.*z.^2 - 40*x.*y.*z.^2 + 10*x.*z.^2 - 5) - ((60*x.^2.*y.^2 - 10*x + z.*(12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2) - 80*x.^2.*y + 20*x.^2).*(- 60*z.*x.^2.*y.^2 + 80*z.*x.^2.*y - 20*z.*x.^2 + 60*z.*x.*y.^2 - 80*z.*x.*y + 20*z.*x))./(5*w).^2).*(y>0.5); %! dFdw(2,:,:,:) = ((6*x.^2.*y.^2 + 3)./(- 10*x.^2.*y.^2.*z.^2 + 10*x.^2.*y.^2 + 10*x.*y.^2.*z.^2 + 5) - ((20*x.*y.^2.*z - 20*x.^2.*y.^2.*z).*(20*y + 20*x.^2.*y.^2 + z.*(6*x.^2.*y.^2 + 3) - 10*y.^2))./(5*w).^2).*(y<0.5) + ... %! ((18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 3)./(- 30*x.^2.*y.^2.*z.^2 + 30*x.^2.*y.^2 + 40*x.^2.*y.*z.^2 - 40*x.^2.*y - 10*x.^2.*z.^2 + 10*x.^2 + 30*x.*y.^2.*z.^2 - 40*x.*y.*z.^2 + 10*x.*z.^2 - 5) - ((- 60*z.*x.^2.*y.^2 + 80*z.*x.^2.*y - 20*z.*x.^2 + 60*z.*x.*y.^2 - 80*z.*x.*y + 20*z.*x).*(60*x.^2.*y.^2 + z.*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 3) - 80*x.^2.*y + 20*x.^2 - 10*y.^2 - 5))./(5*w).^2).*(y>0.5); %! dFdw(3,:,:,:) = ((4*x.^2.*y.^2 + 2)./(2*x.^2.*y.^2 - z.^2.*(2*x.^2.*y.^2 - 2*x.*y.^2) + 1) + (2*z.*(2*x.^2.*y.^2 - 2*x.*y.^2).*(4*y - 6*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2) - 8*x.*y + 12*x.*y.^2 + 4*x.^2.*y - 6*y.^2))./w.^2).*(y<0.5) + ... %! ((12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2)./(6*x.^2.*y.^2 + z.^2.*(- 6*x.^2.*y.^2 + 8*x.^2.*y - 2*x.^2 + 6*x.*y.^2 - 8*x.*y + 2*x) - 8*x.^2.*y + 2*x.^2 - 1) + (2*z.*(- 6*x.^2.*y.^2 + 8*x.^2.*y - 2*x.^2 + 6*x.*y.^2 - 8*x.*y + 2*x).*(2*z - 4*y - 4*x + 2*x.^2.*y.^2 + 8*x.*y - 4*x.*y.^2 - 4*x.^2.*y - 4*x.^2.*z + 2*x.^2 + 2*y.^2 + 16*x.^2.*y.*z - 12*x.^2.*y.^2.*z + 2))./w.^2).*(y>0.5); %! d2Fduu = zeros ([3, size(x)]); %! d2Fduu(1,:,:,:) = (((8*y.^2.*z.^3)/5 + 2*x.*(z.^2.*(8*y.^4 + 4*y.^2) + (8*y.^4.*z.^3)/5 - 4*y.^2) + 8*y.^2)./w.^2 - (2*(2*y.^2.*z.^2 + 4*x.*y.^2 - 4*x.*y.^2.*z.^2).*(x.*((8*y.^2.*z.^3)/5 + 8*y.^2) - (4*y.^2.*z.^3)/5 + x.^2.*(z.^2.*(8*y.^4 + 4*y.^2) + (8*y.^4.*z.^3)/5 - 4*y.^2) + 2))./w.^3).*(y<0.5) + ... %! ((32*y + 2*x.*(12*y.^2 - 16*y + 4) + z.^3.*((32*y)/5 + 2*x.*((72*y.^4)/5 - (192*y.^3)/5 + (176*y.^2)/5 - (64*y)/5 + 8/5) - (24*y.^2)/5 - 8/5) - 24*y.^2 + 2*x.*z.^2.*(72*y.^4 - 192*y.^3 + 164*y.^2 - 48*y + 4) - 8)./w.^2 - (2*(z.^3.*(x.^2.*((72*y.^4)/5 - (192*y.^3)/5 + (176*y.^2)/5 - (64*y)/5 + 8/5) - (16*y)/5 - x.*((24*y.^2)/5 - (32*y)/5 + 8/5) + (12*y.^2)/5 + 4/5) - x.*(24*y.^2 - 32*y + 8) + x.^2.*(12*y.^2 - 16*y + 4) + x.^2.*z.^2.*(72*y.^4 - 192*y.^3 + 164*y.^2 - 48*y + 4) + 2).*(4*x + 6*y.^2.*z.^2 - 16*x.*y + 12*x.*y.^2 - 4*x.*z.^2 - 8*y.*z.^2 + 2*z.^2 + 16*x.*y.*z.^2 - 12*x.*y.^2.*z.^2))./(-w).^3).*(y>0.5); %! d2Fduu(2,:,:,:) = ((z.^3.*((24*x.*y.^4)/5 + (12*y.^2)/5) + (2*y.^2.*(20*y.^2 - 40*y + 20))/5 + z.^2.*((2*y.^2.*(40*y - 20*y.^2))/5 + 16*x.*y.^4))./w.^2 - (2*(z.^2.*(8*x.^2.*y.^4 - y.^2.*(8*y - 4*y.^2) + (2*x.*y.^2.*(40*y - 20*y.^2))/5) + z.^3.*((12*x.^2.*y.^4)/5 + (12*x.*y.^2)/5 - (6*y.^2)/5) + (2*x.*y.^2.*(20*y.^2 - 40*y + 20))/5).*(2*y.^2.*z.^2 + 4*x.*y.^2 - 4*x.*y.^2.*z.^2))./w.^3).*(y<0.5) + ... %! (((2*(3*y.^2 - 4*y + 1).*(36*x.*y.^2 - 48*x.*y + 12*x - 6).*z.^3)/5 - (2*(3*y.^2 - 4*y + 1).*(160*x.*y - 40*x - 120*x.*y.^2 + 20*y.^2 + 10).*z.^2)/5 + (2*(20*y.^2 - 10).*(3*y.^2 - 4*y + 1))/5)./w.^2 - (2*((2*(3*y.^2 - 4*y + 1).*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 6*x + 3).*z.^3)/5 + (2*(3*y.^2 - 4*y + 1).*(60*x.^2.*y.^2 - 80*x.^2.*y + 20*x.^2 - 20*x.*y.^2 - 10*x + 10*y.^2 + 5).*z.^2)/5 - (2*(10*x - 20*x.*y.^2).*(3*y.^2 - 4*y + 1))/5).*(4*x + 6*y.^2.*z.^2 - 16*x.*y + 12*x.*y.^2 - 4*x.*z.^2 - 8*y.*z.^2 + 2*z.^2 + 16*x.*y.*z.^2 - 12*x.*y.^2.*z.^2))./(-w).^3).*(y>0.5); %! d2Fduu(3,:,:,:) = (((16*x.*y.^4 + 8*y.^2).*z.^3 + (4*y.*(4*y.^2 - 6*y.^3) - 8*x.*y.*(2*y.^2 - 3*y.^3)).*z.^2 - 4*y.*(- 6*y.^3 + 4*y.^2 + 3*y - 2) + 8*x.*y.*(4*y.^2 - 6*y.^3))./w.^2 - (2*(2*y.^2.*z.^2 + 4*x.*y.^2 - 4*x.*y.^2.*z.^2).*(4*y.*(3*y - 2) + z.^3.*(8*x.^2.*y.^4 + 8*x.*y.^2 - 4*y.^2) - z.^2.*(4*y.*(2*y.^2 - 3*y.^3).*x.^2 - 4*y.*(4*y.^2 - 6*y.^3).*x + 4*y.*(2*y.^2 - 3*y.^3)) + 4*x.^2.*y.*(4*y.^2 - 6*y.^3) - 4*x.*y.*(- 6*y.^3 + 4*y.^2 + 3*y - 2)))./w.^3).*(y<0.5) + ... %! (-((4*(6*y - 2).*(y - 1) - 8*x.*(y - 1).*(18*y.^3 - 30*y.^2 + 14*y - 2)).*z.^3 + (4*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2) - 8*x.*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1)).*z.^2 - 4*(y - 1).*(6*y.^3 - 14*y.^2 + 11*y - 3) + 8*x.*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2))./w.^2 - (2*(z.^2.*(4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1).*x.^2 - 4*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2).*x + 4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1)) - 4*(y - 1).^2 + z.^3.*(4*(y - 1).*(18*y.^3 - 30*y.^2 + 14*y - 2).*x.^2 - 4*(6*y - 2).*(y - 1).*x + 4*(3*y - 1).*(y - 1)) + 4*x.*(y - 1).*(6*y.^3 - 14*y.^2 + 11*y - 3) - 4*x.^2.*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2)).*(4*x + 6*y.^2.*z.^2 - 16*x.*y + 12*x.*y.^2 - 4*x.*z.^2 - 8*y.*z.^2 + 2*z.^2 + 16*x.*y.*z.^2 - 12*x.*y.^2.*z.^2))./(-w).^3) .* (y>0.5); %! d2Fduv = zeros ([3, size(x)]); %! d2Fduv(1,:,:,:) = ((((8.*x.^2.*(6.*z.^3 - 6.*z.^5))/5 + (8.*x.^4.*(10.*z.^4 - 20.*z.^2 + 10))/5 - (8.*x.^3.*(- 4.*z.^5 + 10.*z.^4 + 4.*z.^3 - 30.*z.^2 + 20))/5 + (16.*x.*z.^5)/5).*y.^3 + ((8.*x.*(2.*z.^3 - 10.*z.^2 + 10))/5 + (8.*x.^2.*(15.*z.^2 - 15))/5 - (8.*z.^3)/5).*y)./w.^3) .* (y<0.5) + ... %! (-(x.^4.*((8.*(3.*y - 2).*(30.*y.^2 - 40.*y + 10).*z.^4)/5 - (8.*(3.*y - 2).*(60.*y.^2 - 80.*y + 20).*z.^2)/5 + (8.*(3.*y - 2).*(30.*y.^2 - 40.*y + 10))/5) - x.^3.*(- (8.*(3.*y - 2).*(12.*y.^2 - 16.*y + 4).*z.^5)/5 + (8.*(3.*y - 2).*(30.*y.^2 - 40.*y + 10).*z.^4)/5 + (8.*(3.*y - 2).*(12.*y.^2 - 16.*y + 4).*z.^3)/5 - (8.*(3.*y - 2).*(90.*y.^2 - 120.*y + 30).*z.^2)/5 + (8.*(3.*y - 2).*(60.*y.^2 - 80.*y + 20))/5) + z.^3.*((24.*y)/5 - 16/5) - x.^2.*((8.*(3.*y - 2).*(18.*y.^2 - 24.*y + 6).*z.^5)/5 - (8.*(3.*y - 2).*(18.*y.^2 - 24.*y + 6).*z.^3)/5 + (72.*y - 48).*z.^2 - 72.*y + 48) + x.*((8.*(3.*y - 2).*(6.*y.^2 - 8.*y + 2).*z.^5)/5 + (32/5 - (48.*y)/5).*z.^3 + (48.*y - 32).*z.^2 - 48.*y + 32))./(-w).^3) .* (y>0.5); %! d2Fduv(2,:,:,:) = ((((4.*x.^2.*(60.*z.^2 - 60.*z.^4))/5 + (4.*x.^3.*(40.*z.^4 - 80.*z.^2 + 40))/5 + 16.*x.*z.^4).*y.^4 + ((4.*x.^2.*(18.*z.^3 - 18.*z.^5))/5 + (4.*x.^3.*(12.*z.^5 - 12.*z.^3 + 40.*z.^2 - 40))/5 + (4.*x.*(6.*z.^5 - 40.*z.^2 + 40))/5 + 16.*z.^2).*y.^3 + ((4.*x.*(60.*z.^2 - 60))/5 - 24.*z.^2).*y.^2 + ((4.*x.*(6.*z.^3 + 20))/5 - (12.*z.^3)/5).*y)./w.^3) .* (y<0.5) + ... %! ((z.^3.*(((432.*y.^3)/5 - (864.*y.^2)/5 + (528.*y)/5 - 96/5).*x.^3 + (- (648.*y.^3)/5 + (1296.*y.^2)/5 - (792.*y)/5 + 144/5).*x.^2 + ((72.*y)/5 - 48/5).*x - (36.*y)/5 + 24/5) - x.^3.*(192.*y.^4 - 496.*y.^3 + 480.*y.^2 - 208.*y + 32) + z.^4.*((- 192.*y.^4 + 208.*y.^3 + 96.*y.^2 - 144.*y + 32).*x.^3 + (288.*y.^4 - 312.*y.^3 - 144.*y.^2 + 216.*y - 48).*x.^2 + (- 96.*y.^4 + 104.*y.^3 + 48.*y.^2 - 72.*y + 16).*x) + x.*(- 96.*y.^3 + 96.*y.^2 + 8.*y - 16) + z.^2.*(x.^2.*(- 288.*y.^4 + 312.*y.^3 + 144.*y.^2 - 216.*y + 48) - 20.*y - x.^3.*(- 384.*y.^4 + 704.*y.^3 - 384.*y.^2 + 64.*y) + x.*(96.*y.^3 - 96.*y.^2 + 40.*y - 16) + 48.*y.^2 - 48.*y.^3 + 8) - z.^5.*(((432.*y.^3)/5 - (864.*y.^2)/5 + (528.*y)/5 - 96/5).*x.^3 + (- (648.*y.^3)/5 + (1296.*y.^2)/5 - (792.*y)/5 + 144/5).*x.^2 + ((216.*y.^3)/5 - (432.*y.^2)/5 + (264.*y)/5 - 48/5).*x))./(-w).^3) .* (y>0.5); %! d2Fduv(3,:,:,:) = (((x.^2.*(48.*z.^2 - 48.*z.^4) - x.^4.*(16.*z.^4 - 48.*z.^2 + 32) + x.^3.*(48.*z.^4 - 96.*z.^2 + 32) + 16.*x.*z.^4).*y.^4 + (x.^2.*(- 48.*z.^5 + 48.*z.^3 + 144.*z.^2 - 144) - x.^3.*(- 32.*z.^5 + 32.*z.^3 + 48.*z.^2 - 48) + x.*(16.*z.^5 - 144.*z.^2 + 96) + 48.*z.^2).*y.^3 + (x.*(96.*z.^2 - 48) + x.^3.*(48.*z.^2 - 48) - x.^2.*(120.*z.^2 - 96) - 24.*z.^2).*y.^2 + (x.*(16.*z.^3 - 24) - 8.*z.^3 + 24).*y + 8.*x - 8)./w.^3) .* (y<0.5) + ... %! ((8.*y - x.^4.*(96.*y.^4 - 320.*y.^3 + 384.*y.^2 - 192.*y + 32) + x.^3.*(96.*y.^4 - 368.*y.^3 + 528.*y.^2 - 336.*y + 80) + z.^3.*((288.*y.^3 - 576.*y.^2 + 352.*y - 64).*x.^3 + (- 432.*y.^3 + 864.*y.^2 - 528.*y + 96).*x.^2 + (48.*y - 32).*x - 24.*y + 16) - x.*(96.*y.^3 - 240.*y.^2 + 200.*y - 56) - z.^4.*((48.*y.^4 - 160.*y.^3 + 192.*y.^2 - 96.*y + 16).*x.^4 + (- 144.*y.^4 + 480.*y.^3 - 576.*y.^2 + 288.*y - 48).*x.^3 + (144.*y.^4 - 480.*y.^3 + 576.*y.^2 - 288.*y + 48).*x.^2 + (- 48.*y.^4 + 160.*y.^3 - 192.*y.^2 + 96.*y - 16).*x) + z.^2.*(x.^4.*(144.*y.^4 - 480.*y.^3 + 576.*y.^2 - 288.*y + 48) - 96.*y + x.^2.*(144.*y.^4 - 624.*y.^3 + 984.*y.^2 - 672.*y + 168) - x.^3.*(288.*y.^4 - 1008.*y.^3 + 1296.*y.^2 - 720.*y + 144) + x.*(144.*y.^3 - 384.*y.^2 + 336.*y - 96) + 120.*y.^2 - 48.*y.^3 + 24) - z.^5.*((288.*y.^3 - 576.*y.^2 + 352.*y - 64).*x.^3 + (- 432.*y.^3 + 864.*y.^2 - 528.*y + 96).*x.^2 + (144.*y.^3 - 288.*y.^2 + 176.*y - 32).*x) + x.^2.*(144.*y.^3 - 384.*y.^2 + 336.*y - 96) - 8)./(-w).^3) .* (y>0.5); %! d2Fduw = zeros ([3, size(x)]); %! d2Fduw(1,:,:,:) = ((x.^2.*((24.*y.^4.*z.^2)/5 + 2.*z.*(8.*y.^4 + 4.*y.^2)) - (12.*y.^2.*z.^2)/5 + (24.*x.*y.^2.*z.^2)/5)./w.^2 - (2.*(4.*x.*y.^2.*z - 4.*x.^2.*y.^2.*z).*(x.*((8.*y.^2.*z.^3)/5 + 8.*y.^2) - (4.*y.^2.*z.^3)/5 + x.^2.*(z.^2.*(8.*y.^4 + 4.*y.^2) + (8.*y.^4.*z.^3)/5 - 4.*y.^2) + 2))./w.^3) .* (y<0.5) + ... %! (-((- (4.*(3.*y - 1).*(y - 1).*(36.*y.^4 - 96.*y.^3 + 88.*y.^2 - 32.*y + 4).*x.^4)/5 + (4.*(3.*y - 1).*(y - 1).*(36.*y.^4 - 96.*y.^3 + 100.*y.^2 - 48.*y + 8).*x.^3)/5 - (4.*(3.*y - 1).*(y - 1).*(18.*y.^2 - 24.*y + 6).*x.^2)/5 + (4.*(3.*y - 1).*(y - 1).*(6.*y.^2 - 8.*y + 2).*x)/5).*z.^4 + ((4.*x.^3.*(3.*y - 1).*(y - 1).*(360.*y.^4 - 960.*y.^3 + 820.*y.^2 - 240.*y + 20))/5 - (4.*x.^4.*(3.*y - 1).*(y - 1).*(360.*y.^4 - 960.*y.^3 + 820.*y.^2 - 240.*y + 20))/5).*z.^3 + (- (4.*(3.*y - 1).*(y - 1).*(108.*y.^4 - 288.*y.^3 + 264.*y.^2 - 96.*y + 12).*x.^4)/5 + (4.*(3.*y - 1).*(y - 1).*(36.*y.^2 - 48.*y + 12).*x.^3)/5 - (24.*(3.*y - 1).*(y - 1).*x)/5 + (12.*(3.*y - 1).*(y - 1))/5).*z.^2 + (- (4.*(3.*y - 1).*(y - 1).*(360.*y.^4 - 960.*y.^3 + 940.*y.^2 - 400.*y + 60).*x.^4)/5 + (4.*(3.*y - 1).*(y - 1).*(360.*y.^2 - 480.*y + 120).*x.^3)/5 - (4.*(3.*y - 1).*(y - 1).*(180.*y.^2 - 240.*y + 90).*x.^2)/5 + 16.*(3.*y - 1).*(y - 1).*x).*z)./(-w).^3) .* (y>0.5); %! d2Fduw(2,:,:,:) = ((2.*z.*(8.*x.^2.*y.^4 - y.^2.*(8.*y - 4.*y.^2) + (2.*x.*y.^2.*(40.*y - 20.*y.^2))/5) + 3.*z.^2.*((12.*x.^2.*y.^4)/5 + (12.*x.*y.^2)/5 - (6.*y.^2)/5))./w.^2 - (2.*(4.*x.*y.^2.*z - 4.*x.^2.*y.^2.*z).*(z.^2.*(8.*x.^2.*y.^4 - y.^2.*(8.*y - 4.*y.^2) + (2.*x.*y.^2.*(40.*y - 20.*y.^2))/5) + z.^3.*((12.*x.^2.*y.^4)/5 + (12.*x.*y.^2)/5 - (6.*y.^2)/5) + (2.*x.*y.^2.*(20.*y.^2 - 40.*y + 20))/5))./w.^3) .* (y<0.5) + ... %! (((6.*(3.*y.^2 - 4.*y + 1).*(18.*x.^2.*y.^2 - 24.*x.^2.*y + 6.*x.^2 - 6.*x + 3).*z.^2)/5 + (4.*(3.*y.^2 - 4.*y + 1).*(60.*x.^2.*y.^2 - 80.*x.^2.*y + 20.*x.^2 - 20.*x.*y.^2 - 10.*x + 10.*y.^2 + 5).*z)/5)./w.^2 - (2.*((2.*(3.*y.^2 - 4.*y + 1).*(18.*x.^2.*y.^2 - 24.*x.^2.*y + 6.*x.^2 - 6.*x + 3).*z.^3)/5 + (2.*(3.*y.^2 - 4.*y + 1).*(60.*x.^2.*y.^2 - 80.*x.^2.*y + 20.*x.^2 - 20.*x.*y.^2 - 10.*x + 10.*y.^2 + 5).*z.^2)/5 - (2.*(10.*x - 20.*x.*y.^2).*(3.*y.^2 - 4.*y + 1))/5).*(- 12.*z.*x.^2.*y.^2 + 16.*z.*x.^2.*y - 4.*z.*x.^2 + 12.*z.*x.*y.^2 - 16.*z.*x.*y + 4.*z.*x))./(-w).^3) .* (y>0.5); %! d2Fduw(3,:,:,:) = (- (2.*z.*(4.*y.*(2.*y.^2 - 3.*y.^3).*x.^2 - 4.*y.*(4.*y.^2 - 6.*y.^3).*x + 4.*y.*(2.*y.^2 - 3.*y.^3)) - 3.*z.^2.*(8.*x.^2.*y.^4 + 8.*x.*y.^2 - 4.*y.^2))./w.^2 - (2.*(4.*x.*y.^2.*z - 4.*x.^2.*y.^2.*z).*(4.*y.*(3.*y - 2) + z.^3.*(8.*x.^2.*y.^4 + 8.*x.*y.^2 - 4.*y.^2) - z.^2.*(4.*y.*(2.*y.^2 - 3.*y.^3).*x.^2 - 4.*y.*(4.*y.^2 - 6.*y.^3).*x + 4.*y.*(2.*y.^2 - 3.*y.^3)) + 4.*x.^2.*y.*(4.*y.^2 - 6.*y.^3) - 4.*x.*y.*(- 6.*y.^3 + 4.*y.^2 + 3.*y - 2)))./w.^3) .* (y<0.5) + ... %! ((2.*z.*(4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1).*x.^2 - 4.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 10.*y - 2).*x + 4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1)) + 3.*z.^2.*(4.*(y - 1).*(18.*y.^3 - 30.*y.^2 + 14.*y - 2).*x.^2 - 4.*(6.*y - 2).*(y - 1).*x + 4.*(3.*y - 1).*(y - 1)))./w.^2 - (2.*(z.^2.*(4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1).*x.^2 - 4.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 10.*y - 2).*x + 4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1)) - 4.*(y - 1).^2 + z.^3.*(4.*(y - 1).*(18.*y.^3 - 30.*y.^2 + 14.*y - 2).*x.^2 - 4.*(6.*y - 2).*(y - 1).*x + 4.*(3.*y - 1).*(y - 1)) + 4.*x.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 11.*y - 3) - 4.*x.^2.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 10.*y - 2)).*(- 12.*z.*x.^2.*y.^2 + 16.*z.*x.^2.*y - 4.*z.*x.^2 + 12.*z.*x.*y.^2 - 16.*z.*x.*y + 4.*z.*x))./(-w).^3) .* (y>0.5); %! d2Fdvv = zeros ([3, size(x)]); %! d2Fdvv(1,:,:,:) = (-(8.*x.*(x - 1).*(z.^3 + 5.*x.*z.^2 - 5.*x).*(- 6.*x.^2.*y.^2.*z.^2 + 6.*x.^2.*y.^2 + 6.*x.*y.^2.*z.^2 - 1))/5./w.^3) .* (y<0.5) + ... %! ((8.*x.*(x - 1).*(z.^3 + 5.*x.*z.^2 - 5.*x).*(- 54.*x.^2.*y.^2.*z.^2 + 54.*x.^2.*y.^2 + 72.*x.^2.*y.*z.^2 - 72.*x.^2.*y - 26.*x.^2.*z.^2 + 26.*x.^2 + 54.*x.*y.^2.*z.^2 - 72.*x.*y.*z.^2 + 26.*x.*z.^2 + 3))/5./(-w).^3) .* (y>0.5); %! d2Fdvv(2,:,:,:) = ((2.*((8.*x.*z.^2 - x.^2.*(8.*z.^2 - 8)).*y.^2 + ((12.*x.*z.^3)/5 - x.^2.*((12.*z.^3)/5 + 8) + 4).*y - 4).*(- 4.*y.*x.^2.*z.^2 + 4.*y.*x.^2 + 4.*y.*x.*z.^2))./w.^3 - ((12.*x.*z.^3)/5 + 2.*y.*(8.*x.*z.^2 - x.^2.*(8.*z.^2 - 8)) - x.^2.*((12.*z.^3)/5 + 8) + 4)./w.^2) .* (y<0.5) + ... %! ((z.^2.*(x.*(32.*y + 4) - x.^2.*(32.*y + 4)) + x.^2.*(32.*y - 20) + z.^3.*((36.*x)/5 - (36.*x.^2)/5) + 4)./w.^2 - (2.*(4.*y + z.^3.*(x.*((36.*y)/5 - 24/5) - x.^2.*((36.*y)/5 - 24/5)) + z.^2.*(x.*(16.*y.^2 + 4.*y - 8) - x.^2.*(16.*y.^2 + 4.*y - 8)) + x.^2.*(16.*y.^2 - 20.*y + 8)).*(8.*x.^2.*z.^2 + 12.*x.^2.*y - 8.*x.*z.^2 - 8.*x.^2 + 12.*x.*y.*z.^2 - 12.*x.^2.*y.*z.^2))./(-w).^3) .* (y>0.5); %! d2Fdvv(3,:,:,:) = ((2.*y.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 4.*(3.*x - 3).*(x - 1) + 8.*x.*z.^3.*(x - 1))./w.^2 - (2.*(4.*(x - 1).^2 - y.*(4.*(3.*x - 3).*(x - 1) - 8.*x.*z.^3.*(x - 1)) + y.^2.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1))).*(- 4.*y.*x.^2.*z.^2 + 4.*y.*x.^2 + 4.*y.*x.*z.^2))./w.^3) .* (y<0.5) + ... %! ((4.*(x - 1).*(4.*x.^3 - 4.*x.^2 + x - 1) + 2.*y.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 24.*x.*z.^3.*(x - 1) - 4.*z.^2.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x))./w.^2 - (2.*(y.^2.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 4.*(x - 1).*(2.*x.^3 - 2.*x.^2 + x - 1) - y.*(24.*x.*(x - 1).*z.^3 + 4.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x).*z.^2 - 4.*(x - 1).*(4.*x.^3 - 4.*x.^2 + x - 1)) + 16.*x.*z.^3.*(x - 1) + 4.*z.^2.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x)).*(8.*x.^2.*z.^2 + 12.*x.^2.*y - 8.*x.*z.^2 - 8.*x.^2 + 12.*x.*y.*z.^2 - 12.*x.^2.*y.*z.^2))./(-w).^3) .* (y>0.5); %! d2Fdvw = zeros ([3, size(x)]); %! d2Fdvw(1,:,:,:) = (((8.*x.*z.*(x - 1).*(20.*x.^3.*z.^2 - 20.*x.^3 + 2.*x.^2.*z.^3 - 20.*x.^2.*z.^2 + 6.*x.^2.*z + 40.*x.^2 - 2.*x.*z.^3).*y.^3)/5 + (8.*x.*z.*(10.*x + 3.*z).*(x - 1).*y)/5)./w.^3) .* (y<0.5) + ... %! (((8.*x.*(3.*y - 2).*(x - 1).*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).*z.^4)/5 + (8.*x.*(3.*y - 2).*(x - 1).*(- 60.*x.^3.*y.^2 + 80.*x.^3.*y - 20.*x.^3 + 60.*x.^2.*y.^2 - 80.*x.^2.*y + 20.*x.^2).*z.^3)/5 - (8.*x.*(3.*y - 2).*(x - 1).*(18.*x.^2.*y.^2 - 24.*x.^2.*y + 6.*x.^2 - 3).*z.^2)/5 + (8.*x.*(3.*y - 2).*(x - 1).*(60.*x.^3.*y.^2 - 80.*x.^3.*y + 20.*x.^3 - 120.*x.^2.*y.^2 + 160.*x.^2.*y - 40.*x.^2 + 10.*x).*z)/5)./(-w).^3) .* (y>0.5); %! d2Fdvw(2,:,:,:) = ((4.*x.*y.*z.*(x - 1).*(40.*x.^2.*y.^3.*z.^2 - 40.*x.^2.*y.^3 + 6.*x.^2.*y.^2.*z.^3 + 18.*x.^2.*y.^2.*z + 80.*x.^2.*y.^2 - 40.*x.*y.^3.*z.^2 - 6.*x.*y.^2.*z.^3 - 40.*y.^2 + 60.*y + 9.*z))/5./w.^3) .* (y<0.5) + ... %! (-((4.*x.*(x - 1).*(54.*x.^2.*y.^3 - 108.*x.^2.*y.^2 + 66.*x.^2.*y - 12.*x.^2 - 54.*x.*y.^3 + 108.*x.*y.^2 - 66.*x.*y + 12.*x).*z.^4)/5 + (4.*x.*(x - 1).*(240.*x.^2.*y.^4 - 260.*x.^2.*y.^3 - 120.*x.^2.*y.^2 + 180.*x.^2.*y - 40.*x.^2 - 240.*x.*y.^4 + 260.*x.*y.^3 + 120.*x.*y.^2 - 180.*x.*y + 40.*x).*z.^3)/5 - (4.*x.*(x - 1).*(- 162.*x.^2.*y.^3 + 324.*x.^2.*y.^2 - 198.*x.^2.*y + 36.*x.^2 + 27.*y - 18).*z.^2)/5 - (4.*x.*(x - 1).*(240.*x.^2.*y.^4 - 980.*x.^2.*y.^3 + 1320.*x.^2.*y.^2 - 700.*x.^2.*y + 120.*x.^2 + 120.*y.^3 - 120.*y.^2 + 50.*y - 20).*z)/5)./(-w).^3) .* (y>0.5); %! d2Fdvw(3,:,:,:) = (-(y.^3.*(8.*x.*z.*(x - 1).*(12.*x.^2 - 24.*x + 12) - 48.*x.^3.*z.^2.*(x - 1) + 8.*x.*z.^4.*(2.*x - 2.*x.^2).*(x - 1)) + y.^4.*(8.*x.*(x - 1).*(- 4.*x.^4 + 12.*x.^3 - 12.*x.^2 + 4.*x).*z.^3 + 8.*x.*(x - 1).*(4.*x.^4 - 8.*x.^3 + 4.*x.^2).*z) - 24.*x.*y.*z.^2.*(x - 1) - 8.*x.*y.^2.*z.*(x - 1).*(6.*x.^2 - 12.*x + 6))./w.^3) .* (y<0.5) + ... %! ((8.*z.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x) - y.*(72.*x.*(x - 1).*z.^2 + 8.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x).*z) + 48.*x.*z.^2.*(x - 1) + 8.*y.^2.*z.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x))./w.^2 - (2.*(y.^2.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 4.*(x - 1).*(2.*x.^3 - 2.*x.^2 + x - 1) - y.*(24.*x.*(x - 1).*z.^3 + 4.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x).*z.^2 - 4.*(x - 1).*(4.*x.^3 - 4.*x.^2 + x - 1)) + 16.*x.*z.^3.*(x - 1) + 4.*z.^2.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x)).*(- 12.*z.*x.^2.*y.^2 + 16.*z.*x.^2.*y - 4.*z.*x.^2 + 12.*z.*x.*y.^2 - 16.*z.*x.*y + 4.*z.*x))./(-w).^3) .* (y>0.5); %! d2Fdww = zeros ([3, size(x)]); %! d2Fdww(1,:,:,:) = ((32.*x.*y.^2.*(2.*x.^2.*y.^2 + 1).*(x - 1).*(5.*x + z + 10.*x.^2.*y.^2 + 2.*x.^2.*y.^2.*z))./(5.*w.^3) - (8.*x.*y.^2.*(x - 1).*(15.*x + z + 30.*x.^2.*y.^2 + 2.*x.^2.*y.^2.*z))/5./w.^2) .* (y<0.5) + ... %! (((8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(36.*x.^4.*y.^4 - 96.*x.^4.*y.^3 + 88.*x.^4.*y.^2 - 32.*x.^4.*y + 4.*x.^4 - 36.*x.^3.*y.^4 + 96.*x.^3.*y.^3 - 88.*x.^3.*y.^2 + 32.*x.^3.*y - 4.*x.^3 - 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).*z.^3)/5 + (8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(540.*x.^4.*y.^4 - 1440.*x.^4.*y.^3 + 1320.*x.^4.*y.^2 - 480.*x.^4.*y + 60.*x.^4 - 540.*x.^3.*y.^4 + 1440.*x.^3.*y.^3 - 1410.*x.^3.*y.^2 + 600.*x.^3.*y - 90.*x.^3 + 90.*x.^2.*y.^2 - 120.*x.^2.*y + 30.*x.^2).*z.^2)/5 + (8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(108.*x.^4.*y.^4 - 288.*x.^4.*y.^3 + 264.*x.^4.*y.^2 - 96.*x.^4.*y + 12.*x.^4 - 36.*x.^2.*y.^2 + 48.*x.^2.*y - 12.*x.^2 + 3).*z)/5 + (8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(180.*x.^4.*y.^4 - 480.*x.^4.*y.^3 + 440.*x.^4.*y.^2 - 160.*x.^4.*y + 20.*x.^4 - 30.*x.^3.*y.^2 + 40.*x.^3.*y - 10.*x.^3 - 30.*x.^2.*y.^2 + 40.*x.^2.*y - 10.*x.^2 + 5.*x))/5)./(-w).^3) .* (y>0.5); %! d2Fdww(2,:,:,:) = ((16.*x.*y.^2.*(2.*x.^2.*y.^2 + 1).*(x - 1).*(20.*y + 3.*z + 20.*x.^2.*y.^2 - 10.*y.^2 + 6.*x.^2.*y.^2.*z))./(5.*w.^3) - (12.*x.*y.^2.*(x - 1).*(20.*y + z + 20.*x.^2.*y.^2 - 10.*y.^2 + 2.*x.^2.*y.^2.*z))/5./w.^2) .* (y<0.5) + ... %! (((4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(108.*x.^4.*y.^4 - 288.*x.^4.*y.^3 + 264.*x.^4.*y.^2 - 96.*x.^4.*y + 12.*x.^4 - 108.*x.^3.*y.^4 + 288.*x.^3.*y.^3 - 264.*x.^3.*y.^2 + 96.*x.^3.*y - 12.*x.^3 - 18.*x.^2.*y.^2 + 24.*x.^2.*y - 6.*x.^2 + 18.*x.*y.^2 - 24.*x.*y + 6.*x).*z.^3)/5 + (4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(1080.*x.^4.*y.^4 - 2880.*x.^4.*y.^3 + 2640.*x.^4.*y.^2 - 960.*x.^4.*y + 120.*x.^4 - 1080.*x.^3.*y.^4 + 2880.*x.^3.*y.^3 - 2640.*x.^3.*y.^2 + 960.*x.^3.*y - 120.*x.^3 - 180.*x.^2.*y.^4 + 240.*x.^2.*y.^3 - 150.*x.^2.*y.^2 + 120.*x.^2.*y - 30.*x.^2 + 180.*x.*y.^4 - 240.*x.*y.^3 + 150.*x.*y.^2 - 120.*x.*y + 30.*x).*z.^2)/5 + (4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(324.*x.^4.*y.^4 - 864.*x.^4.*y.^3 + 792.*x.^4.*y.^2 - 288.*x.^4.*y + 36.*x.^4 - 108.*x.^2.*y.^2 + 144.*x.^2.*y - 36.*x.^2 + 9).*z)/5 + (4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(360.*x.^4.*y.^4 - 960.*x.^4.*y.^3 + 880.*x.^4.*y.^2 - 320.*x.^4.*y + 40.*x.^4 - 60.*x.^2.*y.^4 + 80.*x.^2.*y.^3 - 110.*x.^2.*y.^2 + 120.*x.^2.*y - 30.*x.^2 + 10.*y.^2 + 5))/5)./(-w).^3) .* (y>0.5); %! d2Fdww(3,:,:,:) = ((32.*x.*y.^2.*(2.*x.^2.*y.^2 + 1).*(x - 1).*(2.*y + z - 3.*x.^2.*y.^2 - 4.*x.*y + 6.*x.*y.^2 + 2.*x.^2.*y - 3.*y.^2 + 2.*x.^2.*y.^2.*z))./w.^3 - (8.*x.*y.^2.*(x - 1).*(6.*y + z - 9.*x.^2.*y.^2 - 12.*x.*y + 18.*x.*y.^2 + 6.*x.^2.*y - 9.*y.^2 + 2.*x.^2.*y.^2.*z))./w.^2) .* (y<0.5) + ... %! ((2.*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).*(2.*z - 4.*y - 4.*x + 2.*x.^2.*y.^2 + 8.*x.*y - 4.*x.*y.^2 - 4.*x.^2.*y - 4.*x.^2.*z + 2.*x.^2 + 2.*y.^2 + 16.*x.^2.*y.*z - 12.*x.^2.*y.^2.*z + 2))./w.^2 - (8.*z.^2.*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).^2.*(2.*z - 4.*y - 4.*x + 2.*x.^2.*y.^2 + 8.*x.*y - 4.*x.*y.^2 - 4.*x.^2.*y - 4.*x.^2.*z + 2.*x.^2 + 2.*y.^2 + 16.*x.^2.*y.*z - 12.*x.^2.*y.^2.*z + 2))./(-w).^3 - (4.*z.*(12.*x.^2.*y.^2 - 16.*x.^2.*y + 4.*x.^2 - 2).*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x))./w.^2) .* (y>0.5); %! assert (F, pnt, 1e3*eps) %! assert (dFdu, jac{1}, 1e3*eps) %! assert (dFdv, jac{2}, 1e3*eps) %! assert (dFdw, jac{3}, 1e3*eps) %! assert (d2Fduu, hess{1,1}, 1e3*eps) %! assert (d2Fduv, hess{1,2}, 1e3*eps) %! assert (d2Fduw, hess{1,3}, 1e3*eps) %! assert (d2Fduv, hess{2,1}, 1e3*eps) %! assert (d2Fdvv, hess{2,2}, 1e3*eps) %! assert (d2Fdvw, hess{2,3}, 1e3*eps) %! assert (d2Fduw, hess{3,1}, 1e3*eps) %! assert (d2Fdvw, hess{3,2}, 1e3*eps) %! assert (d2Fdww, hess{3,3}, 1e3*eps) %!test %! nrb = nrbextrude (nrb4surf ([0 0], [1 0], [0 1], [1 1]), [0 0 1]); %! nrb = nrbdegelev (nrb, [1 1 1]); %! nrb.coefs (4,2,2,2) = 1.1; %! [dnrb, dnrb2] = nrbderiv (nrb); %! X = linspace (0, 1, 24); Y = linspace (0, 1, 24); Z = linspace (0, 1, 24); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, {X Y Z}); %! [y, x, z] = meshgrid (X, Y, Z); %! F = zeros ([3, size(x)]); %! F(1,:,:,:) = (5.*x)./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5); %! F(2,:,:,:) = (5.*y)./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5); %! F(3,:,:,:) = (5.*z)./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5); %! dFdu = zeros ([3, size(x)]); %! dFdu(1,:,:,:) = ((z.*(20.*y - 20.*y.^2) - z.^2.*(20.*y - 20.*y.^2)).*x.^2 + 25)./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^2; %! dFdu(2,:,:,:) = (y.^2.*(5.*z.*(8.*x - 4) - 5.*z.^2.*(8.*x - 4)) - y.^3.*(5.*z.*(8.*x - 4) - 5.*z.^2.*(8.*x - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdu(3,:,:,:) = (z.^2.*(5.*y.*(8.*x - 4) - 5.*y.^2.*(8.*x - 4)) - z.^3.*(5.*y.*(8.*x - 4) - 5.*y.^2.*(8.*x - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdv = zeros ([3, size(x)]); %! dFdv(1,:,:,:) = (x.^2.*(5.*z.*(8.*y - 4) - 5.*z.^2.*(8.*y - 4)) - x.^3.*(5.*z.*(8.*y - 4) - 5.*z.^2.*(8.*y - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdv(2,:,:,:) = ((z.*(20.*x - 20.*x.^2) - z.^2.*(20.*x - 20.*x.^2)).*y.^2 + 25)./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^2; %! dFdv(3,:,:,:) = (z.^2.*(5.*x.*(8.*y - 4) - 5.*x.^2.*(8.*y - 4)) - z.^3.*(5.*x.*(8.*y - 4) - 5.*x.^2.*(8.*y - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdw = zeros ([3, size(x)]); %! dFdw(1,:,:,:) = (x.^2.*(y.*(40.*z - 20) - y.^2.*(40.*z - 20)) - x.^3.*(y.*(40.*z - 20) - y.^2.*(40.*z - 20)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdw(2,:,:,:) = (y.^2.*(x.*(40.*z - 20) - x.^2.*(40.*z - 20)) - y.^3.*(x.*(40.*z - 20) - x.^2.*(40.*z - 20)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdw(3,:,:,:) = ((y.*(20.*x - 20.*x.^2) - y.^2.*(20.*x - 20.*x.^2)).*z.^2 + 25)./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^2; %! d2Fduu = zeros ([3, size(x)]); %! d2Fduu(1,:,:,:) = (40.*y.*z.*(y - 1).*(z - 1).*(4.*x.^3.*y.^2.*z.^2 - 4.*x.^3.*y.^2.*z - 4.*x.^3.*y.*z.^2 + 4.*x.^3.*y.*z + 15.*x - 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduu(2,:,:,:) = (40.*y.^2.*z.*(y - 1).*(z - 1).*(4.*y.^2.*z.^2 - 4.*y.^2.*z - 4.*y.*z.^2 + 4.*y.*z + 5) - 40.*x.*y.^2.*z.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z) + 40.*x.^2.*y.^2.*z.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduu(3,:,:,:) = (40.*y.*z.^2.*(y - 1).*(z - 1).*(4.*y.^2.*z.^2 - 4.*y.^2.*z - 4.*y.*z.^2 + 4.*y.*z + 5) - 40.*x.*y.*z.^2.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z) + 40.*x.^2.*y.*z.^2.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduv = zeros ([3, size(x)]); %! d2Fduv(1,:,:,:) = (20.*x.*z.*(2.*y - 1).*(z - 1).*(4.*x.^3.*y.^2.*z.^2 - 4.*x.^3.*y.^2.*z - 4.*x.^3.*y.*z.^2 + 4.*x.^3.*y.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 15.*x - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduv(2,:,:,:) = (20.*y.*z.*(2.*x - 1).*(z - 1).*(4.*x.^2.*y.^3.*z.^2 - 4.*x.^2.*y.^3.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z - 4.*x.*y.^3.*z.^2 + 4.*x.*y.^3.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z + 15.*y - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduv(3,:,:,:) = (20.*z.^2.*(2.*x - 1).*(2.*y - 1).*(z - 1).*(4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.^2.*z - 4.*x.^2.*y.*z.^2 + 4.*x.^2.*y.*z - 4.*x.*y.^2.*z.^2 + 4.*x.*y.^2.*z + 4.*x.*y.*z.^2 - 4.*x.*y.*z + 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduw = zeros ([3, size(x)]); %! d2Fduw(1,:,:,:) = (20.*x.*y.*(2.*z - 1).*(y - 1).*(4.*x.^3.*y.^2.*z.^2 - 4.*x.^3.*y.^2.*z - 4.*x.^3.*y.*z.^2 + 4.*x.^3.*y.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 15.*x - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduw(2,:,:,:) = (20.*y.^2.*(2.*x - 1).*(2.*z - 1).*(y - 1).*(4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.^2.*z - 4.*x.^2.*y.*z.^2 + 4.*x.^2.*y.*z - 4.*x.*y.^2.*z.^2 + 4.*x.*y.^2.*z + 4.*x.*y.*z.^2 - 4.*x.*y.*z + 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduw(3,:,:,:) = (20.*y.*z.*(2.*x - 1).*(y - 1).*(4.*x.^2.*y.^2.*z.^3 - 4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.*z.^3 + 4.*x.^2.*y.*z.^2 - 4.*x.*y.^2.*z.^3 + 4.*x.*y.^2.*z.^2 + 4.*x.*y.*z.^3 - 4.*x.*y.*z.^2 + 15.*z - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvv = zeros ([3, size(x)]); %! d2Fdvv(1,:,:,:) = (40.*x.^2.*z.*(x - 1).*(z - 1).*(4.*x.^2.*z.^2 - 4.*x.^2.*z - 4.*x.*z.^2 + 4.*x.*z + 5) + 40.*x.^2.*y.^2.*z.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z) - 40.*x.^2.*y.*z.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvv(2,:,:,:) = (40.*x.*z.*(x - 1).*(z - 1).*(4.*x.^2.*y.^3.*z.^2 - 4.*x.^2.*y.^3.*z - 4.*x.*y.^3.*z.^2 + 4.*x.*y.^3.*z + 15.*y - 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvv(3,:,:,:) = (40.*x.*z.^2.*(x - 1).*(z - 1).*(4.*x.^2.*z.^2 - 4.*x.^2.*z - 4.*x.*z.^2 + 4.*x.*z + 5) + 40.*x.*y.^2.*z.^2.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z) - 40.*x.*y.*z.^2.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvw = zeros ([3, size(x)]); %! d2Fdvw(1,:,:,:) = (20.*x.^2.*(2.*y - 1).*(2.*z - 1).*(x - 1).*(4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.^2.*z - 4.*x.^2.*y.*z.^2 + 4.*x.^2.*y.*z - 4.*x.*y.^2.*z.^2 + 4.*x.*y.^2.*z + 4.*x.*y.*z.^2 - 4.*x.*y.*z + 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvw(2,:,:,:) = (20.*x.*y.*(2.*z - 1).*(x - 1).*(4.*x.^2.*y.^3.*z.^2 - 4.*x.^2.*y.^3.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z - 4.*x.*y.^3.*z.^2 + 4.*x.*y.^3.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z + 15.*y - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvw(3,:,:,:) = (20.*x.*z.*(2.*y - 1).*(x - 1).*(4.*x.^2.*y.^2.*z.^3 - 4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.*z.^3 + 4.*x.^2.*y.*z.^2 - 4.*x.*y.^2.*z.^3 + 4.*x.*y.^2.*z.^2 + 4.*x.*y.*z.^3 - 4.*x.*y.*z.^2 + 15.*z - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdww = zeros ([3, size(x)]); %! d2Fdww(1,:,:,:) = (40.*x.^2.*y.*(x - 1).*(y - 1).*(4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y + 5) + 40.*x.^2.*y.*z.^2.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y) - 40.*x.^2.*y.*z.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdww(2,:,:,:) = (40.*x.*y.^2.*(x - 1).*(y - 1).*(4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y + 5) + 40.*x.*y.^2.*z.^2.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y) - 40.*x.*y.^2.*z.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdww(3,:,:,:) = (40.*x.*y.*(x - 1).*(y - 1).*(4.*x.^2.*y.^2.*z.^3 - 4.*x.^2.*y.*z.^3 - 4.*x.*y.^2.*z.^3 + 4.*x.*y.*z.^3 + 15.*z - 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %!test %! knots = [0 0 0 0.5 1 1 1]; %! coefs(1,:) = [0 2 4 2]; %! coefs(2,:) = [0 2 2 0]; %! coefs(3,:) = [0 4 2 0]; %! coefs(4,:) = [1 2 2 1]; %! nrb = nrbmak (coefs, knots); %! [dnrb, dnrb2] = nrbderiv (nrb); %! x = linspace (0, 1, 10); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, x); %! w = -4*x.^2 + 4*x + 1; %! F = zeros (3,numel(x)); DF = zeros (3, numel(x)); D2F = zeros (3, numel(x)); %! F(1,:) = (-4*x.*(x-2)./w) .* (x<0.5) + ((4*x - 5)./w + 3) .* (x>0.5); %! F(2,:) = (2-2./w); %! F(3,:) = (-4*x.*(5*x-4)./w) .* (x<0.5) + (-4*(x.^2 - 1)./w) .* (x>0.5); %! DF(1,:) = (8*(2*x.^2-x+1)./w.^2) .* (x<0.5) + (8*(2*x-3).*(x-1)./w.^2) .* (x>0.5); %! DF(2,:) = -8*(2*x-1)./w.^2; %! DF(3,:) = -(8*(2*x.^2+5*x-2)./w.^2) .* (x<0.5) - (8*(2*x.^2-3*x+2)./w.^2) .* (x>0.5); %! D2F(1,:) = 8*(16*x.^3-12*x.^2+24*x-9)./w.^3 .* (x<0.5) + 8*(16*x.^3-60*x.^2+72*x-29)./w.^3 .* (x>0.5); %! D2F(2,:) = -16*(12*x.^2-12*x+5)./w.^3; %! D2F(3,:) = -8*(16*x.^3+60*x.^2-48*x+21)./w.^3 .* (x<0.5) -8*(16*x.^3-36*x.^2+48*x-19)./w.^3 .* (x>0.5); %! assert (F, pnt, 1e3*eps) %! assert (DF, jac, 1e3*eps) %! assert (D2F, hess, 1e3*eps) %!test %! knots = {[0 0 0 1 1 1], [0 0 0 0.5 1 1 1]}; %! coefs = ones (4,3,4); %! coefs(1,:,:) = reshape ([0 0 0 0; 1 1 1 1; 2 2 4 2], 1, 3, 4); %! coefs(2,:,:) = reshape ([0 1 2 3; 0 1 2 3; 0 1 4 3], 1, 3, 4); %! coefs(3,:,:) = reshape ([0 1 0 0; 0 0 0 0; 0 0 0 0], 1, 3, 4); %! coefs(4,:,:) = reshape ([1 1 1 1; 1 1 1 1; 1 1 2 1], 1, 3, 4); %! nrb = nrbmak (coefs, knots); %! [dnrb, dnrb2] = nrbderiv (nrb); %! X = linspace (0, 1, 4); Y = linspace (0, 1, 4); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, {X Y}); %! [y, x] = meshgrid (X, Y); %! w = (2*x.^2.*y.^2 + 1) .* (y < 0.5) + (-6*x.^2.*y.^2 + 8*x.^2.*y - 2*x.^2 + 1) .* (y > 0.5); %! F = zeros ([3,size(x)]); %! F(1,:,:) = ((2*x - 2) ./w + 2) .* (y<0.5) + (2 + (2*x-2)./w) .* (y > 0.5); %! F(2,:,:) = (2 - (2*(y-1).^2)./w).*(y<0.5) + ... %! ((-12*x.^2.*y.^2 + 16*x.^2.*y - 4*x.^2 + 2*y.^2 + 1)./w).*(y>0.5); %! F(3,:,:) = (-2*y.*(3*y - 2).*(x - 1).^2./w) .* (y<0.5) + ... %! (2*(x - 1).^2.*(y - 1).^2./w) .* (y>0.5); %! dFdu = zeros ([3,size(x)]); %! dFdu(1,:,:) = (((8*x - 4*x.^2).*y.^2 + 2)./w.^2).*(y<0.5) + ... %! (((12*y.^2 - 16*y + 4).*x.^2 + (-24*y.^2 + 32*y - 8).*x + 2)./w.^2).*(y>0.5); %! dFdu(2,:,:) = (8*x.*y.^2.*(y - 1).^2./w.^2).*(y<0.5) + ... %! ((4*x.*(3*y - 1).*(2*y.^2 - 1).*(y - 1))./w.^2).*(y>0.5); %! dFdu(3,:,:) = (-4*y.*(2.*x.*y.^2 + 1).*(3*y - 2).*(x - 1)./w.^2).*(y<0.5) + ... %! ((-4*(x - 1).*(y - 1).^2.*(6*x.*y.^2 - 8*x.*y + 2*x - 1))./w.^2).*(y>0.5); %! dFdv = zeros ([3,size(x)]); %! dFdv(1,:,:) = (-8*x.^2.*y.*(x - 1)./w.^2).*(y<0.5) + ... %! (8*x.^2.*(3*y - 2).*(x - 1)./w.^2).*(y>0.5); %! dFdv(2,:,:) = (-4*(2*y.*x.^2 + 1).*(y - 1)./w.^2).*(y<0.5) + ... %! (((16*y.^2 - 20*y + 8).*x.^2 + 4*y)./w.^2).*(y>0.5); %! dFdv(3,:,:) = (-4*(x - 1).^2.*(2*x.^2.*y.^2 + 3*y - 1)./w.^2).*(y<0.5) + ... %! (4*(x - 1).^2.*(y - 1).*(2*x.^2 - 2*x.^2.*y + 1)./w.^2).*(y>0.5); %! d2Fduu = zeros ([3, size(x)]); %! d2Fduu(1,:,:) = (-((48*x.^2 - 16*x.^3).*y.^4 + (24*x - 8).*y.^2)./w.^3).*(y<0.5) + ... %! (((32*(3*y - 1).*(x - 1).*(y - 1))-(8*(3*y - 1).*(x - 3).*(y - 1).*w))./w.^3).*(y>0.5); %! d2Fduu(2,:,:) = (-(8*y.^2.*(6*x.^2.*y.^2 - 1).*(y - 1).^2)./w.^3).*(y<0.5) + ... %! ((4*(3*y - 1).*(2*y.^2 - 1).*(y - 1).*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 + 1))./w.^3).*(y>0.5); %! d2Fduu(3,:,:) = ((4*y.*(3*y - 2).*(8*x.^3.*y.^4 - 12*x.^2.*y.^4 + 6*x.^2.*y.^2 - 12*x.*y.^2 + 2*y.^2 - 1))./w.^3).*(y<0.5) + ... %! ((4*(y - 1).^2.*(6*y.^2 - 8*y + 3) - 4*x.^3.*(y - 1).^2.*(72*y.^4 - 192*y.^3 + 176*y.^2 - 64*y + 8) + 4*x.^2.*(y - 1).^2.*(108*y.^4 - 288*y.^3 + 282*y.^2 - 120*y + 18) - 4*x.*(y - 1).^2.*(36*y.^2 - 48*y + 12))./w.^3) .* (y>0.5); %! d2Fdvv = zeros ([3, size(x)]); %! d2Fdvv(1,:,:) = (8*x.^2.*(6*x.^2.*y.^2 - 1).*(x - 1)./w.^3) .* (y<0.5) + ... %! (8*x.^2.*(x - 1).*(54*x.^2.*y.^2 - 72*x.^2.*y + 26*x.^2 + 3)./w.^3) .* (y>0.5); %! d2Fdvv(2,:,:) = (-((48*y.^2 - 32*y.^3).*x.^4 + (- 24*y.^2 + 48*y - 8).*x.^2 + 4)./w.^3) .*(y<0.5) + ... %! (((192*y.^3 - 360*y.^2 + 288*y - 88).*x.^4 + (72*y.^2 - 28).*x.^2 + 4)./w.^3) .* (y>0.5); %! d2Fdvv(3,:,:) = (4*(x - 1).^2.*(8*x.^4.*y.^3 + 18*x.^2.*y.^2 - 12*x.^2.*y - 3))./w.^3 .* (y<0.5) + ... %! ((4*(x - 1).^2.*(24*x.^4 + 18*x.^2 + 1) + 4*y.^2.*(72*x.^4 + 18*x.^2).*(x - 1).^2 - 96*x.^4.*y.^3.*(x - 1).^2 - 4*y.*(72*x.^4 + 36*x.^2).*(x - 1).^2)./w.^3) .* (y>0.5); %! d2Fduv = zeros ([3, size(x)]); %! d2Fduv(1,:,:) = (-(y.^3.*(32*x.^3 - 16*x.^4) - y.*(16*x - 24*x.^2))./w.^3) .* (y<0.5) + ... %! (-(-8*(3*y - 2).*(6*y.^2 - 8*y + 2).*x.^4 + 8*(3*y - 2).*(12*y.^2 - 16*y + 4).*x.^3 + (48 - 72*y).*x.^2 + (48*y - 32).*x)./w.^3) .* (y>0.5); %! d2Fduv(2,:,:) = (16*x.*y.*(y - 1).*(2*x.^2.*y.^2 + 2*y - 1)./w.^3) .* (y<0.5) + ... %! (-(8*x.*(4*y.^2 - 5*y + 2))./w.^2 + (16*x.*(3*y - 2).*(2*y.^2 - 1))./w.^3) .* (y>0.5); %! d2Fduv(3,:,:) = (-(8*(x - 1).*(4*x.^3.*y.^4 - 6*x.^2.*y.^3 + 6*x.^2.*y.^2 + 12*x.*y.^3 - 6*x.*y.^2 + 3*y - 1))./w.^3) .* (y<0.5) + ... %! ((8*(x - 1).*(y - 1).*(12*x.^3.*y.^3 - 28*x.^3.*y.^2 + 20*x.^3.*y - 4*x.^3 + 6*x.^2.*y.^2 - 12*x.^2.*y + 6*x.^2 - 12*x.*y.^2 + 18*x.*y - 6*x + 1))./w.^3) .* (y>0.5); %! assert (F, pnt, 1e3*eps) %! assert (dFdu, jac{1}, 1e3*eps) %! assert (dFdv, jac{2}, 1e3*eps) %! assert (d2Fduu, hess{1,1}, 1e3*eps) %! assert (d2Fduv, hess{1,2}, 1e3*eps) %! assert (d2Fduv, hess{2,1}, 1e3*eps) %! assert (d2Fdvv, hess{2,2}, 1e3*eps) %!test %! knots = {[0 0 0 1 1 1], [0 0 0 0.5 1 1 1]}; %! coefs = ones (4,3,4); %! coefs(1,:,:) = reshape ([0 0 0 0; 1 1 1 1; 2 2 4 2], 1, 3, 4); %! coefs(2,:,:) = reshape ([0 1 2 3; 0 1 2 3; 0 1 4 3], 1, 3, 4); %! coefs(3,:,:) = reshape ([0 1 0 0; 0 0 0 0; 0 0 0 0], 1, 3, 4); %! coefs(4,:,:) = reshape ([1 1 1 1; 1 1 1 1; 1 1 2 1], 1, 3, 4); %! nrb = nrbmak (coefs, knots); %! nrb = nrbdegelev (nrbextrude (nrb, [0.4 0.6 2]), [0 0 1]); %! nrb.coefs(4,2,3,3) = 1.5; %! [dnrb, dnrb2] = nrbderiv (nrb); %! X = linspace (0, 1, 4); Y = linspace (0, 1, 4); Z = linspace (0, 1, 4); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, {X Y Z}); %! [y, x, z] = meshgrid (X, Y, Z); %! w = (-2*x.^2.*y.^2.*z.^2 + 2*x.^2.*y.^2 + 2*x.*y.^2.*z.^2 + 1) .* (y < 0.5) + ... %! (6*x.^2.*y.^2.*z.^2 - 6*x.^2.*y.^2 - 8*x.^2.*y.*z.^2 + 8*x.^2.*y + 2*x.^2.*z.^2 - 2*x.^2 - 6*x.*y.^2.*z.^2 + 8*x.*y.*z.^2 - 2*x.*z.^2 + 1) .* (y > 0.5); %! F = zeros ([3,size(x)]); %! F(1,:,:,:) = ((10*x + 20*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2))./(5*w)) .* (y<0.5) + ... %! (60*x.^2.*y.^2 - 10*x + z.*(12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2) - 80*x.^2.*y + 20*x.^2)./(-5*w) .* (y > 0.5); %! F(2,:,:,:) = ((20*y + 20*x.^2.*y.^2 + z.*(6*x.^2.*y.^2 + 3) - 10*y.^2)./(5*w)).*(y<0.5) + ... %! ((60*x.^2.*y.^2 + z.*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 3) - 80*x.^2.*y + 20*x.^2 - 10*y.^2 - 5)./(-5*w)).*(y>0.5); %! F(3,:,:,:) = ((4*y - 6*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2) - 8*x.*y + 12*x.*y.^2 + 4*x.^2.*y - 6*y.^2)./w) .* (y<0.5) + ... %! ((2*z - 4*y - 4*x + 2*x.^2.*y.^2 + 8*x.*y - 4*x.*y.^2 - 4*x.^2.*y - 4*x.^2.*z + 2*x.^2 + 2*y.^2 + 16*x.^2.*y.*z - 12*x.^2.*y.^2.*z + 2)./w) .* (y>0.5); %! dFdu = zeros ([3,size(x)]); %! dFdu(1,:,:,:) = ((x.*((8*y.^2.*z.^3)/5 + 8*y.^2) - (4*y.^2.*z.^3)/5 + x.^2.*(z.^2.*(8*y.^4 + 4*y.^2) + (8*y.^4.*z.^3)/5 - 4*y.^2) + 2)./w.^2).*(y<0.5) + ... %! ((z.^3.*(x.^2.*((72*y.^4)/5 - (192*y.^3)/5 + (176*y.^2)/5 - (64*y)/5 + 8/5) - (16*y)/5 - x.*((24*y.^2)/5 - (32*y)/5 + 8/5) + (12*y.^2)/5 + 4/5) - x.*(24*y.^2 - 32*y + 8) + x.^2.*(12*y.^2 - 16*y + 4) + x.^2.*z.^2.*(72*y.^4 - 192*y.^3 + 164*y.^2 - 48*y + 4) + 2)./w.^2).*(y>0.5); %! dFdu(2,:,:,:) = ((z.^2.*(8*x.^2.*y.^4 - y.^2.*(8*y - 4*y.^2) + (2*x.*y.^2.*(40*y - 20*y.^2))/5) + z.^3.*((12*x.^2.*y.^4)/5 + (12*x.*y.^2)/5 - (6*y.^2)/5) + (2*x.*y.^2.*(20*y.^2 - 40*y + 20))/5)./w.^2).*(y<0.5) + ... %! (((2*(3*y.^2 - 4*y + 1).*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 6*x + 3).*z.^3)/5 + (2*(3*y.^2 - 4*y + 1).*(60*x.^2.*y.^2 - 80*x.^2.*y + 20*x.^2 - 20*x.*y.^2 - 10*x + 10*y.^2 + 5).*z.^2)/5 - (2*(10*x - 20*x.*y.^2).*(3*y.^2 - 4*y + 1))/5)./w.^2).*(y>0.5); %! dFdu(3,:,:,:) = ((4*y.*(3*y - 2) + z.^3.*(8*x.^2.*y.^4 + 8*x.*y.^2 - 4*y.^2) - z.^2.*(4*y.*(2*y.^2 - 3*y.^3).*x.^2 - 4*y.*(4*y.^2 - 6*y.^3).*x + 4*y.*(2*y.^2 - 3*y.^3)) + 4*x.^2.*y.*(4*y.^2 - 6*y.^3) - 4*x.*y.*(- 6*y.^3 + 4*y.^2 + 3*y - 2)) ./w.^2).*(y<0.5) + ... %! ((z.^2.*(4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1).*x.^2 - 4*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2).*x + 4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1)) - 4*(y - 1).^2 + z.^3.*(4*(y - 1).*(18*y.^3 - 30*y.^2 + 14*y - 2).*x.^2 - 4*(6*y - 2).*(y - 1).*x + 4*(3*y - 1).*(y - 1)) + 4*x.*(y - 1).*(6*y.^3 - 14*y.^2 + 11*y - 3) - 4*x.^2.*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2))./w.^2) .* (y > 0.5); %! dFdv = zeros ([3,size(x)]); %! dFdv(1,:,:,:) = ((8*x.*y.*(x - 1).*(z.^3 + 5*x.*z.^2 - 5*x))/5./w.^2).*(y<0.5) + ... %! (-(8*x.*(3*y - 2).*(x - 1).*(z.^3 + 5*x.*z.^2 - 5*x))/5./w.^2).*(y>0.5); %! dFdv(2,:,:,:) = (-((8*x.*z.^2 - x.^2.*(8*z.^2 - 8)).*y.^2 + ((12*x.*z.^3)/5 - x.^2.*((12*z.^3)/5 + 8) + 4).*y - 4)./w.^2).*(y<0.5) + ... %! ((4*y + z.^3.*(x.*((36*y)/5 - 24/5) - x.^2.*((36*y)/5 - 24/5)) + z.^2.*(x.*(16*y.^2 + 4*y - 8) - x.^2.*(16*y.^2 + 4*y - 8)) + x.^2.*(16*y.^2 - 20*y + 8))./w.^2).*(y>0.5); %! dFdv(3,:,:,:) = ((4*(x - 1).^2 - y.*(4*(3*x - 3).*(x - 1) - 8*x.*z.^3.*(x - 1)) + y.^2.*(4*(x - 1).*(2*x.^3 - 4*x.^2 + 2*x).*z.^2 + 4*(2*x.^2 - 2*x.^3).*(x - 1)))./w.^2).*(y<0.5) + ... %! ((y.^2.*(4*(x - 1).*(2*x.^3 - 4*x.^2 + 2*x).*z.^2 + 4*(2*x.^2 - 2*x.^3).*(x - 1)) - 4*(x - 1).*(2*x.^3 - 2*x.^2 + x - 1) - y.*(24*x.*(x - 1).*z.^3 + 4*(x - 1).*(4*x.^3 - 8*x.^2 + 4*x).*z.^2 - 4*(x - 1).*(4*x.^3 - 4*x.^2 + x - 1)) + 16*x.*z.^3.*(x - 1) + 4*z.^2.*(x - 1).*(2*x.^3 - 4*x.^2 + 2*x))./w.^2).*(y>0.5); %! dFdw = zeros ([3,size(x)]); %! dFdw(1,:,:,:) = ((4*x.^2.*y.^2 + 2)./(- 10*x.^2.*y.^2.*z.^2 + 10*x.^2.*y.^2 + 10*x.*y.^2.*z.^2 + 5) - ((20*x.*y.^2.*z - 20*x.^2.*y.^2.*z).*(10*x + 20*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2)))./(5*w).^2).*(y<0.5) + ... %! ((12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2)./(- 30*x.^2.*y.^2.*z.^2 + 30*x.^2.*y.^2 + 40*x.^2.*y.*z.^2 - 40*x.^2.*y - 10*x.^2.*z.^2 + 10*x.^2 + 30*x.*y.^2.*z.^2 - 40*x.*y.*z.^2 + 10*x.*z.^2 - 5) - ((60*x.^2.*y.^2 - 10*x + z.*(12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2) - 80*x.^2.*y + 20*x.^2).*(- 60*z.*x.^2.*y.^2 + 80*z.*x.^2.*y - 20*z.*x.^2 + 60*z.*x.*y.^2 - 80*z.*x.*y + 20*z.*x))./(5*w).^2).*(y>0.5); %! dFdw(2,:,:,:) = ((6*x.^2.*y.^2 + 3)./(- 10*x.^2.*y.^2.*z.^2 + 10*x.^2.*y.^2 + 10*x.*y.^2.*z.^2 + 5) - ((20*x.*y.^2.*z - 20*x.^2.*y.^2.*z).*(20*y + 20*x.^2.*y.^2 + z.*(6*x.^2.*y.^2 + 3) - 10*y.^2))./(5*w).^2).*(y<0.5) + ... %! ((18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 3)./(- 30*x.^2.*y.^2.*z.^2 + 30*x.^2.*y.^2 + 40*x.^2.*y.*z.^2 - 40*x.^2.*y - 10*x.^2.*z.^2 + 10*x.^2 + 30*x.*y.^2.*z.^2 - 40*x.*y.*z.^2 + 10*x.*z.^2 - 5) - ((- 60*z.*x.^2.*y.^2 + 80*z.*x.^2.*y - 20*z.*x.^2 + 60*z.*x.*y.^2 - 80*z.*x.*y + 20*z.*x).*(60*x.^2.*y.^2 + z.*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 3) - 80*x.^2.*y + 20*x.^2 - 10*y.^2 - 5))./(5*w).^2).*(y>0.5); %! dFdw(3,:,:,:) = ((4*x.^2.*y.^2 + 2)./(2*x.^2.*y.^2 - z.^2.*(2*x.^2.*y.^2 - 2*x.*y.^2) + 1) + (2*z.*(2*x.^2.*y.^2 - 2*x.*y.^2).*(4*y - 6*x.^2.*y.^2 + z.*(4*x.^2.*y.^2 + 2) - 8*x.*y + 12*x.*y.^2 + 4*x.^2.*y - 6*y.^2))./w.^2).*(y<0.5) + ... %! ((12*x.^2.*y.^2 - 16*x.^2.*y + 4*x.^2 - 2)./(6*x.^2.*y.^2 + z.^2.*(- 6*x.^2.*y.^2 + 8*x.^2.*y - 2*x.^2 + 6*x.*y.^2 - 8*x.*y + 2*x) - 8*x.^2.*y + 2*x.^2 - 1) + (2*z.*(- 6*x.^2.*y.^2 + 8*x.^2.*y - 2*x.^2 + 6*x.*y.^2 - 8*x.*y + 2*x).*(2*z - 4*y - 4*x + 2*x.^2.*y.^2 + 8*x.*y - 4*x.*y.^2 - 4*x.^2.*y - 4*x.^2.*z + 2*x.^2 + 2*y.^2 + 16*x.^2.*y.*z - 12*x.^2.*y.^2.*z + 2))./w.^2).*(y>0.5); %! d2Fduu = zeros ([3, size(x)]); %! d2Fduu(1,:,:,:) = (((8*y.^2.*z.^3)/5 + 2*x.*(z.^2.*(8*y.^4 + 4*y.^2) + (8*y.^4.*z.^3)/5 - 4*y.^2) + 8*y.^2)./w.^2 - (2*(2*y.^2.*z.^2 + 4*x.*y.^2 - 4*x.*y.^2.*z.^2).*(x.*((8*y.^2.*z.^3)/5 + 8*y.^2) - (4*y.^2.*z.^3)/5 + x.^2.*(z.^2.*(8*y.^4 + 4*y.^2) + (8*y.^4.*z.^3)/5 - 4*y.^2) + 2))./w.^3).*(y<0.5) + ... %! ((32*y + 2*x.*(12*y.^2 - 16*y + 4) + z.^3.*((32*y)/5 + 2*x.*((72*y.^4)/5 - (192*y.^3)/5 + (176*y.^2)/5 - (64*y)/5 + 8/5) - (24*y.^2)/5 - 8/5) - 24*y.^2 + 2*x.*z.^2.*(72*y.^4 - 192*y.^3 + 164*y.^2 - 48*y + 4) - 8)./w.^2 - (2*(z.^3.*(x.^2.*((72*y.^4)/5 - (192*y.^3)/5 + (176*y.^2)/5 - (64*y)/5 + 8/5) - (16*y)/5 - x.*((24*y.^2)/5 - (32*y)/5 + 8/5) + (12*y.^2)/5 + 4/5) - x.*(24*y.^2 - 32*y + 8) + x.^2.*(12*y.^2 - 16*y + 4) + x.^2.*z.^2.*(72*y.^4 - 192*y.^3 + 164*y.^2 - 48*y + 4) + 2).*(4*x + 6*y.^2.*z.^2 - 16*x.*y + 12*x.*y.^2 - 4*x.*z.^2 - 8*y.*z.^2 + 2*z.^2 + 16*x.*y.*z.^2 - 12*x.*y.^2.*z.^2))./(-w).^3).*(y>0.5); %! d2Fduu(2,:,:,:) = ((z.^3.*((24*x.*y.^4)/5 + (12*y.^2)/5) + (2*y.^2.*(20*y.^2 - 40*y + 20))/5 + z.^2.*((2*y.^2.*(40*y - 20*y.^2))/5 + 16*x.*y.^4))./w.^2 - (2*(z.^2.*(8*x.^2.*y.^4 - y.^2.*(8*y - 4*y.^2) + (2*x.*y.^2.*(40*y - 20*y.^2))/5) + z.^3.*((12*x.^2.*y.^4)/5 + (12*x.*y.^2)/5 - (6*y.^2)/5) + (2*x.*y.^2.*(20*y.^2 - 40*y + 20))/5).*(2*y.^2.*z.^2 + 4*x.*y.^2 - 4*x.*y.^2.*z.^2))./w.^3).*(y<0.5) + ... %! (((2*(3*y.^2 - 4*y + 1).*(36*x.*y.^2 - 48*x.*y + 12*x - 6).*z.^3)/5 - (2*(3*y.^2 - 4*y + 1).*(160*x.*y - 40*x - 120*x.*y.^2 + 20*y.^2 + 10).*z.^2)/5 + (2*(20*y.^2 - 10).*(3*y.^2 - 4*y + 1))/5)./w.^2 - (2*((2*(3*y.^2 - 4*y + 1).*(18*x.^2.*y.^2 - 24*x.^2.*y + 6*x.^2 - 6*x + 3).*z.^3)/5 + (2*(3*y.^2 - 4*y + 1).*(60*x.^2.*y.^2 - 80*x.^2.*y + 20*x.^2 - 20*x.*y.^2 - 10*x + 10*y.^2 + 5).*z.^2)/5 - (2*(10*x - 20*x.*y.^2).*(3*y.^2 - 4*y + 1))/5).*(4*x + 6*y.^2.*z.^2 - 16*x.*y + 12*x.*y.^2 - 4*x.*z.^2 - 8*y.*z.^2 + 2*z.^2 + 16*x.*y.*z.^2 - 12*x.*y.^2.*z.^2))./(-w).^3).*(y>0.5); %! d2Fduu(3,:,:,:) = (((16*x.*y.^4 + 8*y.^2).*z.^3 + (4*y.*(4*y.^2 - 6*y.^3) - 8*x.*y.*(2*y.^2 - 3*y.^3)).*z.^2 - 4*y.*(- 6*y.^3 + 4*y.^2 + 3*y - 2) + 8*x.*y.*(4*y.^2 - 6*y.^3))./w.^2 - (2*(2*y.^2.*z.^2 + 4*x.*y.^2 - 4*x.*y.^2.*z.^2).*(4*y.*(3*y - 2) + z.^3.*(8*x.^2.*y.^4 + 8*x.*y.^2 - 4*y.^2) - z.^2.*(4*y.*(2*y.^2 - 3*y.^3).*x.^2 - 4*y.*(4*y.^2 - 6*y.^3).*x + 4*y.*(2*y.^2 - 3*y.^3)) + 4*x.^2.*y.*(4*y.^2 - 6*y.^3) - 4*x.*y.*(- 6*y.^3 + 4*y.^2 + 3*y - 2)))./w.^3).*(y<0.5) + ... %! (-((4*(6*y - 2).*(y - 1) - 8*x.*(y - 1).*(18*y.^3 - 30*y.^2 + 14*y - 2)).*z.^3 + (4*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2) - 8*x.*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1)).*z.^2 - 4*(y - 1).*(6*y.^3 - 14*y.^2 + 11*y - 3) + 8*x.*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2))./w.^2 - (2*(z.^2.*(4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1).*x.^2 - 4*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2).*x + 4*(y - 1).*(3*y.^3 - 7*y.^2 + 5*y - 1)) - 4*(y - 1).^2 + z.^3.*(4*(y - 1).*(18*y.^3 - 30*y.^2 + 14*y - 2).*x.^2 - 4*(6*y - 2).*(y - 1).*x + 4*(3*y - 1).*(y - 1)) + 4*x.*(y - 1).*(6*y.^3 - 14*y.^2 + 11*y - 3) - 4*x.^2.*(y - 1).*(6*y.^3 - 14*y.^2 + 10*y - 2)).*(4*x + 6*y.^2.*z.^2 - 16*x.*y + 12*x.*y.^2 - 4*x.*z.^2 - 8*y.*z.^2 + 2*z.^2 + 16*x.*y.*z.^2 - 12*x.*y.^2.*z.^2))./(-w).^3) .* (y>0.5); %! d2Fduv = zeros ([3, size(x)]); %! d2Fduv(1,:,:,:) = ((((8.*x.^2.*(6.*z.^3 - 6.*z.^5))/5 + (8.*x.^4.*(10.*z.^4 - 20.*z.^2 + 10))/5 - (8.*x.^3.*(- 4.*z.^5 + 10.*z.^4 + 4.*z.^3 - 30.*z.^2 + 20))/5 + (16.*x.*z.^5)/5).*y.^3 + ((8.*x.*(2.*z.^3 - 10.*z.^2 + 10))/5 + (8.*x.^2.*(15.*z.^2 - 15))/5 - (8.*z.^3)/5).*y)./w.^3) .* (y<0.5) + ... %! (-(x.^4.*((8.*(3.*y - 2).*(30.*y.^2 - 40.*y + 10).*z.^4)/5 - (8.*(3.*y - 2).*(60.*y.^2 - 80.*y + 20).*z.^2)/5 + (8.*(3.*y - 2).*(30.*y.^2 - 40.*y + 10))/5) - x.^3.*(- (8.*(3.*y - 2).*(12.*y.^2 - 16.*y + 4).*z.^5)/5 + (8.*(3.*y - 2).*(30.*y.^2 - 40.*y + 10).*z.^4)/5 + (8.*(3.*y - 2).*(12.*y.^2 - 16.*y + 4).*z.^3)/5 - (8.*(3.*y - 2).*(90.*y.^2 - 120.*y + 30).*z.^2)/5 + (8.*(3.*y - 2).*(60.*y.^2 - 80.*y + 20))/5) + z.^3.*((24.*y)/5 - 16/5) - x.^2.*((8.*(3.*y - 2).*(18.*y.^2 - 24.*y + 6).*z.^5)/5 - (8.*(3.*y - 2).*(18.*y.^2 - 24.*y + 6).*z.^3)/5 + (72.*y - 48).*z.^2 - 72.*y + 48) + x.*((8.*(3.*y - 2).*(6.*y.^2 - 8.*y + 2).*z.^5)/5 + (32/5 - (48.*y)/5).*z.^3 + (48.*y - 32).*z.^2 - 48.*y + 32))./(-w).^3) .* (y>0.5); %! d2Fduv(2,:,:,:) = ((((4.*x.^2.*(60.*z.^2 - 60.*z.^4))/5 + (4.*x.^3.*(40.*z.^4 - 80.*z.^2 + 40))/5 + 16.*x.*z.^4).*y.^4 + ((4.*x.^2.*(18.*z.^3 - 18.*z.^5))/5 + (4.*x.^3.*(12.*z.^5 - 12.*z.^3 + 40.*z.^2 - 40))/5 + (4.*x.*(6.*z.^5 - 40.*z.^2 + 40))/5 + 16.*z.^2).*y.^3 + ((4.*x.*(60.*z.^2 - 60))/5 - 24.*z.^2).*y.^2 + ((4.*x.*(6.*z.^3 + 20))/5 - (12.*z.^3)/5).*y)./w.^3) .* (y<0.5) + ... %! ((z.^3.*(((432.*y.^3)/5 - (864.*y.^2)/5 + (528.*y)/5 - 96/5).*x.^3 + (- (648.*y.^3)/5 + (1296.*y.^2)/5 - (792.*y)/5 + 144/5).*x.^2 + ((72.*y)/5 - 48/5).*x - (36.*y)/5 + 24/5) - x.^3.*(192.*y.^4 - 496.*y.^3 + 480.*y.^2 - 208.*y + 32) + z.^4.*((- 192.*y.^4 + 208.*y.^3 + 96.*y.^2 - 144.*y + 32).*x.^3 + (288.*y.^4 - 312.*y.^3 - 144.*y.^2 + 216.*y - 48).*x.^2 + (- 96.*y.^4 + 104.*y.^3 + 48.*y.^2 - 72.*y + 16).*x) + x.*(- 96.*y.^3 + 96.*y.^2 + 8.*y - 16) + z.^2.*(x.^2.*(- 288.*y.^4 + 312.*y.^3 + 144.*y.^2 - 216.*y + 48) - 20.*y - x.^3.*(- 384.*y.^4 + 704.*y.^3 - 384.*y.^2 + 64.*y) + x.*(96.*y.^3 - 96.*y.^2 + 40.*y - 16) + 48.*y.^2 - 48.*y.^3 + 8) - z.^5.*(((432.*y.^3)/5 - (864.*y.^2)/5 + (528.*y)/5 - 96/5).*x.^3 + (- (648.*y.^3)/5 + (1296.*y.^2)/5 - (792.*y)/5 + 144/5).*x.^2 + ((216.*y.^3)/5 - (432.*y.^2)/5 + (264.*y)/5 - 48/5).*x))./(-w).^3) .* (y>0.5); %! d2Fduv(3,:,:,:) = (((x.^2.*(48.*z.^2 - 48.*z.^4) - x.^4.*(16.*z.^4 - 48.*z.^2 + 32) + x.^3.*(48.*z.^4 - 96.*z.^2 + 32) + 16.*x.*z.^4).*y.^4 + (x.^2.*(- 48.*z.^5 + 48.*z.^3 + 144.*z.^2 - 144) - x.^3.*(- 32.*z.^5 + 32.*z.^3 + 48.*z.^2 - 48) + x.*(16.*z.^5 - 144.*z.^2 + 96) + 48.*z.^2).*y.^3 + (x.*(96.*z.^2 - 48) + x.^3.*(48.*z.^2 - 48) - x.^2.*(120.*z.^2 - 96) - 24.*z.^2).*y.^2 + (x.*(16.*z.^3 - 24) - 8.*z.^3 + 24).*y + 8.*x - 8)./w.^3) .* (y<0.5) + ... %! ((8.*y - x.^4.*(96.*y.^4 - 320.*y.^3 + 384.*y.^2 - 192.*y + 32) + x.^3.*(96.*y.^4 - 368.*y.^3 + 528.*y.^2 - 336.*y + 80) + z.^3.*((288.*y.^3 - 576.*y.^2 + 352.*y - 64).*x.^3 + (- 432.*y.^3 + 864.*y.^2 - 528.*y + 96).*x.^2 + (48.*y - 32).*x - 24.*y + 16) - x.*(96.*y.^3 - 240.*y.^2 + 200.*y - 56) - z.^4.*((48.*y.^4 - 160.*y.^3 + 192.*y.^2 - 96.*y + 16).*x.^4 + (- 144.*y.^4 + 480.*y.^3 - 576.*y.^2 + 288.*y - 48).*x.^3 + (144.*y.^4 - 480.*y.^3 + 576.*y.^2 - 288.*y + 48).*x.^2 + (- 48.*y.^4 + 160.*y.^3 - 192.*y.^2 + 96.*y - 16).*x) + z.^2.*(x.^4.*(144.*y.^4 - 480.*y.^3 + 576.*y.^2 - 288.*y + 48) - 96.*y + x.^2.*(144.*y.^4 - 624.*y.^3 + 984.*y.^2 - 672.*y + 168) - x.^3.*(288.*y.^4 - 1008.*y.^3 + 1296.*y.^2 - 720.*y + 144) + x.*(144.*y.^3 - 384.*y.^2 + 336.*y - 96) + 120.*y.^2 - 48.*y.^3 + 24) - z.^5.*((288.*y.^3 - 576.*y.^2 + 352.*y - 64).*x.^3 + (- 432.*y.^3 + 864.*y.^2 - 528.*y + 96).*x.^2 + (144.*y.^3 - 288.*y.^2 + 176.*y - 32).*x) + x.^2.*(144.*y.^3 - 384.*y.^2 + 336.*y - 96) - 8)./(-w).^3) .* (y>0.5); %! d2Fduw = zeros ([3, size(x)]); %! d2Fduw(1,:,:,:) = ((x.^2.*((24.*y.^4.*z.^2)/5 + 2.*z.*(8.*y.^4 + 4.*y.^2)) - (12.*y.^2.*z.^2)/5 + (24.*x.*y.^2.*z.^2)/5)./w.^2 - (2.*(4.*x.*y.^2.*z - 4.*x.^2.*y.^2.*z).*(x.*((8.*y.^2.*z.^3)/5 + 8.*y.^2) - (4.*y.^2.*z.^3)/5 + x.^2.*(z.^2.*(8.*y.^4 + 4.*y.^2) + (8.*y.^4.*z.^3)/5 - 4.*y.^2) + 2))./w.^3) .* (y<0.5) + ... %! (-((- (4.*(3.*y - 1).*(y - 1).*(36.*y.^4 - 96.*y.^3 + 88.*y.^2 - 32.*y + 4).*x.^4)/5 + (4.*(3.*y - 1).*(y - 1).*(36.*y.^4 - 96.*y.^3 + 100.*y.^2 - 48.*y + 8).*x.^3)/5 - (4.*(3.*y - 1).*(y - 1).*(18.*y.^2 - 24.*y + 6).*x.^2)/5 + (4.*(3.*y - 1).*(y - 1).*(6.*y.^2 - 8.*y + 2).*x)/5).*z.^4 + ((4.*x.^3.*(3.*y - 1).*(y - 1).*(360.*y.^4 - 960.*y.^3 + 820.*y.^2 - 240.*y + 20))/5 - (4.*x.^4.*(3.*y - 1).*(y - 1).*(360.*y.^4 - 960.*y.^3 + 820.*y.^2 - 240.*y + 20))/5).*z.^3 + (- (4.*(3.*y - 1).*(y - 1).*(108.*y.^4 - 288.*y.^3 + 264.*y.^2 - 96.*y + 12).*x.^4)/5 + (4.*(3.*y - 1).*(y - 1).*(36.*y.^2 - 48.*y + 12).*x.^3)/5 - (24.*(3.*y - 1).*(y - 1).*x)/5 + (12.*(3.*y - 1).*(y - 1))/5).*z.^2 + (- (4.*(3.*y - 1).*(y - 1).*(360.*y.^4 - 960.*y.^3 + 940.*y.^2 - 400.*y + 60).*x.^4)/5 + (4.*(3.*y - 1).*(y - 1).*(360.*y.^2 - 480.*y + 120).*x.^3)/5 - (4.*(3.*y - 1).*(y - 1).*(180.*y.^2 - 240.*y + 90).*x.^2)/5 + 16.*(3.*y - 1).*(y - 1).*x).*z)./(-w).^3) .* (y>0.5); %! d2Fduw(2,:,:,:) = ((2.*z.*(8.*x.^2.*y.^4 - y.^2.*(8.*y - 4.*y.^2) + (2.*x.*y.^2.*(40.*y - 20.*y.^2))/5) + 3.*z.^2.*((12.*x.^2.*y.^4)/5 + (12.*x.*y.^2)/5 - (6.*y.^2)/5))./w.^2 - (2.*(4.*x.*y.^2.*z - 4.*x.^2.*y.^2.*z).*(z.^2.*(8.*x.^2.*y.^4 - y.^2.*(8.*y - 4.*y.^2) + (2.*x.*y.^2.*(40.*y - 20.*y.^2))/5) + z.^3.*((12.*x.^2.*y.^4)/5 + (12.*x.*y.^2)/5 - (6.*y.^2)/5) + (2.*x.*y.^2.*(20.*y.^2 - 40.*y + 20))/5))./w.^3) .* (y<0.5) + ... %! (((6.*(3.*y.^2 - 4.*y + 1).*(18.*x.^2.*y.^2 - 24.*x.^2.*y + 6.*x.^2 - 6.*x + 3).*z.^2)/5 + (4.*(3.*y.^2 - 4.*y + 1).*(60.*x.^2.*y.^2 - 80.*x.^2.*y + 20.*x.^2 - 20.*x.*y.^2 - 10.*x + 10.*y.^2 + 5).*z)/5)./w.^2 - (2.*((2.*(3.*y.^2 - 4.*y + 1).*(18.*x.^2.*y.^2 - 24.*x.^2.*y + 6.*x.^2 - 6.*x + 3).*z.^3)/5 + (2.*(3.*y.^2 - 4.*y + 1).*(60.*x.^2.*y.^2 - 80.*x.^2.*y + 20.*x.^2 - 20.*x.*y.^2 - 10.*x + 10.*y.^2 + 5).*z.^2)/5 - (2.*(10.*x - 20.*x.*y.^2).*(3.*y.^2 - 4.*y + 1))/5).*(- 12.*z.*x.^2.*y.^2 + 16.*z.*x.^2.*y - 4.*z.*x.^2 + 12.*z.*x.*y.^2 - 16.*z.*x.*y + 4.*z.*x))./(-w).^3) .* (y>0.5); %! d2Fduw(3,:,:,:) = (- (2.*z.*(4.*y.*(2.*y.^2 - 3.*y.^3).*x.^2 - 4.*y.*(4.*y.^2 - 6.*y.^3).*x + 4.*y.*(2.*y.^2 - 3.*y.^3)) - 3.*z.^2.*(8.*x.^2.*y.^4 + 8.*x.*y.^2 - 4.*y.^2))./w.^2 - (2.*(4.*x.*y.^2.*z - 4.*x.^2.*y.^2.*z).*(4.*y.*(3.*y - 2) + z.^3.*(8.*x.^2.*y.^4 + 8.*x.*y.^2 - 4.*y.^2) - z.^2.*(4.*y.*(2.*y.^2 - 3.*y.^3).*x.^2 - 4.*y.*(4.*y.^2 - 6.*y.^3).*x + 4.*y.*(2.*y.^2 - 3.*y.^3)) + 4.*x.^2.*y.*(4.*y.^2 - 6.*y.^3) - 4.*x.*y.*(- 6.*y.^3 + 4.*y.^2 + 3.*y - 2)))./w.^3) .* (y<0.5) + ... %! ((2.*z.*(4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1).*x.^2 - 4.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 10.*y - 2).*x + 4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1)) + 3.*z.^2.*(4.*(y - 1).*(18.*y.^3 - 30.*y.^2 + 14.*y - 2).*x.^2 - 4.*(6.*y - 2).*(y - 1).*x + 4.*(3.*y - 1).*(y - 1)))./w.^2 - (2.*(z.^2.*(4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1).*x.^2 - 4.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 10.*y - 2).*x + 4.*(y - 1).*(3.*y.^3 - 7.*y.^2 + 5.*y - 1)) - 4.*(y - 1).^2 + z.^3.*(4.*(y - 1).*(18.*y.^3 - 30.*y.^2 + 14.*y - 2).*x.^2 - 4.*(6.*y - 2).*(y - 1).*x + 4.*(3.*y - 1).*(y - 1)) + 4.*x.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 11.*y - 3) - 4.*x.^2.*(y - 1).*(6.*y.^3 - 14.*y.^2 + 10.*y - 2)).*(- 12.*z.*x.^2.*y.^2 + 16.*z.*x.^2.*y - 4.*z.*x.^2 + 12.*z.*x.*y.^2 - 16.*z.*x.*y + 4.*z.*x))./(-w).^3) .* (y>0.5); %! d2Fdvv = zeros ([3, size(x)]); %! d2Fdvv(1,:,:,:) = (-(8.*x.*(x - 1).*(z.^3 + 5.*x.*z.^2 - 5.*x).*(- 6.*x.^2.*y.^2.*z.^2 + 6.*x.^2.*y.^2 + 6.*x.*y.^2.*z.^2 - 1))/5./w.^3) .* (y<0.5) + ... %! ((8.*x.*(x - 1).*(z.^3 + 5.*x.*z.^2 - 5.*x).*(- 54.*x.^2.*y.^2.*z.^2 + 54.*x.^2.*y.^2 + 72.*x.^2.*y.*z.^2 - 72.*x.^2.*y - 26.*x.^2.*z.^2 + 26.*x.^2 + 54.*x.*y.^2.*z.^2 - 72.*x.*y.*z.^2 + 26.*x.*z.^2 + 3))/5./(-w).^3) .* (y>0.5); %! d2Fdvv(2,:,:,:) = ((2.*((8.*x.*z.^2 - x.^2.*(8.*z.^2 - 8)).*y.^2 + ((12.*x.*z.^3)/5 - x.^2.*((12.*z.^3)/5 + 8) + 4).*y - 4).*(- 4.*y.*x.^2.*z.^2 + 4.*y.*x.^2 + 4.*y.*x.*z.^2))./w.^3 - ((12.*x.*z.^3)/5 + 2.*y.*(8.*x.*z.^2 - x.^2.*(8.*z.^2 - 8)) - x.^2.*((12.*z.^3)/5 + 8) + 4)./w.^2) .* (y<0.5) + ... %! ((z.^2.*(x.*(32.*y + 4) - x.^2.*(32.*y + 4)) + x.^2.*(32.*y - 20) + z.^3.*((36.*x)/5 - (36.*x.^2)/5) + 4)./w.^2 - (2.*(4.*y + z.^3.*(x.*((36.*y)/5 - 24/5) - x.^2.*((36.*y)/5 - 24/5)) + z.^2.*(x.*(16.*y.^2 + 4.*y - 8) - x.^2.*(16.*y.^2 + 4.*y - 8)) + x.^2.*(16.*y.^2 - 20.*y + 8)).*(8.*x.^2.*z.^2 + 12.*x.^2.*y - 8.*x.*z.^2 - 8.*x.^2 + 12.*x.*y.*z.^2 - 12.*x.^2.*y.*z.^2))./(-w).^3) .* (y>0.5); %! d2Fdvv(3,:,:,:) = ((2.*y.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 4.*(3.*x - 3).*(x - 1) + 8.*x.*z.^3.*(x - 1))./w.^2 - (2.*(4.*(x - 1).^2 - y.*(4.*(3.*x - 3).*(x - 1) - 8.*x.*z.^3.*(x - 1)) + y.^2.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1))).*(- 4.*y.*x.^2.*z.^2 + 4.*y.*x.^2 + 4.*y.*x.*z.^2))./w.^3) .* (y<0.5) + ... %! ((4.*(x - 1).*(4.*x.^3 - 4.*x.^2 + x - 1) + 2.*y.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 24.*x.*z.^3.*(x - 1) - 4.*z.^2.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x))./w.^2 - (2.*(y.^2.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 4.*(x - 1).*(2.*x.^3 - 2.*x.^2 + x - 1) - y.*(24.*x.*(x - 1).*z.^3 + 4.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x).*z.^2 - 4.*(x - 1).*(4.*x.^3 - 4.*x.^2 + x - 1)) + 16.*x.*z.^3.*(x - 1) + 4.*z.^2.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x)).*(8.*x.^2.*z.^2 + 12.*x.^2.*y - 8.*x.*z.^2 - 8.*x.^2 + 12.*x.*y.*z.^2 - 12.*x.^2.*y.*z.^2))./(-w).^3) .* (y>0.5); %! d2Fdvw = zeros ([3, size(x)]); %! d2Fdvw(1,:,:,:) = (((8.*x.*z.*(x - 1).*(20.*x.^3.*z.^2 - 20.*x.^3 + 2.*x.^2.*z.^3 - 20.*x.^2.*z.^2 + 6.*x.^2.*z + 40.*x.^2 - 2.*x.*z.^3).*y.^3)/5 + (8.*x.*z.*(10.*x + 3.*z).*(x - 1).*y)/5)./w.^3) .* (y<0.5) + ... %! (((8.*x.*(3.*y - 2).*(x - 1).*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).*z.^4)/5 + (8.*x.*(3.*y - 2).*(x - 1).*(- 60.*x.^3.*y.^2 + 80.*x.^3.*y - 20.*x.^3 + 60.*x.^2.*y.^2 - 80.*x.^2.*y + 20.*x.^2).*z.^3)/5 - (8.*x.*(3.*y - 2).*(x - 1).*(18.*x.^2.*y.^2 - 24.*x.^2.*y + 6.*x.^2 - 3).*z.^2)/5 + (8.*x.*(3.*y - 2).*(x - 1).*(60.*x.^3.*y.^2 - 80.*x.^3.*y + 20.*x.^3 - 120.*x.^2.*y.^2 + 160.*x.^2.*y - 40.*x.^2 + 10.*x).*z)/5)./(-w).^3) .* (y>0.5); %! d2Fdvw(2,:,:,:) = ((4.*x.*y.*z.*(x - 1).*(40.*x.^2.*y.^3.*z.^2 - 40.*x.^2.*y.^3 + 6.*x.^2.*y.^2.*z.^3 + 18.*x.^2.*y.^2.*z + 80.*x.^2.*y.^2 - 40.*x.*y.^3.*z.^2 - 6.*x.*y.^2.*z.^3 - 40.*y.^2 + 60.*y + 9.*z))/5./w.^3) .* (y<0.5) + ... %! (-((4.*x.*(x - 1).*(54.*x.^2.*y.^3 - 108.*x.^2.*y.^2 + 66.*x.^2.*y - 12.*x.^2 - 54.*x.*y.^3 + 108.*x.*y.^2 - 66.*x.*y + 12.*x).*z.^4)/5 + (4.*x.*(x - 1).*(240.*x.^2.*y.^4 - 260.*x.^2.*y.^3 - 120.*x.^2.*y.^2 + 180.*x.^2.*y - 40.*x.^2 - 240.*x.*y.^4 + 260.*x.*y.^3 + 120.*x.*y.^2 - 180.*x.*y + 40.*x).*z.^3)/5 - (4.*x.*(x - 1).*(- 162.*x.^2.*y.^3 + 324.*x.^2.*y.^2 - 198.*x.^2.*y + 36.*x.^2 + 27.*y - 18).*z.^2)/5 - (4.*x.*(x - 1).*(240.*x.^2.*y.^4 - 980.*x.^2.*y.^3 + 1320.*x.^2.*y.^2 - 700.*x.^2.*y + 120.*x.^2 + 120.*y.^3 - 120.*y.^2 + 50.*y - 20).*z)/5)./(-w).^3) .* (y>0.5); %! d2Fdvw(3,:,:,:) = (-(y.^3.*(8.*x.*z.*(x - 1).*(12.*x.^2 - 24.*x + 12) - 48.*x.^3.*z.^2.*(x - 1) + 8.*x.*z.^4.*(2.*x - 2.*x.^2).*(x - 1)) + y.^4.*(8.*x.*(x - 1).*(- 4.*x.^4 + 12.*x.^3 - 12.*x.^2 + 4.*x).*z.^3 + 8.*x.*(x - 1).*(4.*x.^4 - 8.*x.^3 + 4.*x.^2).*z) - 24.*x.*y.*z.^2.*(x - 1) - 8.*x.*y.^2.*z.*(x - 1).*(6.*x.^2 - 12.*x + 6))./w.^3) .* (y<0.5) + ... %! ((8.*z.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x) - y.*(72.*x.*(x - 1).*z.^2 + 8.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x).*z) + 48.*x.*z.^2.*(x - 1) + 8.*y.^2.*z.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x))./w.^2 - (2.*(y.^2.*(4.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x).*z.^2 + 4.*(2.*x.^2 - 2.*x.^3).*(x - 1)) - 4.*(x - 1).*(2.*x.^3 - 2.*x.^2 + x - 1) - y.*(24.*x.*(x - 1).*z.^3 + 4.*(x - 1).*(4.*x.^3 - 8.*x.^2 + 4.*x).*z.^2 - 4.*(x - 1).*(4.*x.^3 - 4.*x.^2 + x - 1)) + 16.*x.*z.^3.*(x - 1) + 4.*z.^2.*(x - 1).*(2.*x.^3 - 4.*x.^2 + 2.*x)).*(- 12.*z.*x.^2.*y.^2 + 16.*z.*x.^2.*y - 4.*z.*x.^2 + 12.*z.*x.*y.^2 - 16.*z.*x.*y + 4.*z.*x))./(-w).^3) .* (y>0.5); %! d2Fdww = zeros ([3, size(x)]); %! d2Fdww(1,:,:,:) = ((32.*x.*y.^2.*(2.*x.^2.*y.^2 + 1).*(x - 1).*(5.*x + z + 10.*x.^2.*y.^2 + 2.*x.^2.*y.^2.*z))./(5.*w.^3) - (8.*x.*y.^2.*(x - 1).*(15.*x + z + 30.*x.^2.*y.^2 + 2.*x.^2.*y.^2.*z))/5./w.^2) .* (y<0.5) + ... %! (((8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(36.*x.^4.*y.^4 - 96.*x.^4.*y.^3 + 88.*x.^4.*y.^2 - 32.*x.^4.*y + 4.*x.^4 - 36.*x.^3.*y.^4 + 96.*x.^3.*y.^3 - 88.*x.^3.*y.^2 + 32.*x.^3.*y - 4.*x.^3 - 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).*z.^3)/5 + (8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(540.*x.^4.*y.^4 - 1440.*x.^4.*y.^3 + 1320.*x.^4.*y.^2 - 480.*x.^4.*y + 60.*x.^4 - 540.*x.^3.*y.^4 + 1440.*x.^3.*y.^3 - 1410.*x.^3.*y.^2 + 600.*x.^3.*y - 90.*x.^3 + 90.*x.^2.*y.^2 - 120.*x.^2.*y + 30.*x.^2).*z.^2)/5 + (8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(108.*x.^4.*y.^4 - 288.*x.^4.*y.^3 + 264.*x.^4.*y.^2 - 96.*x.^4.*y + 12.*x.^4 - 36.*x.^2.*y.^2 + 48.*x.^2.*y - 12.*x.^2 + 3).*z)/5 + (8.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(180.*x.^4.*y.^4 - 480.*x.^4.*y.^3 + 440.*x.^4.*y.^2 - 160.*x.^4.*y + 20.*x.^4 - 30.*x.^3.*y.^2 + 40.*x.^3.*y - 10.*x.^3 - 30.*x.^2.*y.^2 + 40.*x.^2.*y - 10.*x.^2 + 5.*x))/5)./(-w).^3) .* (y>0.5); %! d2Fdww(2,:,:,:) = ((16.*x.*y.^2.*(2.*x.^2.*y.^2 + 1).*(x - 1).*(20.*y + 3.*z + 20.*x.^2.*y.^2 - 10.*y.^2 + 6.*x.^2.*y.^2.*z))./(5.*w.^3) - (12.*x.*y.^2.*(x - 1).*(20.*y + z + 20.*x.^2.*y.^2 - 10.*y.^2 + 2.*x.^2.*y.^2.*z))/5./w.^2) .* (y<0.5) + ... %! (((4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(108.*x.^4.*y.^4 - 288.*x.^4.*y.^3 + 264.*x.^4.*y.^2 - 96.*x.^4.*y + 12.*x.^4 - 108.*x.^3.*y.^4 + 288.*x.^3.*y.^3 - 264.*x.^3.*y.^2 + 96.*x.^3.*y - 12.*x.^3 - 18.*x.^2.*y.^2 + 24.*x.^2.*y - 6.*x.^2 + 18.*x.*y.^2 - 24.*x.*y + 6.*x).*z.^3)/5 + (4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(1080.*x.^4.*y.^4 - 2880.*x.^4.*y.^3 + 2640.*x.^4.*y.^2 - 960.*x.^4.*y + 120.*x.^4 - 1080.*x.^3.*y.^4 + 2880.*x.^3.*y.^3 - 2640.*x.^3.*y.^2 + 960.*x.^3.*y - 120.*x.^3 - 180.*x.^2.*y.^4 + 240.*x.^2.*y.^3 - 150.*x.^2.*y.^2 + 120.*x.^2.*y - 30.*x.^2 + 180.*x.*y.^4 - 240.*x.*y.^3 + 150.*x.*y.^2 - 120.*x.*y + 30.*x).*z.^2)/5 + (4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(324.*x.^4.*y.^4 - 864.*x.^4.*y.^3 + 792.*x.^4.*y.^2 - 288.*x.^4.*y + 36.*x.^4 - 108.*x.^2.*y.^2 + 144.*x.^2.*y - 36.*x.^2 + 9).*z)/5 + (4.*x.*(3.*y - 1).*(x - 1).*(y - 1).*(360.*x.^4.*y.^4 - 960.*x.^4.*y.^3 + 880.*x.^4.*y.^2 - 320.*x.^4.*y + 40.*x.^4 - 60.*x.^2.*y.^4 + 80.*x.^2.*y.^3 - 110.*x.^2.*y.^2 + 120.*x.^2.*y - 30.*x.^2 + 10.*y.^2 + 5))/5)./(-w).^3) .* (y>0.5); %! d2Fdww(3,:,:,:) = ((32.*x.*y.^2.*(2.*x.^2.*y.^2 + 1).*(x - 1).*(2.*y + z - 3.*x.^2.*y.^2 - 4.*x.*y + 6.*x.*y.^2 + 2.*x.^2.*y - 3.*y.^2 + 2.*x.^2.*y.^2.*z))./w.^3 - (8.*x.*y.^2.*(x - 1).*(6.*y + z - 9.*x.^2.*y.^2 - 12.*x.*y + 18.*x.*y.^2 + 6.*x.^2.*y - 9.*y.^2 + 2.*x.^2.*y.^2.*z))./w.^2) .* (y<0.5) + ... %! ((2.*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).*(2.*z - 4.*y - 4.*x + 2.*x.^2.*y.^2 + 8.*x.*y - 4.*x.*y.^2 - 4.*x.^2.*y - 4.*x.^2.*z + 2.*x.^2 + 2.*y.^2 + 16.*x.^2.*y.*z - 12.*x.^2.*y.^2.*z + 2))./w.^2 - (8.*z.^2.*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x).^2.*(2.*z - 4.*y - 4.*x + 2.*x.^2.*y.^2 + 8.*x.*y - 4.*x.*y.^2 - 4.*x.^2.*y - 4.*x.^2.*z + 2.*x.^2 + 2.*y.^2 + 16.*x.^2.*y.*z - 12.*x.^2.*y.^2.*z + 2))./(-w).^3 - (4.*z.*(12.*x.^2.*y.^2 - 16.*x.^2.*y + 4.*x.^2 - 2).*(- 6.*x.^2.*y.^2 + 8.*x.^2.*y - 2.*x.^2 + 6.*x.*y.^2 - 8.*x.*y + 2.*x))./w.^2) .* (y>0.5); %! assert (F, pnt, 1e3*eps) %! assert (dFdu, jac{1}, 1e3*eps) %! assert (dFdv, jac{2}, 1e3*eps) %! assert (dFdw, jac{3}, 1e3*eps) %! assert (d2Fduu, hess{1,1}, 1e3*eps) %! assert (d2Fduv, hess{1,2}, 1e3*eps) %! assert (d2Fduw, hess{1,3}, 1e3*eps) %! assert (d2Fduv, hess{2,1}, 1e3*eps) %! assert (d2Fdvv, hess{2,2}, 1e3*eps) %! assert (d2Fdvw, hess{2,3}, 1e3*eps) %! assert (d2Fduw, hess{3,1}, 1e3*eps) %! assert (d2Fdvw, hess{3,2}, 1e3*eps) %! assert (d2Fdww, hess{3,3}, 1e3*eps) %!test %! nrb = nrbextrude (nrb4surf ([0 0], [1 0], [0 1], [1 1]), [0 0 1]); %! nrb = nrbdegelev (nrb, [1 1 1]); %! nrb.coefs (4,2,2,2) = 1.1; %! [dnrb, dnrb2] = nrbderiv (nrb); %! X = linspace (0, 1, 24); Y = linspace (0, 1, 24); Z = linspace (0, 1, 24); %! [pnt, jac, hess] = nrbdeval (nrb, dnrb, dnrb2, {X Y Z}); %! [y, x, z] = meshgrid (X, Y, Z); %! F = zeros ([3, size(x)]); %! F(1,:,:,:) = (5.*x)./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5); %! F(2,:,:,:) = (5.*y)./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5); %! F(3,:,:,:) = (5.*z)./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5); %! dFdu = zeros ([3, size(x)]); %! dFdu(1,:,:,:) = ((z.*(20.*y - 20.*y.^2) - z.^2.*(20.*y - 20.*y.^2)).*x.^2 + 25)./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^2; %! dFdu(2,:,:,:) = (y.^2.*(5.*z.*(8.*x - 4) - 5.*z.^2.*(8.*x - 4)) - y.^3.*(5.*z.*(8.*x - 4) - 5.*z.^2.*(8.*x - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdu(3,:,:,:) = (z.^2.*(5.*y.*(8.*x - 4) - 5.*y.^2.*(8.*x - 4)) - z.^3.*(5.*y.*(8.*x - 4) - 5.*y.^2.*(8.*x - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdv = zeros ([3, size(x)]); %! dFdv(1,:,:,:) = (x.^2.*(5.*z.*(8.*y - 4) - 5.*z.^2.*(8.*y - 4)) - x.^3.*(5.*z.*(8.*y - 4) - 5.*z.^2.*(8.*y - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdv(2,:,:,:) = ((z.*(20.*x - 20.*x.^2) - z.^2.*(20.*x - 20.*x.^2)).*y.^2 + 25)./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^2; %! dFdv(3,:,:,:) = (z.^2.*(5.*x.*(8.*y - 4) - 5.*x.^2.*(8.*y - 4)) - z.^3.*(5.*x.*(8.*y - 4) - 5.*x.^2.*(8.*y - 4)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdw = zeros ([3, size(x)]); %! dFdw(1,:,:,:) = (x.^2.*(y.*(40.*z - 20) - y.^2.*(40.*z - 20)) - x.^3.*(y.*(40.*z - 20) - y.^2.*(40.*z - 20)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdw(2,:,:,:) = (y.^2.*(x.*(40.*z - 20) - x.^2.*(40.*z - 20)) - y.^3.*(x.*(40.*z - 20) - x.^2.*(40.*z - 20)))./((- 4.*x.^2.*y.^2 + 4.*x.^2.*y + 4.*x.*y.^2 - 4.*x.*y).*z.^2 + (4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y).*z + 5).^2; %! dFdw(3,:,:,:) = ((y.*(20.*x - 20.*x.^2) - y.^2.*(20.*x - 20.*x.^2)).*z.^2 + 25)./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^2; %! d2Fduu = zeros ([3, size(x)]); %! d2Fduu(1,:,:,:) = (40.*y.*z.*(y - 1).*(z - 1).*(4.*x.^3.*y.^2.*z.^2 - 4.*x.^3.*y.^2.*z - 4.*x.^3.*y.*z.^2 + 4.*x.^3.*y.*z + 15.*x - 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduu(2,:,:,:) = (40.*y.^2.*z.*(y - 1).*(z - 1).*(4.*y.^2.*z.^2 - 4.*y.^2.*z - 4.*y.*z.^2 + 4.*y.*z + 5) - 40.*x.*y.^2.*z.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z) + 40.*x.^2.*y.^2.*z.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduu(3,:,:,:) = (40.*y.*z.^2.*(y - 1).*(z - 1).*(4.*y.^2.*z.^2 - 4.*y.^2.*z - 4.*y.*z.^2 + 4.*y.*z + 5) - 40.*x.*y.*z.^2.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z) + 40.*x.^2.*y.*z.^2.*(y - 1).*(z - 1).*(12.*y.^2.*z.^2 - 12.*y.^2.*z - 12.*y.*z.^2 + 12.*y.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduv = zeros ([3, size(x)]); %! d2Fduv(1,:,:,:) = (20.*x.*z.*(2.*y - 1).*(z - 1).*(4.*x.^3.*y.^2.*z.^2 - 4.*x.^3.*y.^2.*z - 4.*x.^3.*y.*z.^2 + 4.*x.^3.*y.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 15.*x - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduv(2,:,:,:) = (20.*y.*z.*(2.*x - 1).*(z - 1).*(4.*x.^2.*y.^3.*z.^2 - 4.*x.^2.*y.^3.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z - 4.*x.*y.^3.*z.^2 + 4.*x.*y.^3.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z + 15.*y - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduv(3,:,:,:) = (20.*z.^2.*(2.*x - 1).*(2.*y - 1).*(z - 1).*(4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.^2.*z - 4.*x.^2.*y.*z.^2 + 4.*x.^2.*y.*z - 4.*x.*y.^2.*z.^2 + 4.*x.*y.^2.*z + 4.*x.*y.*z.^2 - 4.*x.*y.*z + 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduw = zeros ([3, size(x)]); %! d2Fduw(1,:,:,:) = (20.*x.*y.*(2.*z - 1).*(y - 1).*(4.*x.^3.*y.^2.*z.^2 - 4.*x.^3.*y.^2.*z - 4.*x.^3.*y.*z.^2 + 4.*x.^3.*y.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 15.*x - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduw(2,:,:,:) = (20.*y.^2.*(2.*x - 1).*(2.*z - 1).*(y - 1).*(4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.^2.*z - 4.*x.^2.*y.*z.^2 + 4.*x.^2.*y.*z - 4.*x.*y.^2.*z.^2 + 4.*x.*y.^2.*z + 4.*x.*y.*z.^2 - 4.*x.*y.*z + 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fduw(3,:,:,:) = (20.*y.*z.*(2.*x - 1).*(y - 1).*(4.*x.^2.*y.^2.*z.^3 - 4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.*z.^3 + 4.*x.^2.*y.*z.^2 - 4.*x.*y.^2.*z.^3 + 4.*x.*y.^2.*z.^2 + 4.*x.*y.*z.^3 - 4.*x.*y.*z.^2 + 15.*z - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvv = zeros ([3, size(x)]); %! d2Fdvv(1,:,:,:) = (40.*x.^2.*z.*(x - 1).*(z - 1).*(4.*x.^2.*z.^2 - 4.*x.^2.*z - 4.*x.*z.^2 + 4.*x.*z + 5) + 40.*x.^2.*y.^2.*z.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z) - 40.*x.^2.*y.*z.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvv(2,:,:,:) = (40.*x.*z.*(x - 1).*(z - 1).*(4.*x.^2.*y.^3.*z.^2 - 4.*x.^2.*y.^3.*z - 4.*x.*y.^3.*z.^2 + 4.*x.*y.^3.*z + 15.*y - 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvv(3,:,:,:) = (40.*x.*z.^2.*(x - 1).*(z - 1).*(4.*x.^2.*z.^2 - 4.*x.^2.*z - 4.*x.*z.^2 + 4.*x.*z + 5) + 40.*x.*y.^2.*z.^2.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z) - 40.*x.*y.*z.^2.*(x - 1).*(z - 1).*(12.*x.^2.*z.^2 - 12.*x.^2.*z - 12.*x.*z.^2 + 12.*x.*z))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvw = zeros ([3, size(x)]); %! d2Fdvw(1,:,:,:) = (20.*x.^2.*(2.*y - 1).*(2.*z - 1).*(x - 1).*(4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.^2.*z - 4.*x.^2.*y.*z.^2 + 4.*x.^2.*y.*z - 4.*x.*y.^2.*z.^2 + 4.*x.*y.^2.*z + 4.*x.*y.*z.^2 - 4.*x.*y.*z + 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvw(2,:,:,:) = (20.*x.*y.*(2.*z - 1).*(x - 1).*(4.*x.^2.*y.^3.*z.^2 - 4.*x.^2.*y.^3.*z - 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z - 4.*x.*y.^3.*z.^2 + 4.*x.*y.^3.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z + 15.*y - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdvw(3,:,:,:) = (20.*x.*z.*(2.*y - 1).*(x - 1).*(4.*x.^2.*y.^2.*z.^3 - 4.*x.^2.*y.^2.*z.^2 - 4.*x.^2.*y.*z.^3 + 4.*x.^2.*y.*z.^2 - 4.*x.*y.^2.*z.^3 + 4.*x.*y.^2.*z.^2 + 4.*x.*y.*z.^3 - 4.*x.*y.*z.^2 + 15.*z - 10))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdww = zeros ([3, size(x)]); %! d2Fdww(1,:,:,:) = (40.*x.^2.*y.*(x - 1).*(y - 1).*(4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y + 5) + 40.*x.^2.*y.*z.^2.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y) - 40.*x.^2.*y.*z.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdww(2,:,:,:) = (40.*x.*y.^2.*(x - 1).*(y - 1).*(4.*x.^2.*y.^2 - 4.*x.^2.*y - 4.*x.*y.^2 + 4.*x.*y + 5) + 40.*x.*y.^2.*z.^2.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y) - 40.*x.*y.^2.*z.*(x - 1).*(y - 1).*(12.*x.^2.*y.^2 - 12.*x.^2.*y - 12.*x.*y.^2 + 12.*x.*y))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; %! d2Fdww(3,:,:,:) = (40.*x.*y.*(x - 1).*(y - 1).*(4.*x.^2.*y.^2.*z.^3 - 4.*x.^2.*y.*z.^3 - 4.*x.*y.^2.*z.^3 + 4.*x.*y.*z.^3 + 15.*z - 5))./(- 4.*x.^2.*y.^2.*z.^2 + 4.*x.^2.*y.^2.*z + 4.*x.^2.*y.*z.^2 - 4.*x.^2.*y.*z + 4.*x.*y.^2.*z.^2 - 4.*x.*y.^2.*z - 4.*x.*y.*z.^2 + 4.*x.*y.*z + 5).^3; nurbs-1.4.3/inst/PaxHeaders.16248/nrbglue.m0000644000000000000000000000013014030323741015175 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbglue.m0000644000175000017500000001257314030323741017306 0ustar00rvazquezrvazquez00000000000000function glued = nrbglue(nrb1, nrb2, side1, side2) % % NRBGLUE: Glues two NURBS patches together with C^0-continuity at the % interface. % % Calling Sequence: % % glued = nrbglue(nrb1, nrb2, [side1, side2]) % % INPUT: % % nrb1 : first NURBS struct. % nrb2 : second NURBS struct. % side1 : index of the boundary side of nrb1 to be glued (optional). % side2 : index of the boundary side of nrb2 to be glued (optional). % % OUTPUT: % % glued : Glued NURBS struct. % % The resulting NURBS struct has the same orientation as nrb1. % % Description: % % The orientation of the NURBS boundaries: % % ^ V direction % | % | % | % | % /---------> U direction % / % / % v W direction % % The indices of the NURBS boundaries (see also nrbextract): % % 1 : U = 0 % 2 : U = 1 % 3 : V = 0 % 4 : V = 1 % 5 : W = 0 % 6 : W = 1 % % Copyright (C) 2019 Ondine Chanon, Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . interfaces = nrbmultipatch ([nrb1, nrb2]); if (isempty (interfaces)) error ('The patches cannot be glued together: non-compatible sides. Check degree and knot vectors') end if iscell(nrb1.knots) ndim = length(nrb1.knots); else ndim = 1; end if (nargin == 4) if (side1 > 2*ndim || side2 > 2*ndim || side1 < 1 || side2 < 1) error ('Wrong number of the sides'); end sides1 = [interfaces.side1]; interface_index = find (sides1 == side1); if (isempty(interface_index) || interfaces(interface_index).side2 ~= side2) error ('The input sides are not correct') end else if (numel (interfaces) > 1) error ('The two patches can be glued in more than one way (a ring?). Specify the sides') end side1 = interfaces.side1; side2 = interfaces.side2; end dir1 = ceil(side1/2); dir2 = ceil(side2/2); if (nrb1.order(dir1) < nrb2.order(dir2)) deg_raise = zeros(1, ndim); deg_raise(dir1) = nrb2.order(dir2) - nrb1.order(dir1); nrb1 = nrbdegelev(nrb1, deg_raise); warning ('The order of nrb1 has been raised along the gluing direction') elseif (nrb1.order(dir1) > nrb2.order(dir2)) deg_raise = zeros(1, ndim); deg_raise(dir2) = nrb1.order(dir1) - nrb2.order(dir2); nrb2 = nrbdegelev(nrb2, deg_raise); warning ('The order of nrb2 has been raised along the gluing direction') end % Decide which patch will play the role of the first one if (mod(side1,2) == 0) id1 = 1; id2 = 2; else id1 = 2; id2 = 1; end if (mod(side1,2) == mod(side2,2)) nrb2 = nrbreverse (nrb2, dir2); end % Make the gluing direction the first one if (ndim > 1) dir_change = [dir1, setdiff(1:ndim, dir1)]; nrb1 = nrbpermute (nrb1, dir_change); nrb2 = nrbpermute (nrb2, [dir2, setdiff(1:ndim, dir2)]); end nrbs = [nrb1, nrb2]; interfaces = nrbmultipatch (nrbs); if (ndim == 1) nrbs(1).knots = {nrbs(1).knots}; nrbs(2).knots = {nrbs(2).knots}; end % Recompute, to simplify things if (numel(interfaces) > 1) side = mod(side1-1,2) + 1; sides1 = [interfaces.side1]; interfaces = interfaces(sides1 == side); end if (ndim == 2 && interfaces.ornt == -1) nrbs(2) = nrbreverse (nrbs(2), 2); elseif (ndim == 3) if (interfaces.flag == -1) nrbs(2) = nrbpermute (nrbs(2), [1 3 2]); end if (interfaces.ornt1 == -1) nrbs(2) = nrbreverse (nrbs(2), 2); end if (interfaces.ornt2 == -1) nrbs(2) = nrbreverse (nrbs(2), 3); end end knots = nrbs(id1).knots; knots{1} = [nrbs(id1).knots{1}(1:end-1), ... nrbs(id2).knots{1}(nrbs(id2).order(1)+1:end)+nrbs(id1).knots{1}(end)]; coefs = cat(2, nrbs(id1).coefs, nrbs(id2).coefs(:,2:end,:,:)); if (ndim == 1) knots = knots{1}; end glued = nrbmak(coefs,knots); % Recover the parametric directions of nrb1 if (ndim > 1) if (dir_change(1) ~= 3) glued = nrbpermute (glued, dir_change); else glued = nrbpermute (glued, [2 3 1]); end end %!demo %! nrb1 = nrbline([0,0],[1,1]); %! nrb2 = nrbline([1,1],[3,4]); %! glued = nrbglue(nrb1, nrb2); %! nrbkntplot(glued) %! title ('Gluing of two straight lines') %!demo %! nrb1 = nrbsquare([0,0],1,1,[1,1],[1,2]); %! nrb2 = nrbsquare([1,0],3,1,[1,1],[2,2]); %! glued = nrbglue(nrb1, nrb2); %! nrbkntplot(glued) %! title ('Gluing of two bilinear patches') %!demo %! nrb1 = nrbsquare([0,0],1,1,[1,1],[1,2]); %! nrb2 = nrbsquare([1,0],3,1,[1,1],[2,2]); %! nrb1 = nrbextrude(nrb1,[0,0,1]); %! nrb2 = nrbextrude(nrb2,[0,0,1]); %! glued = nrbglue(nrb1, nrb2); %! nrbkntplot(glued) %! title ('Gluing of two trilinear patches') nurbs-1.4.3/inst/PaxHeaders.16248/nrbrect.m0000644000000000000000000000013214030323741015200 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbrect.m0000644000175000017500000000333114030323741017277 0ustar00rvazquezrvazquez00000000000000function curve = nrbrect(w,h) % % NRBRECT: Construct NURBS representation of a rectangular curve. % % Calling Sequence: % % crv = nrbrect() % crv = nrbrect(size) % crv = nrbrect(width, height) % % INPUT: % % size : Size of the square (width = height). % % width : Width of the rectangle (along x-axis). % % height : Height of the rectangle (along y-axis). % % OUTPUT: % % crv : NURBS curve, see nrbmak. % % % Description: % % Construct a rectangle or square in the x-y plane with the bottom % lhs corner at (0,0,0). If no rhs arguments provided the function % constructs a unit square. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 1 w = 1; h = 1; end if nargin < 2 h = w; end coefs = [0 w w w w 0 0 0; 0 0 0 h h h h 0; 0 0 0 0 0 0 0 0; 1 1 1 1 1 1 1 1]; knots = [0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1]; curve = nrbmak(coefs, knots); end %!demo %! crv = nrbtform(nrbrect(2,1), vecrotz(35*pi/180)); %! nrbplot(crv,4); %! axis equal %! title('Construction and rotation of a rectangular curve.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/vecrotz.m0000644000000000000000000000013214030323741015235 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecrotz.m0000644000175000017500000000333214030323741017335 0ustar00rvazquezrvazquez00000000000000function rz = vecrotz(angle) % % VECROTZ: Transformation matrix for a rotation around the z axis. % % Calling Sequence: % % rz = vecrotz(angle); % % INPUT: % % angle : rotation angle defined in radians % % OUTPUT: % % rz : (4x4) Transformation matrix. % % % Description: % % Return the (4x4) Transformation matrix for a rotation about the z axis % by the defined angle. % % The matrix is: % % [ cos(angle) -sin(angle) 0 0] % [ -sin(angle) cos(angle) 0 0] % [ 0 0 1 0] % [ 0 0 0 1] % % Examples: % % Rotate the NURBS line (0.0 0.0 0.0) - (3.0 3.0 3.0) by 45 degrees % around the z-axis % % line = nrbline([0.0 0.0 0.0],[3.0 3.0 3.0]); % trans = vecrotz(%pi/4); % rline = nrbtform(line, trans); % % See also: % % nrbtform % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . sn = sin(angle); cn = cos(angle); rz = [cn -sn 0 0; sn cn 0 0; 0 0 1 0; 0 0 0 1]; end nurbs-1.4.3/inst/PaxHeaders.16248/bspderiv.m0000644000000000000000000000013214030323741015357 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/bspderiv.m0000644000175000017500000000652414030323741017465 0ustar00rvazquezrvazquez00000000000000function [dc,dk] = bspderiv(d,c,k) % BSPDERIV: B-Spline derivative. % % MATLAB SYNTAX: % % [dc,dk] = bspderiv(d,c,k) % % INPUT: % % d - degree of the B-Spline % c - control points double matrix(mc,nc) % k - knot sequence double vector(nk) % % OUTPUT: % % dc - control points of the derivative double matrix(mc,nc) % dk - knot sequence of the derivative double vector(nk) % % Modified version of Algorithm A3.3 from 'The NURBS BOOK' pg98. % % Copyright (C) 2000 Mark Spink, 2007 Daniel Claxton % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . [mc,nc] = size(c); nk = numel(k); % % int bspderiv(int d, double *c, int mc, int nc, double *k, int nk, double *dc, % double *dk) % { % int ierr = 0; % int i, j, tmp; % % // control points % double **ctrl = vec2mat(c,mc,nc); % % // control points of the derivative dc = zeros(mc,nc-1); % double **dctrl = vec2mat(dc,mc,nc-1); % for i=0:nc-2 % for (i = 0; i < nc-1; i++) { tmp = d / (k(i+d+2) - k(i+2)); % tmp = d / (k[i+d+1] - k[i+1]); dc(1:mc,i+1) = tmp * (c(1:mc,i+2) - c(1:mc,i+1)); % for (j = 0; j < mc; j++) { % dctrl[i][j] = tmp * (ctrl[i+1][j] - ctrl[i][j]); end % } % } % dk = zeros(1,nk-2); % j = 0; dk(1:nk-2) = k(2:nk-1); % for (i = 1; i < nk-1; i++) % dk[j++] = k[i]; % % freevec2mat(dctrl); % freevec2mat(ctrl); % % return ierr; end % } nurbs-1.4.3/inst/PaxHeaders.16248/bspinterpsurf.m0000644000000000000000000000013214030323741016447 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/bspinterpsurf.m0000644000175000017500000001010014030323741020536 0ustar00rvazquezrvazquez00000000000000function srf = bspinterpsurf (X, Y, Z, p, method) % % BSPINTERPSURF: B-Spline surface interpolation. % % Calling Sequence: % % srf = bspinterpsurf (Q, p, method); % % INPUT: % % X, Y, Z - grid of points to be interpolated. (See ndgrid) % p - degree of the interpolating curve ([degree_x, degree_y]). % method - parametrization method. The available choices are: % 'equally_spaced' % 'chord_length' (default) % % OUTPUT: % % srf - the B-Spline surface. % % See The NURBS book pag. 376 for more information. As of now only the % chord length method is implemented. % % Copyright (C) 2015 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % if (nargin < 5 || isempty (method)) method = 'chord_length'; end [n, m] = size (X); Q = zeros (3, n, m); Q(1,:,:) = X; Q(2,:,:) = Y; Q(3,:,:) = Z; if (strcmpi (method, 'equally_spaced')) u = linspace (0, 1, n); v = linspace (0, 1, m); elseif (strcmp (method, 'chord_length')) u = zeros (m, n); for ii = 1:m d = sum (sqrt (sum (diff (squeeze(Q(:,:,ii))')'.^2,1))); u(ii,2:n) = cumsum (sqrt (sum (diff(Q(:,:,ii), [], 2).^2, 1)))/d; % for jj = 2:n-1 % u(ii,jj) = u(ii,jj-1) + norm (Q(:,jj,ii) - Q(:,jj-1,ii)) / d; % end u(ii,end) = 1; end u = mean (u); v = zeros (n, m); for ii = 1:n d = sum (sqrt (sum (diff (squeeze(Q(:,ii,:))')'.^2,1))); v(ii,2:m) = cumsum (sqrt (sum (diff(Q(:,ii,:), [], 3).^2, 1)))/d; % for jj = 2:m-1 % v(ii,jj) = v(ii,jj-1) + norm (Q(:,ii,jj) - Q(:,ii,jj-1)) / d; % end v(ii,end) = 1; end v = mean (v); end % TODO: implement centripetal method % Compute knot vectors knts{1} = zeros (1, n+p(1)+1); for jj = 2:n-p(1) knts{1}(jj+p(1)) = 1/p(1) * sum (u(jj:jj+p(1)-1)); end knts{1}(end-p(1):end) = ones (1, p(1)+1); knts{2} = zeros (1, m+p(2)+1); for jj = 2:m-p(2) knts{2}(jj+p(2)) = 1/p(2) * sum (v(jj:jj+p(2)-1)); end knts{2}(end-p(2):end) = ones (1, p(2)+1); % Interpolation R = zeros (size (Q)); P = zeros (4, n, m); for ii = 1:m A = zeros (n, n); A(1,1) = 1; A(n,n) = 1; for jj=2:n-1 span = findspan (n, p(1), u(jj), knts{1}); A(jj,span-p(1)+1:span+1) = basisfun (span, u(jj), p(1), knts{1}); end R(1,:,ii) = A \ squeeze(Q(1,:,ii))'; R(2,:,ii) = A \ squeeze(Q(2,:,ii))'; R(3,:,ii) = A \ squeeze(Q(3,:,ii))'; end for ii = 1:n A = zeros (m, m); A(1,1) = 1; A(m,m) = 1; for jj=2:m-1 span = findspan (m, p(2), v(jj), knts{2}); A(jj,span-p(2)+1:span+1) = basisfun (span, v(jj), p(2), knts{2}); end P(1,ii,:) = A \ squeeze(R(1,ii,:)); P(2,ii,:) = A \ squeeze(R(2,ii,:)); P(3,ii,:) = A \ squeeze(R(3,ii,:)); end P(4,:,:) = ones (n, m); % Create B-Spline interpolant srf = nrbmak (P, knts); end %!demo %! x = linspace (-3, 3, 40); %! y = linspace (-3, 3, 40); %! [X, Y] = meshgrid (x, y); %! Z = peaks (X, Y); %! %! srf1 = bspinterpsurf (X, Y, Z, [2 2], 'equally_spaced'); %! srf2 = bspinterpsurf (X, Y, Z, [2 2], 'chord_length'); %! figure %! nrbkntplot(srf1) %! title ('Approximation of the peaks functions, with the equally spaced method') %! figure %! nrbkntplot(srf2) %! title ('Approximation of the peaks functions, with the chord length method') nurbs-1.4.3/inst/PaxHeaders.16248/nrbctrlplot.m0000644000000000000000000000013214030323741016106 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbctrlplot.m0000644000175000017500000001026214030323741020206 0ustar00rvazquezrvazquez00000000000000function nrbctrlplot (nurbs, nsub) % NRBCTRLPLOT: Plot a NURBS entity along with its control points. % % Calling Sequence: % % nrbctrlplot (nurbs) % nrbkntplot(nurbs, npnts) % % INPUT: % % nurbs: NURBS curve, surface or volume, see nrbmak. % npnts: Number of evaluation points, for a surface or volume, a row % vector with the number of points along each direction. % % Example: % % Plot the test curve and test surface with their control polygon and % control net, respectively % % nrbctrlplot(nrbtestcrv) % nrbctrlplot(nrbtestsrf) % % See also: % % nrbkntplot % % Copyright (C) 2011, 2012 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 1) error ('nrbctrlplot: Need a NURBS to plot!'); end % Default values light='on'; cmap='summer'; colormap (cmap); hold_flag = ishold; if (iscell (nurbs.knots)) if (size (nurbs.knots,2) == 3) if (nargin < 2) nsub = [25 25 25]; elseif (numel(nsub) == 1) nsub = [nsub nsub nsub]; end nrbplot (nurbs, nsub, 'light', light, 'colormap', cmap); hold on % Plot the control points coefs = bsxfun (@rdivide, nurbs.coefs(1:3,:,:,:), nurbs.coefs(4,:,:,:)); coefs = reshape (coefs, 3, []); plot3 (coefs(1,:), coefs(2,:), coefs(3,:), 'r.','MarkerSize',20); % Plot the control net for ii = 1:size (nurbs.coefs, 2) for jj = 1:size (nurbs.coefs, 3) coefs = reshape (nurbs.coefs(1:3,ii,jj,:), 3, []); weights = reshape (nurbs.coefs(4,ii,jj,:), 1, []); plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'k--') end for kk = 1:size (nurbs.coefs, 4) coefs = reshape (nurbs.coefs(1:3,ii,:,kk), 3, []); weights = reshape (nurbs.coefs(4,ii,:,kk), 1, []); plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'k--') end end for jj = 1:size (nurbs.coefs, 3) for kk = 1:size (nurbs.coefs, 4) coefs = reshape (nurbs.coefs(1:3,:,jj,kk), 3, []); weights = reshape (nurbs.coefs(4,:,jj,kk), 1, []); plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'k--') end end elseif (size (nurbs.knots,2) == 2) % plot a NURBS surface if (nargin < 2) nsub = [50 50]; elseif (numel(nsub) == 1) nsub = [nsub nsub]; end nrbplot (nurbs, nsub, 'light', light, 'colormap', cmap); hold on % And plot the control net for ii = 1:size (nurbs.coefs, 2) coefs = reshape (nurbs.coefs(1:3,ii,:), 3, []); weights = reshape (nurbs.coefs(4,ii,:), 1, []); plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'k--') plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'r.','MarkerSize',20) end for jj = 1:size (nurbs.coefs, 3) coefs = reshape (nurbs.coefs(1:3,:,jj), 3, []); weights = reshape (nurbs.coefs(4,:,jj), 1, []); plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'k--') plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'r.','MarkerSize',20) end end else % plot a NURBS curve if (nargin < 2) nsub = 1000; end nrbplot (nurbs, nsub); hold on % And plot the control polygon coefs = nurbs.coefs(1:3,:); weights = nurbs.coefs(4,:); plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'k--') plot3 (coefs(1,:)./weights, coefs(2,:)./weights, coefs(3,:)./weights,'r.','MarkerSize',20) end if (~hold_flag) hold off end end %!demo %! crv = nrbtestcrv; %! nrbctrlplot(crv) %! title('Test curve') %! hold off %!demo %! srf = nrbtestsrf; %! nrbctrlplot(srf) %! title('Test surface') %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/nrbkntins.m0000644000000000000000000000013014030323741015547 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbkntins.m0000644000175000017500000001367714030323741017666 0ustar00rvazquezrvazquez00000000000000function inurbs = nrbkntins(nurbs,iknots) % % NRBKNTINS: Insert a single or multiple knots into a NURBS curve, % surface or volume. % % Calling Sequence: % % icrv = nrbkntins(crv,iuknots); % isrf = nrbkntins(srf,{iuknots ivknots}); % ivol = nrbkntins(vol,{iuknots ivknots iwknots}); % % INPUT: % % crv : NURBS curve, see nrbmak. % % srf : NURBS surface, see nrbmak. % % srf : NURBS volume, see nrbmak. % % iuknots : Knots to be inserted along U direction. % % ivknots : Knots to be inserted along V direction. % % iwknots : Knots to be inserted along W direction. % % OUTPUT: % % icrv : new NURBS structure for a curve with knots inserted. % % isrf : new NURBS structure for a surface with knots inserted. % % ivol : new NURBS structure for a volume with knots inserted. % % Description: % % Inserts knots into the NURBS data structure, these can be knots at % new positions or at the location of existing knots to increase the % multiplicity. The knot multiplicity can be increased up to the order of % the spline. Any further increase of the multiplicity will generate zero % basis functions, but not cause any error in the code. % This function use the B-Spline function bspkntins, which interfaces to % an internal 'C' routine. % % Examples: % % Insert two knots into a curve, one at 0.3 and another % twice at 0.4 % % icrv = nrbkntins(crv, [0.3 0.4 0.4]) % % Insert into a surface two knots as (1) into the U knot % sequence and one knot into the V knot sequence at 0.5. % % isrf = nrbkntins(srf, {[0.3 0.4 0.4] [0.5]}) % % See also: % % bspkntins % % Note: % % The knot multiplicty can be increased beyond the order of the spline % without causing errors, but the added basis functions will be equal to % zero. % % Copyright (C) 2000 Mark Spink, 2010 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 2 error('Input argument must include the NURBS and knots to be inserted'); end if ~isstruct(nurbs) error('NURBS representation is not structure!'); end if ~strcmp(nurbs.form,'B-NURBS') error('Not a recognised NURBS representation'); end degree = nurbs.order-1; if iscell(nurbs.knots) fmax = @(x,y) any (y > max(x)); fmin = @(x,y) any (y < min(x)); if (any(cellfun(fmax, nurbs.knots, iknots)) || any(cellfun(fmin, nurbs.knots, iknots))) error ('Trying to insert a knot outside the interval of definition') end if size(nurbs.knots,2)==3 % NURBS represents a volume num1 = nurbs.number(1); num2 = nurbs.number(2); num3 = nurbs.number(3); % Insert knots along the w direction if isempty(iknots{3}) coefs = nurbs.coefs; knots{3} = nurbs.knots{3}; else coefs = reshape(nurbs.coefs,4*num1*num2,num3); [coefs,knots{3}] = bspkntins(degree(3),coefs,nurbs.knots{3},iknots{3}); num3 = size(coefs,2); coefs = reshape(coefs,[4 num1 num2 num3]); end % Insert knots along the v direction if isempty(iknots{2}) knots{2} = nurbs.knots{2}; else coefs = permute(coefs,[1 2 4 3]); coefs = reshape(coefs,4*num1*num3,num2); [coefs,knots{2}] = bspkntins(degree(2),coefs,nurbs.knots{2},iknots{2}); num2 = size(coefs,2); coefs = reshape(coefs,[4 num1 num3 num2]); coefs = permute(coefs,[1 2 4 3]); end % Insert knots along the u direction if isempty(iknots{1}) knots{1} = nurbs.knots{1}; else coefs = permute(coefs,[1 3 4 2]); coefs = reshape(coefs,4*num2*num3,num1); [coefs,knots{1}] = bspkntins(degree(1),coefs,nurbs.knots{1},iknots{1}); coefs = reshape(coefs,[4 num2 num3 size(coefs,2)]); coefs = permute(coefs,[1 4 2 3]); end elseif size(nurbs.knots,2)==2 % NURBS represents a surface num1 = nurbs.number(1); num2 = nurbs.number(2); % Insert knots along the v direction if isempty(iknots{2}) coefs = nurbs.coefs; knots{2} = nurbs.knots{2}; else coefs = reshape(nurbs.coefs,4*num1,num2); [coefs,knots{2}] = bspkntins(degree(2),coefs,nurbs.knots{2},iknots{2}); num2 = size(coefs,2); coefs = reshape(coefs,[4 num1 num2]); end % Insert knots along the u direction if isempty(iknots{1}) knots{1} = nurbs.knots{1}; else coefs = permute(coefs,[1 3 2]); coefs = reshape(coefs,4*num2,num1); [coefs,knots{1}] = bspkntins(degree(1),coefs,nurbs.knots{1},iknots{1}); coefs = reshape(coefs,[4 num2 size(coefs,2)]); coefs = permute(coefs,[1 3 2]); end end else if (any(iknots > max(nurbs.knots)) || any(iknots < min(nurbs.knots))) error ('Trying to insert a knot outside the interval of definition') end % NURBS represents a curve if isempty(iknots) coefs = nurbs.coefs; knots = nurbs.knots; else [coefs,knots] = bspkntins(degree,nurbs.coefs,nurbs.knots,iknots); end end % construct new NURBS inurbs = nrbmak(coefs,knots); end %!demo %! crv = nrbtestcrv; %! plot(crv.coefs(1,:),crv.coefs(2,:),'bo') %! title('Knot insertion along test curve: curve and control polygons.'); %! hold on; %! plot(crv.coefs(1,:),crv.coefs(2,:),'b--'); %! %! nrbplot(crv,48); %! %! icrv = nrbkntins(crv,[0.125 0.375 0.625 0.875] ); %! plot(icrv.coefs(1,:),icrv.coefs(2,:),'ro') %! plot(icrv.coefs(1,:),icrv.coefs(2,:),'r--'); %! hold offnurbs-1.4.3/inst/PaxHeaders.16248/curvederivcpts.m0000644000000000000000000000013214030323741016611 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/curvederivcpts.m0000644000175000017500000000362714030323741020720 0ustar00rvazquezrvazquez00000000000000function pk = curvederivcpts (n, p, U, P, d, r1, r2) % Compute control points of n-th derivatives of a B-spline curve. % % usage: pk = curvederivcpts (n, p, U, P, d) % pk = curvederivcpts (n, p, U, P, d, r1, r2) % % If r1, r2 are not given, all the control points are computed. % % INPUT: % n+1 = number of control points % p = degree of the spline % d = maximum derivative order (d<=p) % U = knots % P = control points % r1 = first control point to compute % r2 = auxiliary index for the last control point to compute % OUTPUT: % pk(k,i) = i-th control point of (k-1)-th derivative, r1 <= i <= r2-k % % Adaptation of algorithm A3.3 from the NURBS book, pg98. % % Copyright (C) 2009 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin <= 5) r1 = 0; r2 = n; end r = r2 - r1; for i=0:r pk(1, i+1) = P(r1+i+1); end for k=1:d tmp = p - k + 1; for i=0:r-k pk (k+1, i+1) = tmp * (pk(k,i+2)-pk(k,i+1)) / ... (U(r1+i+p+2)-U(r1+i+k+1)); end end end %!test %! line = nrbmak([0.0 1.5; 0.0 3.0],[0.0 0.0 1.0 1.0]); %! pk = curvederivcpts (line.number-1, line.order-1, line.knots,... %! line.coefs(1,:), 2); %! assert (pk, [0 3/2; 3/2 0], 100*eps); nurbs-1.4.3/inst/PaxHeaders.16248/bspkntins.m0000644000000000000000000000013214030323741015554 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/bspkntins.m0000644000175000017500000001355614030323741017665 0ustar00rvazquezrvazquez00000000000000function [ic,ik,C] = bspkntins(d,c,k,u) % BSPKNTINS: Insert knots into a B-Spline % % Calling Sequence: % % [ic,ik] = bspkntins(d,c,k,u) % % INPUT: % % d - spline degree integer % c - control points double matrix(mc,nc) % k - knot sequence double vector(nk) % u - new knots double vector(nu) % % OUTPUT: % % ic - new control points double matrix(mc,nc+nu) % ik - new knot sequence double vector(nk+nu) % % Modified version of Algorithm A5.4 from 'The NURBS BOOK' pg164. % % Copyright (C) 2000 Mark Spink, 2007 Daniel Claxton, 2010-2016 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . [mc,nc] = size(c); u = sort(u); nu = numel(u); nk = numel(k); % % int bspkntins(int d, double *c, int mc, int nc, double *k, int nk, % double *u, int nu, double *ic, double *ik) % { % int ierr = 0; % int a, b, r, l, i, j, m, n, s, q, ind; % double alfa; % % double **ctrl = vec2mat(c, mc, nc); ic = zeros(mc,nc+nu); % double **ictrl = vec2mat(ic, mc, nc+nu); ik = zeros(1,nk+nu); % n = nc - 1; % n = nc - 1; r = nu - 1; % r = nu - 1; % m = n + d + 1; % m = n + d + 1; a = findspan(n, d, u(1), k); % a = findspan(n, d, u[0], k); b = findspan(n, d, u(r+1), k); % b = findspan(n, d, u[r], k); b = b+1; % ++b; % % for (q = 0; q < mc; q++) { ic(:,1:a-d+1) = c(:,1:a-d+1); % for (j = 0; j <= a-d; j++) ictrl[j][q] = ctrl[j][q]; ic(:,b+nu:nc+nu) = c(:,b:nc); % for (j = b-1; j <= n; j++) ictrl[j+r+1][q] = ctrl[j][q]; % } ik(1:a+1) = k(1:a+1); % for (j = 0; j <= a; j++) ik[j] = k[j]; ik(b+d+nu+1:m+nu+1) = k(b+d+1:m+1); % for (j = b+d; j <= m; j++) ik[j+r+1] = k[j]; % ii = b + d - 1; % i = b + d - 1; ss = ii + nu; % s = b + d + r; for jj=r:-1:0 % for (j = r; j >= 0; j--) { ind = (a+1):ii; % while (u[j] <= k[i] && i > a) { ind = ind(u(jj+1)<=k(ind+1)); % for (q = 0; q < mc; q++) ic(:,ind+ss-ii-d) = c(:,ind-d); % ictrl[s-d-1][q] = ctrl[i-d-1][q]; ik(ind+ss-ii+1) = k(ind+1); % ik[s] = k[i]; ii = ii - numel(ind); % --i; ss = ss - numel(ind); % --s; % } ic(:,ss-d) = ic(:,ss-d+1); % ictrl[s-d-1][q] = ictrl[s-d][q]; for l=1:d % for (l = 1; l <= d; l++) { ind = ss - d + l; % ind = s - d + l; alfa = ik(ss+l+1) - u(jj+1); % alfa = ik[s+l] - u[j]; if abs(alfa) == 0 % if (fabs(alfa) == 0.0) ic(:,ind) = ic(:,ind+1); % for (q = 0; q < mc; q++) % ictrl[ind-1][q] = ictrl[ind][q]; else % else { alfa = alfa/(ik(ss+l+1) - k(ii-d+l+1)); % alfa /= (ik[s+l] - k[i-d+l]); tmp = (1-alfa) * ic(:,ind+1); % for (q = 0; q < mc; q++) ic(:,ind) = alfa*ic(:,ind) + tmp; % ictrl[ind-1][q] = alfa*ictrl[ind-1][q]+(1.0-alfa)*ictrl[ind][q]; end % } end % } % ik(ss+1) = u(jj+1); % ik[s] = u[j]; ss = ss - 1; end % } % % freevec2mat(ctrl); % freevec2mat(ictrl); % % return ierr; end % } nurbs-1.4.3/inst/PaxHeaders.16248/nrbtform.m0000644000000000000000000000013214030323741015372 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbtform.m0000644000175000017500000000472714030323741017503 0ustar00rvazquezrvazquez00000000000000function nurbs = nrbtform(nurbs,tmat) % % NRBTFORM: Apply transformation matrix to the NURBS. % % Calling Sequence: % % tnurbs = nrbtform(nurbs,tmatrix); % % INPUT: % % nurbs : NURBS data structure (see nrbmak for details). % % tmatrix : Transformation matrix, a matrix of size (4,4) defining % a single or multiple transformations. % % OUTPUT: % % tnurbs : The return transformed NURBS data structure. % % Description: % % The NURBS is transform as defined a transformation matrix of size (4,4), % such as a rotation, translation or change in scale. The transformation % matrix can define a single transformation or multiple series of % transformations. The matrix can be simply constructed by the functions % vecscale, vectrans and vecrot, and also vecrotx, vecroty, and vecrotz. % % Examples: % % Rotate a square by 45 degrees about the z axis. % % rsqr = nrbtform(nrbrect(), vecrotz(45*pi/180)); % nrbplot(rsqr, 1000); % % See also: % % vecscale, vectrans, vecrot, vecrotx, vecroty, vecrotz % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 2 error('Not enough input arguments!'); end; if iscell(nurbs.knots) if size(nurbs.knots,2) == 2 % NURBS is a surface [dim,nu,nv] = size(nurbs.coefs); nurbs.coefs = reshape(tmat*reshape(nurbs.coefs,dim,nu*nv),[dim nu nv]); elseif size(nurbs.knots,2) == 3 % NURBS is a volume [dim,nu,nv,nw] = size(nurbs.coefs); nurbs.coefs = reshape(tmat*reshape(nurbs.coefs,dim,nu*nv*nw),[dim nu nv nw]); end else % NURBS is a curve nurbs.coefs = tmat*nurbs.coefs; end end %!demo %! xx = vectrans([2.0 1.0])*vecroty(pi/8)*vecrotx(pi/4)*vecscale([1.0 2.0]); %! c0 = nrbtform(nrbcirc, xx); %! nrbplot(c0,50); %! grid on %! title('Construction of an ellipse by transforming a unit circle.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/nrbcylind.m0000644000000000000000000000013214030323741015525 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbcylind.m0000644000175000017500000000352314030323741017627 0ustar00rvazquezrvazquez00000000000000function surf = nrbcylind(height,radius,center,sang,eang) % % NRBCYLIND: Construct a cylinder or cylindrical patch. % % Calling Sequence: % % srf = nrbcylind() % srf = nrbcylind(height) % srf = nrbcylind(height,radius) % srf = nrbcylind(height,radius,center) % srf = nrbcylind(height,radius,center,sang,eang) % % INPUT: % % height : Height of the cylinder along the axis, default 1.0 % % radius : Radius of the cylinder, default 1.0 % % center : Center of the cylinder, default (0,0,0) % % sang : Start angle relative to the origin, default 0. % % eang : End angle relative to the origin, default 2*pi. % % OUTPUT: % % srf : cylindrical surface patch % % Description: % % Construct a cylinder or cylindrical patch by extruding a circular arc. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 1 height = 1; end if nargin < 2 radius = 1; end if nargin < 3 center = []; end if nargin < 5 sang = 0; eang = 2*pi; end surf = nrbextrude(nrbcirc(radius,center,sang,eang),[0.0 0.0 height]); end %!demo %! srf = nrbcylind(3,1,[],3*pi/2,pi); %! nrbplot(srf,[20,20]); %! axis equal; %! title('Cylinderical section by extrusion of a circular arc.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/nrbunclamp.m0000644000000000000000000000013214030323741015702 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbunclamp.m0000644000175000017500000000710214030323741020001 0ustar00rvazquezrvazquez00000000000000function ucrv = nrbunclamp (crv, k, xdim) % NRBUNCLAMP: Compute the knot vector and control points of the unclamped curve or surface. % % Calling Sequence: % % ucrv = nrbrunclamp (crv) % ucrv = nrbrunclamp (crv, k) % ucrv = nrbrunclamp (crv, k, dim) % % INPUT: % % crv : NURBS curve or surface, see nrbmak. % k : continuity for the unclamping, from 0 up to p-1 (p-1 by default). % dim : dimension in which to unclamp (all by default). % % OUTPUT: % % ucrv: NURBS curve with unclamped knot vector, see nrbmak % % Description: % % Unclamps a curve, removing the open knot vector. Computes the new % knot vector and control points of the unclamped curve. % % Adapted from Algorithm A12.1 from 'The NURBS BOOK' pg577. % % Copyright (C) 2013, 2014, 2017 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (iscell (crv.knots)) knt = crv.knots; curve = false; else knt = {crv.knots}; curve = true; end ndim = numel (knt); if (nargin < 3) xdim = 1:ndim; end if (nargin < 2 || isempty (k)) k = crv.order - 2; end %if (iscell (crv.knots)) if (numel(k) ~= ndim) k = k * ones(1, ndim); end Pw = crv.coefs; for idim = xdim p = crv.order(idim) - 1; U = knt{idim}; n = crv.number(idim); m = n + p + 1; kk = k(idim); if (kk >= p) warning ('Taking the maximum k allowed, degree - 1') kk = p - 1; end % Unclamp at left end for ii=0:kk U(kk-ii+1) = U(kk-ii+2) - (U(n+1-ii) - U(n-ii)); end Pw = permute (Pw, [1, circshift([2 3 4], [0, 1-idim])]); for ii = p-kk-1:p-2 for jj = ii:-1:0 alpha = (U(p+1) - U(p+jj-ii)) / (U(p+jj+2) - U(p+jj-ii)); Pw(:,jj+1,:,:) = (Pw(:,jj+1,:,:) - alpha*Pw(:,jj+2,:,:))/(1-alpha); end end % Unclamp at right end for ii=0:kk U(m-kk+ii) = U(m-kk+ii-1) + U(p+ii+1+1) - U(p+ii+1); end for ii = p-kk-1:p-2 for jj = ii:-1:0 alpha = (U(n+1)-U(n-jj))/(U(n+2-jj+ii)-U(n-jj)); Pw(:,n-jj,:,:) = (Pw(:,n-jj,:,:) - (1-alpha)*Pw(:,n-jj-1,:,:))/alpha; end end Pw = permute (Pw, [1, circshift([2 3 4], [0, idim-1])]); knt{idim} = U; end if (~curve) ucrv = nrbmak (Pw, knt); else ucrv = nrbmak (Pw, knt{:}); end %!demo %! crv = nrbcirc (1,[],0,2*pi/3); %! crv = nrbdegelev (crv, 2); %! figure %! nrbctrlplot (crv); hold on %! nrbctrlplot (nrbtform (nrbunclamp (crv, 1), vectrans([-0.4, -0.4]))); %! nrbctrlplot (nrbtform (nrbunclamp (crv, 2), vectrans([-0.8, -0.8]))); %! nrbctrlplot (nrbtform (nrbunclamp (crv, 3), vectrans([-1.6, -1.6]))); %! title ('Original curve and unclamped versions') %!test %! crv = nrbdegelev (nrbtestcrv,2); %! x = linspace (0, 1, 100); %! F = nrbeval (crv, x); %! ucrv = nrbunclamp (crv, 0); %! assert (F, nrbeval(ucrv, x)); %! ucrv = nrbunclamp (crv, 1); %! assert (F, nrbeval(ucrv, x), 1e-14); %! ucrv = nrbunclamp (crv, 2); %! assert (F, nrbeval(ucrv, x), 1e-14); %! ucrv = nrbunclamp (crv, 3); %! assert (F, nrbeval(ucrv, x), 1e-14); nurbs-1.4.3/inst/PaxHeaders.16248/bspeval.m0000644000000000000000000000013214030323741015175 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/bspeval.m0000644000175000017500000000737414030323741017307 0ustar00rvazquezrvazquez00000000000000function p = bspeval(d,c,k,u) % BSPEVAL: Evaluate B-Spline at parametric points. % % Calling Sequence: % % p = bspeval(d,c,k,u) % % INPUT: % % d - Degree of the B-Spline. % c - Control Points, matrix of size (dim,nc). % k - Knot sequence, row vector of size nk. % u - Parametric evaluation points, row vector of size nu. % % OUTPUT: % % p - Evaluated points, matrix of size (dim,nu) % % Copyright (C) 2000 Mark Spink, 2007 Daniel Claxton, 2010 C. de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . nu = numel(u); [mc,nc] = size(c); % int bspeval(int d, double *c, int mc, int nc, double *k, int nk, double *u,int nu, double *p){ % int ierr = 0; % int i, s, tmp1, row, col; % double tmp2; % % // Construct the control points % double **ctrl = vec2mat(c,mc,nc); % % // Contruct the evaluated points % double **pnt = vec2mat(p,mc,nu); % % // space for the basis functions %N = zeros(d+1,1); % double *N = (double*) mxMalloc((d+1)*sizeof(double)); % % // for each parametric point i %for col=1:nu % for (col = 0; col < nu; col++) { % // find the span of u[col] s = findspan(nc-1, d, u(:), k); % s = findspan(nc-1, d, u[col], k); N = basisfun(s,u(:),d,k); % basisfun(s, u[col], d, k, N); % tmp1 = s - d + 1; % tmp1 = s - d; %for row=1:mc % for (row = 0; row < mc; row++) { p = zeros (mc, nu); % tmp2 = 0.0; for i=0:d % for (i = 0; i <= d; i++) p = p + repmat (N(:,i+1)', mc, 1).*c(:,tmp1+i); % tmp2 += N[i] * ctrl[tmp1+i][row]; end % %p(row,:) = tmp2; % pnt[col][row] = tmp2; %end % } %end % } % % mxFree(N); % freevec2mat(pnt); % freevec2mat(ctrl); % % return ierr; end % } nurbs-1.4.3/inst/PaxHeaders.16248/nrbrevolve.m0000644000000000000000000000013214030323741015725 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbrevolve.m0000644000175000017500000001277714030323741020042 0ustar00rvazquezrvazquez00000000000000function surf = nrbrevolve(curve,pnt,vec,theta) % % NRBREVOLVE: Construct a NURBS surface by revolving a NURBS curve, or % construct a NURBS volume by revolving a NURBS surface. % % Calling Sequence: % % srf = nrbrevolve(crv,pnt,vec[,ang]) % % INPUT: % % crv : NURBS curve or surface to revolve, see nrbmak. % % pnt : Coordinates of the point used to define the axis % of rotation. % % vec : Vector defining the direction of the rotation axis. % % ang : Angle to revolve the curve, default 2*pi % % OUTPUT: % % srf : constructed surface or volume % % Description: % % Construct a NURBS surface by revolving the profile NURBS curve around % an axis defined by a point and vector. % % Examples: % % Construct a sphere by rotating a semicircle around a x-axis. % % crv = nrbcirc(1.0,[0 0 0],0,pi); % srf = nrbrevolve(crv,[0 0 0],[1 0 0]); % nrbplot(srf,[20 20]); % % NOTE: % % The algorithm: % % 1) vectrans the point to the origin (0,0,0) % 2) rotate the vector into alignment with the z-axis % % for each control point along the curve % % 3) determine the radius and angle of control % point to the z-axis % 4) construct a circular arc in the x-y plane with % this radius and start angle and sweep angle theta % 5) combine the arc and profile, coefs and weights. % % next control point % % 6) rotate and vectrans the surface back into position % by reversing 1 and 2. % % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 3) error('Not enough arguments to construct revolved surface'); end if (nargin < 4) theta = 2.0*pi; end if (iscell (curve.knots) && numel(curve.knots) == 3) error('The function nrbrevolve is not yet ready to create volumes') end % Translate curve the center point to the origin if isempty(pnt) pnt = zeros(3,1); end if length(pnt) ~= 3 error('All point and vector coordinates must be 3D'); end % Translate and rotate the original curve or surface into alignment with the z-axis T = vectrans(-pnt); angx = vecangle(vec(1),vec(3)); RY = vecroty(-angx); vectmp = RY*[vecnormalize(vec(:));1.0]; angy = vecangle(vectmp(2),vectmp(3)); RX = vecrotx(angy); curve = nrbtform(curve,RX*RY*T); % Construct an arc arc = nrbcirc(1.0,[],0.0,theta); if (iscell (curve.knots)) % Construct the revolved volume coefs = zeros([4 arc.number curve.number]); angle = squeeze (vecangle(curve.coefs(2,:,:),curve.coefs(1,:,:))); radius = squeeze (vecmag(curve.coefs(1:2,:,:))); for i = 1:curve.number(1) for j = 1:curve.number(2) coefs(:,:,i,j) = vecrotz(angle(i,j))*vectrans([0.0 0.0 curve.coefs(3,i,j)])*... vecscale([radius(i,j) radius(i,j)])*arc.coefs; coefs(4,:,i,j) = coefs(4,:,i,j)*curve.coefs(4,i,j); end end surf = nrbmak(coefs,{arc.knots, curve.knots{:}}); else % Construct the revolved surface coefs = zeros(4, arc.number, curve.number); angle = vecangle(curve.coefs(2,:),curve.coefs(1,:)); radius = vecmag(curve.coefs(1:2,:)); for i = 1:curve.number coefs(:,:,i) = vecrotz(angle(i))*vectrans([0.0 0.0 curve.coefs(3,i)])*... vecscale([radius(i) radius(i)])*arc.coefs; coefs(4,:,i) = coefs(4,:,i)*curve.coefs(4,i); end surf = nrbmak(coefs,{arc.knots, curve.knots}); end % Rotate and vectrans the surface back into position T = vectrans(pnt); RX = vecrotx(-angy); RY = vecroty(angx); surf = nrbtform(surf,T*RY*RX); end %!demo %! sphere = nrbrevolve(nrbcirc(1,[],0.0,pi),[0.0 0.0 0.0],[1.0 0.0 0.0]); %! nrbplot(sphere,[40 40],'light','on'); %! title('Ball and tori - surface construction by revolution'); %! hold on; %! torus = nrbrevolve(nrbcirc(0.2,[0.9 1.0]),[0.0 0.0 0.0],[1.0 0.0 0.0]); %! nrbplot(torus,[40 40],'light','on'); %! nrbplot(nrbtform(torus,vectrans([-1.8])),[20 10],'light','on'); %! hold off; %!demo %! pnts = [3.0 5.5 5.5 1.5 1.5 4.0 4.5; %! 0.0 0.0 0.0 0.0 0.0 0.0 0.0; %! 0.5 1.5 4.5 3.0 7.5 6.0 8.5]; %! crv = nrbmak(pnts,[0 0 0 1/4 1/2 3/4 3/4 1 1 1]); %! %! xx = vecrotz(25*pi/180)*vecroty(15*pi/180)*vecrotx(20*pi/180); %! nrb = nrbtform(crv,vectrans([5 5])*xx); %! %! pnt = [5 5 0]'; %! vec = xx*[0 0 1 1]'; %! srf = nrbrevolve(nrb,pnt,vec(1:3)); %! %! p = nrbeval(srf,{linspace(0.0,1.0,100) linspace(0.0,1.0,100)}); %! surfl(squeeze(p(1,:,:)),squeeze(p(2,:,:)),squeeze(p(3,:,:))); %! title('Construct of a 3D surface by revolution of a curve.'); %! shading interp; %! colormap(copper); %! axis equal; %! hold off %!demo %! crv1 = nrbcirc(1,[0 0],0, pi/2); %! crv2 = nrbcirc(2,[0 0],0, pi/2); %! srf = nrbruled (crv1, crv2); %! srf = nrbtform (srf, [1 0 0 0; 0 1 0 1; 0 0 1 0; 0 0 0 1]); %! vol = nrbrevolve (srf, [0 0 0], [1 0 0], pi/2); %! nrbplot(vol, [30 30 30], 'light', 'on') nurbs-1.4.3/inst/PaxHeaders.16248/nrbspheretile.m0000644000000000000000000000013214030323741016407 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbspheretile.m0000644000175000017500000001756514030323741020524 0ustar00rvazquezrvazquez00000000000000function [tile] = nrbspheretile(ver3d) % % NRBSPHERETILE: Makes a quadrilateral NURBS tile of the unit sphere % from four vertex points. % % Calling Sequence: % % [tile] = nrbspheretile(ver3d) % % INPUT: % % ver3d: 3-by-4 matrix with the coordinates of 4 (ordered) vertices on % the sphere. Vertices should be ordered clockwise when viewed % from outside the sphere. The tile should be convex and should % contain the south pole. % % OUTPUT: % % tile: NURBS object representing the tile of the sphere. The tile % will be a quartic rational Bezier patch. % % See also: nrbspheretiling % % This function is based on the paper: % J.E. Cobb, Tiling the Sphere with Rational Bezier patches, 1988 % % For more details, see: % Sander Dedoncker, Laurens Coox, Florian Maurin, Francesco Greco, Wim Desmet % Bézier tilings of the sphere and their applications in benchmarking multipatch isogeometric methods % Computer Meth. Appl. Mech. Engrg., 2018 % % Copyright (C) 2017 Sander Dedoncker % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . %% Catch invalid input if nargin > 1 error('Too many input arguments'); end if size(ver3d,2) ~= 4 || size(ver3d,1) ~= 3 error('Give a matrix of four column vectors.'); end % ver3dnorm = round(sum(ver3d.^2,1),14); ver3dnorm = sqrt (sum (ver3d.^2, 1)); if (any (abs (ver3dnorm - 1) > 1e-14)) error('Vertices must lie on the unit sphere.') end %% Initialize variables planes = zeros(3,4); % matrix of a and b coeffs circles = zeros(3,4); % matrix of [Cu;Cv;R] hecon = zeros(4,8); % homogeneous coordinates of the external control points hcon2d = zeros(4,3,3); % homogeneous coordinates for control points of the sgp tile NM = zeros(25,25); % coeff matrix of the collocation system hcon3d = zeros(4,5,5); % homogeneous coordinates for control points of the sphere tile knots_deg2 = [0 0 0 1 1 1]; knots_deg4 = [0 0 0 0 0 1 1 1 1 1]; %% Solve ver3d = [ver3d,ver3d(:,1),ver3d(:,2)]; % repeat first and second point ver2d = istf(ver3d); % representations of the points in the uv plane % Find planes through great circle (ax+by+z=0 or ax+by=0) % planes(:,i) = [ai;bi;1] or [ai;bi;0] % Planes are defined by their inward (wrt the tile) pointing normal vectors npchk = zeros(1,4); spchk = zeros(1,4); for ii = 1:4 nvec1 = cross((ver3d(:,ii+1)-ver3d(:,ii)),ver3d(:,ii)); % inward pointing vector normal to the current plane nvec1 = nvec1./norm(nvec1); npchk(ii) = [0 0 1]*nvec1; % keeps track of location of north pole wrt the planes spchk(ii) = [0 0 -1]*nvec1; % keeps track of location of south pole wrt the planes if nvec1(3,1) == 0 % plane is vertical (ax+by=0) planes(:,ii) = nvec1(1:3,1); else % plane is not vertical (ax+by+z=0) planes(:,ii) = nvec1(1:3,1)./nvec1(3,1); end nvec2 = cross((ver3d(:,ii+2)-ver3d(:,ii+1)),ver3d(:,ii+1)); % inward pointing vector normal to the subsequent plane nvec2 = nvec2./norm(nvec2); tvec = cross(ver3d(:,ii+1),nvec2); % vector tangential to the next plane and normal to the intersection line tvec = tvec./norm(tvec); if nvec1'*tvec<0 || any(isnan([nvec1;tvec])) error('Vertices are invalid. Check that: no two vertices are the same, the internal angles are smaller than 180 degrees, the vertices are ordered correctly.'); end end if any(spchk < zeros(1,4)) error('Tile does not contain the south pole, which may lead to a noninjective mapping. Consider rotating the vertices.') end if all(npchk >= zeros(1,4)) error('Tile contains the north pole. Consider rotating the vertices.') end % Find circle arcs representing edges in the (u,v) plane. % R = 2*sqrt(1+a^2+b^2) and C = (-2a,-2b) circles(1,:) = -2*planes(1,:); circles(2,:) = -2*planes(2,:); circles(3,:) = 2*sqrt(1 + planes(1,:).^2 + planes(2,:).^2); % Find external control points in the z=-1 plane for ii=1:4 P1 = ver2d(:,ii); P3 = ver2d(:,ii+1); Q = (P1+P3)/2; C = circles(1:2,ii); R = circles(3,ii); if planes(3,ii) == 0 %[-ver3d(2,i),ver3d(1,i)]*(ver3d(1:2,i) - ver3d(1:2,i+1)) == 0 % plane is vertical (ax+by=0) P2 = Q; w2 = 1; else P2 = C+(Q-C)*R^2/norm(Q-C)^2; % |CP2| = |CQ|/(cos(alpha)^2) w2 = norm(Q-C)/R; % w2 = cos(alpha) end hecon(:,(ii-1)*2+1:ii*2)=[P1,P2*w2;-1,-1*w2;1,w2];% in 4D homogenous coordinates! Needed to construct patch properly end % Construct control net in the z=-1 plane hcon2d(:,1,1) = hecon(:,1); hcon2d(:,2,1) = hecon(:,2); hcon2d(:,3,1) = hecon(:,3); hcon2d(:,3,2) = hecon(:,4); hcon2d(:,3,3) = hecon(:,5); hcon2d(:,2,3) = hecon(:,6); hcon2d(:,1,3) = hecon(:,7); hcon2d(:,1,2) = hecon(:,8); hcon2d(:,2,2) = [0;0;-1;1]; % central point % Make patch in the z=-1 plane sgp = nrbmak(hcon2d,{knots_deg2, knots_deg2}); % Calculate points corresponding to collocation points xi = 0:1/4:1; % collocation points in parametric space et = xi; [p,w] = nrbeval(sgp,{xi,et}); hcol2d = [p;w]; % Project onto the sphere using the s-mapping hcol3d = pstf(hcol2d); % Evaluate the basis functions for the projective 3 space in the collocation points bfxi = basisfun (findspan(4,4,xi,knots_deg4),xi,4,knots_deg4); bfet = basisfun (findspan(4,4,et,knots_deg4),et,4,knots_deg4); for ii = 1:5 for jj = 1:5 temp = bfxi(ii,:)'*bfet(jj,:); NM(ii + (jj-1)*5,:) = temp(:); % equation for collocation point k end end % Solve for each dimension separately for ii = 1:4 rhs = hcol3d(ii,:,:); temp = NM\rhs(:); hcon3d(ii,:,:) = reshape(temp,[1,5,5]); end % Make tile tile = nrbmak(hcon3d,{knots_deg4, knots_deg4}); end %% Auxiliary functions function [ uv ] = istf( xyz ) % % ISTF: s-mapping of the unit sphere in xyz-space to the uv plane. See % Cobb, 1988 % % Calling Sequences: % % [ xyz ] = istf( uv ) % % INPUT: % % xyz : 3-by-n-by-m array with points in the xyz-space % % OUTPUT: % % uv : 2-by-n-by-m array with points in the uv-plane % x = xyz(1,:); y = xyz(2,:); z = xyz(3,:); dim = size(xyz); dim(1) = 2; u = 2*x./(-z+1); v = 2*y./(-z+1); uv = [u;v]; uv = reshape(uv,dim); end function [ xyzw ] = pstf( uvw ) % % PSTF: s-mapping of the projective uv plane to the unit sphere in % projective xyz-space. See Cobb, 1988 % % Calling Sequences: % % [ xyzw ] = pstf( uvw ) % % INPUT: % % uvw : 3-by-n-by-m array with points in the projective uv-plane % % OUTPUT: % % xyzw : 4-by-n-by-m array with points in the projective xyz-space % u = uvw(1,:); v = uvw(2,:); w = uvw(end,:); dim = size(uvw); dim(1) = 4; x = 4*u.*w; y = 4*v.*w; z = u.*u+v.*v-4*w.*w; w2 = u.*u+v.*v+4*w.*w; xyzw = [x;y;z;w2]; xyzw = reshape(xyzw,dim); end %!demo %! vertices = [1 -1 -1 1;1 1 -1 -1;-1 -1 -1 -1]/sqrt(3); %! tile = nrbspheretile(vertices); %! figure %! nrbkntplot(tile) %! title('Spherical cube tile') %!demo %! vertices = [0 1 sqrt(2)/2 0;0 0 sqrt(2)/2 1; -1 0 0 0] ; %! tile = nrbspheretile(vertices); %! figure %! nrbkntplot(tile) %! title('Single patch octant tile') nurbs-1.4.3/inst/PaxHeaders.16248/nrbline.m0000644000000000000000000000013014030323741015170 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbline.m0000644000175000017500000000301414030323741017267 0ustar00rvazquezrvazquez00000000000000function curve = nrbline(p1,p2) % % NRBLINE: Construct a straight line. % % Calling Sequence: % % crv = nrbline() % crv = nrbline(p1,p2) % % INPUT: % % p1 : 2D or 3D cartesian coordinate of the start point. % % p2 : 2D or 3D cartesian coordinate of the end point. % % OUTPUT: % % crv : NURBS curve for a straight line. % % Description: % % Constructs NURBS data structure for a straight line. If no rhs % coordinates are included the function returns a unit straight % line along the x-axis. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . coefs = [zeros(3,2); ones(1,2)]; if nargin < 2 coefs(1,2) = 1.0; else coefs(1:length(p1),1) = p1(:); coefs(1:length(p2),2) = p2(:); end curve = nrbmak(coefs, [0 0 1 1]); end %!demo %! crv = nrbline([0.0 0.0 0.0]',[5.0 4.0 2.0]'); %! nrbplot(crv,1); %! grid on; %! title('3D straight line.'); %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/vecdot.m0000644000000000000000000000013214030323741015025 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecdot.m0000644000175000017500000000265014030323741017127 0ustar00rvazquezrvazquez00000000000000function dot = vecdot(vec1,vec2) % % VECDOT: The dot product of two vectors. % % Calling Sequence: % % dot = vecdot(vec1,vec2); % % INPUT: % % vec1 : An array of column vectors represented by a matrix of % vec2 size (dim,nv), where is the dimension of the vector and % nv the number of vectors. % % OUTPUT: % % dot : Row vector of scalars, each element corresponding to % the dot product of the respective components in vec1 and % vec2. % % Description: % % Scalar dot product of two vectors. % % Examples: % % Determine the dot product of % (2.3,3.4,5.6) and (1.2,4.5,1.2) % (5.1,0.0,2.3) and (2.5,3.2,4.0) % % dot = vecdot([2.3 5.1; 3.4 0.0; 5.6 2.3],[1.2 2.5; 4.5 3.2; 1.2 4.0]); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . dot = sum(vec1.*vec2); end nurbs-1.4.3/inst/PaxHeaders.16248/surfderiveval.m0000644000000000000000000000013214030323741016422 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/surfderiveval.m0000644000175000017500000000557614030323741020536 0ustar00rvazquezrvazquez00000000000000function skl = surfderiveval (n, p, U, m, q, V, P, u, v, d) % % SURFDERIVEVAL: Compute the derivatives of a B-spline surface. % % usage: skl = surfderiveval (n, p, U, m, q, V, P, u, v, d) % % INPUT: % % n+1, m+1 = number of control points % p, q = spline order % U, V = knots % P = control points % u,v = evaluation points % d = derivative order % % OUTPUT: % % skl (k+1, l+1) = surface differentiated k % times in the u direction and l % times in the v direction % % Adaptation of algorithm A3.8 from the NURBS book, pg115 % % Copyright (C) 2009 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . skl = zeros (d+1, d+1); du = min (d, p); dv = min (d, q); uspan = findspan (n, p, u, U); for ip=0:p Nu(1:ip+1,ip+1) = basisfun (uspan, u, ip, U)'; end vspan = findspan (m, q, v, V); for ip=0:q Nv(1:ip+1,ip+1) = basisfun (vspan, v, ip, V)'; end pkl = surfderivcpts (n, p, U, m, q, V, P, d, uspan-p, uspan, ... vspan-q, vspan); for k = 0:du dd = min (d-k, dv); for l = 0:dd skl(k+1,l+1) =0; for i=0:q-l tmp = 0; for j = 0:p-k tmp = tmp + Nu(j+1,p-k+1) * pkl(k+1,l+1,j+1,i+1); end skl(k+1,l+1) = skl(k+1,l+1) + Nv(i+1,q-l+1)*tmp; end end end end %!shared srf %!test %! k = [0 0 0 1 1 1]; %! c = [0 1/2 1]; %! [coef(2,:,:), coef(1,:,:)] = meshgrid (c, c); %! srf = nrbmak (coef, {k, k}); %! skl = surfderiveval (srf.number(1)-1, ... %! srf.order(1)-1, ... %! srf.knots{1}, ... %! srf.number(2)-1, ... %! srf.order(2)-1, ... %! srf.knots{2},... %! squeeze(srf.coefs(1,:,:)), .5, .5, 1) ; %! assert (skl, [.5 0; 1 0]) %!test %! srf = nrbkntins (srf, {[], rand(1,2)}); %! skl = surfderiveval (srf.number(1)-1,... %! srf.order(1)-1, ... %! srf.knots{1},... %! srf.number(2)-1,... %! srf.order(2)-1, ... %! srf.knots{2},... %! squeeze(srf.coefs(1,:,:)), .5, .5, 1) ; %! assert (skl, [.5 0; 1 0], 100*eps) nurbs-1.4.3/inst/PaxHeaders.16248/vecrot.m0000644000000000000000000000013214030323741015043 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecrot.m0000644000175000017500000000311214030323741017137 0ustar00rvazquezrvazquez00000000000000function rx = vecrot(angle, vector) % % VECROT: Transformation matrix for a rotation around the axis given by a vector. % % Calling Sequence: % % rx = vecrot (angle, vector); % % INPUT: % % angle : rotation angle defined in radians % vector: vector defining the rotation axis % % OUTPUT: % % rx: (4x4) Transformation matrix. % % % Description: % % Return the (4x4) Transformation matrix for a rotation about the axis % defined by vector, and by the given angle. % % See also: % % nrbtform % % Copyright (C) 2011 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % Normalize the vector vec = vector / norm (vector); sn = sin (angle); cn = cos (angle); rx = [cn+vec(1)^2*(1-cn), vec(1)*vec(2)*(1-cn)-vec(3)*sn, vec(1)*vec(3)*(1-cn)+vec(2)*sn, 0; vec(1)*vec(2)*(1-cn)+vec(3)*sn, cn+vec(2)^2*(1-cn), vec(2)*vec(3)*(1-cn)-vec(1)*sn, 0; vec(1)*vec(3)*(1-cn)-vec(2)*sn, vec(2)*vec(3)*(1-cn)+vec(1)*sn, cn+vec(3)^2*(1-cn), 0; 0 0 0 1]; end nurbs-1.4.3/inst/PaxHeaders.16248/aveknt.m0000644000000000000000000000013214030323741015031 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/aveknt.m0000644000175000017500000000522414030323741017133 0ustar00rvazquezrvazquez00000000000000function pts = aveknt (varargin) % AVEKNT: compute the knot averages (Greville points) of a knot vector % % Calling Sequence: % % pts = aveknt (knt, p) % pts = aveknt (nrb) % % INPUT: % % knt - knot sequence % p - spline order (degree + 1) % nrb - NURBS structure (see nrbmak) % % OUTPUT: % % pts - average knots. If the input is a NURBS, it gives a cell-array, % with the average knots in each direction % % See also: % % Copyright (C) 2016 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin == 1) if (isfield (varargin{1}, 'form')) nrb = varargin{1}; knt = nrb.knots; order = nrb.order; else error ('The input should be a NURBS structure, or a knot vector and the order. See the help for details') end elseif (nargin == 2) knt = varargin{1}; order = varargin{2}; else error ('The input should be a NURBS structure, or a knot vector and the order. See the help for details') end onedim = false; if (~iscell (knt)) knt = {knt}; onedim = true; end ndim = numel (knt); pts = cell (ndim, 1); for idim = 1:ndim if (numel (knt{idim}) < order(idim)+1) error ('The knot vector must contain at least p+2 knots, with p the degree') end knt_aux = repmat (knt{idim}(2:end-1), 1, order(idim)-1); knt_aux = [knt_aux(:); zeros(order(idim)-1, 1)]; knt_aux = reshape (knt_aux, [], order(idim)-1); pts{idim} = sum (knt_aux.', 1) / (order(idim)-1); pts{idim} = pts{idim}(1:end-order(idim)+1); end if (onedim) pts = pts{1}; end end %!test %! knt = [0 0 0 0.5 1 1 1]; %! pts = aveknt (knt, 3); %! assert (pts - [0 1/4 3/4 1] < 1e-14) %! %!test %! knt = {[0 0 0 0.5 1 1 1] [0 0 0 0 1/3 2/3 1 1 1 1]}; %! pts = aveknt (knt, [3 4]); %! assert (pts{1} - [0 1/4 3/4 1] < 1e-14); %! assert (pts{2} - [0 1/9 1/3 2/3 8/9 1] < 1e-14); %! %!test %! nrb = nrb4surf([0 0], [1 0], [0 1], [1 1]); %! nrb = nrbkntins (nrbdegelev (nrb, [1 2]), {[1/2] [1/3 2/3]}); %! pts = aveknt (nrb); %! assert (pts{1} - [0 1/4 3/4 1] < 1e-14); %! assert (pts{2} - [0 1/9 1/3 2/3 8/9 1] < 1e-14); nurbs-1.4.3/inst/PaxHeaders.16248/nrb2iges.m0000644000000000000000000000013214030323741015254 xustar0030 mtime=1617012705.783919136 30 atime=1617012705.783919136 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrb2iges.m0000644000175000017500000002425014030323741017356 0ustar00rvazquezrvazquez00000000000000function nrb2iges (nurbs, filename) % NRB2IGES : Write a NURBS curve or surface to an IGES file. % % Calling Sequence: % % nrb2iges (nurbs, filename); % % INPUT: % % nurbs : NURBS curve or surface, see nrbmak. % filename : name of the output file. % % Description: % % The data of the nurbs structure is written in a file following the IGES % format. For a more in-depth explanation see, for example: % . % % Copyright (C) 2014 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % % This file is based on nurbs2iges.m, (C) 2006 Fu Qiang, originally % released under the MIT license. dt = datestr (now, 'yyyy.mm.dd'); dim = numel (nurbs(1).order); % START SECTION S{1} = ''; S{2} = 'IGES obtained from Nurbs toolbox.'; S{3} = 'See .'; S{4} = ''; % GLOBAL SECTION G{1} = '1H,'; % Parameter Deliminator Character G{2} = '1H;'; % Record Delimiter Character G{3} = HString ('Nurbs toolbox'); % Product ID from Sender G{4} = HString (filename); % File Name G{5} = HString ('Octave Nurbs'); % System ID G{6} = HString ('nrb2iges'); % Pre-processor Version G{7} = '32'; % Number of Bits for Integers (No. of bits present in the integer representation of the sending system) G{8} = '75'; % Single Precision Magnitude (Maximum power of 10 which may be represented as a single precision floating point number from the sending system) G{9} = '6'; % Single Precision Significance (No. of significant digits of a single precision floating point number on the sending system) G{10}= '75'; % Double Precision Magnitude (Maximum power of 10 which may be represented as a double precision floating point number from the sending system) G{11}= '15'; % Double Precision Significance (No. of significant digits of a double precision floating point number on the sending system) G{12}= HString('Nurbs from Octave'); % Product ID for Receiver G{13}= '1.0'; % Model Space Scale G{14}= '6'; % Unit Flag (6 = metres) G{15}= HString('M'); % Units (metres = "M") G{16}= '1000'; % Maximum Number of Line Weights G{17}= '1.0'; % Size of Maximum Line Width G{18}= HString(dt); % Date and Time of file generation G{19}= '0.000001'; % Minimum User-intended Resolution G{20}= '10000.0'; % Approximate Maximum Coordinate G{21}= HString('Jacopo Corno'); % Name of Author G{22}= HString('GSCE - TU Darmstadt'); % Author's Organization G{23}= '3'; % IGES Version Number (3 = IGES version 2.0) G{24}= '0'; % Drafting Standard Code (0 = no standard) % Convert section array to lines (maximum lenght 72) SectionG = make_section (G, 72); % DIRECTORY ENTRY SECTION % Each directory entry consists of two, 80 character, fixed formatted lines D = []; for ii = 1:length (nurbs) switch (dim) case 1 % NURBS curve D(ii).type = 126; case 2 % NURBS surface D(ii).type = 128; otherwise error ('Only curves and surfaces can be saved in IGES format.') end D(ii).id = 2*ii - 1; % odd counter (see Parameter data section) D(ii).p_start = 0; D(ii).p_count = 0; end % PARAMETER DATA SECTION % The structure is a free formatted data entry from columns 1 to 64. % Each line of free formatted data consists of the entity type number % followed by the parameter data describing the entity. % Columns 65 to 72 are reserved for a parameter data index which is an % odd number counter, right justified in the field, which begins at the % number 1 and progresses in odd increments for each entity entered. % Column 73 is reserved for the letter "P" to indicate the data element % belongs to the parameter data section. % Columns 74 to 80 are reserved for the sequence number. Each line of % data corresponds to the entity type as specified in the global section. SectionP = {}; for ii = 1:length (nurbs) P = make_section_array (nurbs(ii)); % finish one entity % Convert section array to lines SP = make_section (P, 64); D(ii).p_count = length (SP); if (ii == 1) D(ii).p_start = 1; else D(ii).p_start = D(ii-1).p_start + D(ii-1).p_count; end SectionP{ii} = SP; end % SAVE fid = fopen (filename, 'w'); % Save Start Section for ii = 1:length (S) fprintf (fid, '%-72sS%7d\n', S{ii}, ii); end % Save Global Section for ii = 1:length (SectionG) fprintf (fid, '%-72sG%7d\n', SectionG{ii}, ii); end % Save Directory Entry Section for i = 1:length (D) fprintf (fid, '%8d%8d%8d%8d%8d%8d%8d%8d%8dD%7d\n', ... D(i).type, D(i).p_start, 0, 0 ,0, 0, 0, 0, 0, i*2-1); fprintf (fid, '%8d%8d%8d%8d%8d%8s%8s%8s%8dD%7d\n', ... D(i).type, 0, 0, D(i).p_count, 0, ' ', ' ', ' ', 0, i*2); end % Save Parameter Data Section lines_p = 0; for jj = 1:length (D) sec = SectionP{jj}; for ii = 1:length (sec) lines_p = lines_p + 1; fprintf (fid, '%-64s %7dP%7d\n', sec{ii}, D(jj).id, lines_p); end end % Save Terminate Section sec_t = sprintf ('%7dS%7dG%7dD%7dP%7d', length (S), length(SectionG), 2*length(D), lines_p); fprintf (fid, '%-72sT%7d\n', sec_t, 1); fclose(fid); end function P = make_section_array (nurbs) dim = numel (nurbs.order); % in IGES the control points are stored in the format [x, y, z, w] % instead of [w*x, w*y, w*z, w] for idim = 1:3 nurbs.coefs(idim,:) = nurbs.coefs(idim,:) ./ nurbs.coefs(4,:); end P = {}; switch dim case 1 % Rational B-Spline Curve Entity cp = nurbs.coefs; deg = nurbs.order - 1; knots = nurbs.knots; uspan = [0 1]; isplanar = ~any(cp(3,:)); P{1} = '126'; % NURBS curve P{2} = int2str (size (cp, 2) - 1); % Number of control points P{3} = int2str (deg); % Degree P{4} = int2str (isplanar); % Curve on xy plane P{5} = '0'; P{6} = '0'; P{7} = '0'; index = 8; for ii = 1:length (knots) P{index} = sprintf ('%f', knots(ii)); index = index + 1; end for ii = 1:size (cp, 2) P{index} = sprintf ('%f', cp(4,ii)); index = index + 1; end for ii = 1:size (cp, 2) P{index} = sprintf ('%f', cp(1,ii)); index = index + 1; P{index} = sprintf ('%f', cp(2,ii)); index = index + 1; P{index} = sprintf ('%f', cp(3,ii)); index = index + 1; end P{index} = sprintf ('%f', uspan(1)); index = index +1; P{index} = sprintf ('%f', uspan(2)); index = index +1; P{index} = '0.0'; index = index +1; P{index} = '0.0'; index = index +1; if isplanar P{index} = '1.0'; else P{index} = '0.0'; end index = index + 1; P{index} = '0'; index = index + 1; P{index} = '0'; case 2 % Rational B-Spline Surface Entity cp = nurbs.coefs; degU = nurbs.order(1) - 1; degV = nurbs.order(2) - 1; knotsU = nurbs.knots{1}; knotsV = nurbs.knots{2}; uspan = [0 1]; vspan = [0 1]; P{1} = '128'; % NURBS surface P{2} = int2str (size (cp, 2) - 1); % Number of control points in U P{3} = int2str (size (cp, 3) - 1); % Number of control points in V P{4} = int2str (degU); % Degree in U P{5} = int2str (degV); % Degree in V P{6} = '0'; P{7} = '0'; P{8} = '0'; P{9} = '0'; P{10} = '0'; index = 11; for ii = 1:length (knotsU) P{index} = sprintf ('%f', knotsU(ii)); index = index + 1; end for ii = 1:length (knotsV) P{index} = sprintf ('%f', knotsV(ii)); index = index + 1; end for jj = 1:size (cp, 3) for ii = 1:size (cp, 2) P{index} = sprintf ('%f', cp(4,ii,jj)); index = index + 1; end end for jj = 1:size (cp, 3) for ii = 1:size (cp, 2) P{index} = sprintf ('%f',cp(1,ii,jj)); index = index + 1; P{index} = sprintf ('%f',cp(2,ii,jj)); index = index + 1; P{index} = sprintf ('%f',cp(3,ii,jj)); index = index + 1; end end P{index} = sprintf('%f',uspan(1)); index = index +1; P{index} = sprintf('%f',uspan(2)); index = index +1; P{index} = sprintf('%f',vspan(1)); index = index +1; P{index} = sprintf('%f',vspan(2)); index = index +1; P{index} = '0'; index = index + 1; P{index} = '0'; otherwise end end function hs = HString (str) % HString : Convert the string STR to the Hollerith format. hs = sprintf ('%dH%s', length(str), str); end function sec = make_section (fields, linewidth) sec = {}; index = 1; line = ''; num = length (fields); for i = 1:num if (i < num) newitem = [fields{i} ',']; else newitem = [fields{i} ';']; end len = length (line) + length (newitem); if ( len > linewidth ) % new line sec{index} = line; index = index + 1; line = ''; end line = [line newitem]; end sec{index} = line; end nurbs-1.4.3/inst/PaxHeaders.16248/nrbtransp.m0000644000000000000000000000013214030323741015552 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.803919443 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbtransp.m0000644000175000017500000000361114030323741017652 0ustar00rvazquezrvazquez00000000000000function tsrf = nrbtransp(srf) % % NRBTRANSP: Transpose a NURBS surface, by swapping U and V directions. % % Calling Sequence: % % tsrf = nrbtransp(srf) % % INPUT: % % srf : NURBS surface, see nrbmak. % % OUTPUT: % % tsrf : NURBS surface with U and V diretions transposed. % % Description: % % Utility function that transposes a NURBS surface, by swapping U and % V directions. NURBS curves cannot be transposed. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if ~iscell(srf.knots) error(' A NURBS curve cannot be transposed.'); elseif size(srf.knots,2) == 3 error('The transposition of NURBS volumes has not been implemented.'); end tsrf = nrbmak(permute(srf.coefs,[1 3 2]), fliplr(srf.knots)); end %!demo %! srf = nrb4surf([0 0 0], [1 0 1], [0 1 1], [1 1 2]); %! nrbplot(srf,[20 5]); %! title('Plane surface and its transposed (translated)') %! hold on %! srf.coefs(3,:,:) = srf.coefs(3,:,:) + 10; %! srf = nrbtransp(srf); %! nrbplot(srf,[20 5]); %! hold off %!test %! srf = nrbrevolve(nrbline([1 0],[2 0]), [0 0 0], [0 0 1], pi/2); %! srft = nrbtransp(srf); %! assert (srf.number, fliplr(srft.number)); %! assert (srf.order, fliplr(srft.order)); %! assert (srf.knots, fliplr(srft.knots)); %! assert (srf.coefs, permute(srft.coefs, [1 3 2]));nurbs-1.4.3/inst/PaxHeaders.16248/nrbeval.m0000644000000000000000000000013214030323741015172 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbeval.m0000644000175000017500000002223514030323741017275 0ustar00rvazquezrvazquez00000000000000function [p,w] = nrbeval(nurbs,tt) % % NRBEVAL: Evaluate a NURBS at parametric points. % % Calling Sequences: % % [p,w] = nrbeval(crv,ut) % [p,w] = nrbeval(srf,{ut,vt}) % [p,w] = nrbeval(vol,{ut,vt,wt}) % [p,w] = nrbeval(srf,pts) % % INPUT: % % crv : NURBS curve, see nrbmak. % % srf : NURBS surface, see nrbmak. % % vol : NURBS volume, see nrbmak. % % ut : Parametric evaluation points along U direction. % % vt : Parametric evaluation points along V direction. % % wt : Parametric evaluation points along W direction. % % pts : Array of scattered points in parametric domain % % OUTPUT: % % p : Evaluated points on the NURBS curve, surface or volume as % Cartesian coordinates (x,y,z). If w is included on the lhs argument % list the points are returned as homogeneous coordinates (wx,wy,wz). % % w : Weights of the homogeneous coordinates of the evaluated % points. Note inclusion of this argument changes the type % of coordinates returned in p (see above). % % Description: % % Evaluation of NURBS curves, surfaces or volume at parametric points along % the U, V and W directions. Either homogeneous coordinates are returned % if the weights are requested in the lhs arguments, or as Cartesian coordinates. % This function utilises the 'C' interface bspeval. % % Examples: % % Evaluate the NURBS circle at twenty points from 0.0 to 1.0 % % nrb = nrbcirc; % ut = linspace(0.0,1.0,20); % p = nrbeval(nrb,ut); % % See also: % % bspeval % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Carlo de Falco % Copyright (C) 2010, 2011, 2015 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (nargin < 2) error('Not enough input arguments'); end foption = 1; % output format 3D cartesian coordinates if (nargout == 2) foption = 0; % output format 4D homogenous coordinates end if (~isstruct(nurbs)) error('NURBS representation is not structure!'); end if (~strcmp(nurbs.form,'B-NURBS')) error('Not a recognised NURBS representation'); end if (iscell(nurbs.knots)) if (size(nurbs.knots,2) == 3) %% NURBS structure represents a volume num1 = nurbs.number(1); num2 = nurbs.number(2); num3 = nurbs.number(3); degree = nurbs.order-1; if (iscell(tt)) nt1 = numel (tt{1}); nt2 = numel (tt{2}); nt3 = numel (tt{3}); %% evaluate along the w direction val = reshape (nurbs.coefs, 4*num1*num2, num3); val = bspeval (degree(3), val, nurbs.knots{3}, tt{3}); val = reshape (val, [4 num1 num2 nt3]); %% Evaluate along the v direction val = permute (val, [1 2 4 3]); val = reshape (val, 4*num1*nt3, num2); val = bspeval (degree(2), val, nurbs.knots{2}, tt{2}); val = reshape (val, [4 num1 nt3 nt2]); val = permute (val, [1 2 4 3]); %% Evaluate along the u direction val = permute (val, [1 3 4 2]); val = reshape (val, 4*nt2*nt3, num1); val = bspeval (degree(1), val, nurbs.knots{1}, tt{1}); val = reshape (val, [4 nt2 nt3 nt1]); val = permute (val, [1 4 2 3]); pnts = val; p = pnts(1:3,:,:,:); w = pnts(4,:,:,:); if (foption) p = p./repmat(w,[3 1 1 1]); end else %% Evaluate at scattered points %% tt(1,:) represents the u direction %% tt(2,:) represents the v direction %% tt(3,:) represents the w direction st = size(tt); if (st(1) ~= 3 && st(2) == 3 && numel(st) == 2) tt = tt'; st = size (tt); end nt = prod(st(2:end)); tt = reshape (tt, [3, nt]); %% evaluate along the w direction val = reshape(nurbs.coefs,4*num1*num2,num3); val = bspeval(degree(3),val,nurbs.knots{3},tt(3,:)); val = reshape(val,[4 num1 num2 nt]); %% evaluate along the v direction val2 = zeros(4*num1,nt); for v = 1:nt coefs = reshape(val(:,:,:,v),4*num1,num2); val2(:,v) = bspeval(degree(2),coefs,nurbs.knots{2},tt(2,v)); end val2 = reshape(val2,[4 num1 nt]); %% evaluate along the u direction pnts = zeros(4,nt); for v = 1:nt coefs = reshape (val2(:,:,v), [4 num1]); pnts(:,v) = bspeval(degree(1),coefs,nurbs.knots{1},tt(1,v)); end w = pnts(4,:); p = pnts(1:3,:); if (foption) p = p./repmat(w,[3, 1]); end if (numel(st) ~= 2) w = reshape (w, [st(2:end)]); p = reshape (p, [3, st(2:end)]); end end elseif (size(nurbs.knots,2) == 2) %% NURBS structure represents a surface num1 = nurbs.number(1); num2 = nurbs.number(2); degree = nurbs.order-1; if (iscell(tt)) %% Evaluate over a [u,v] grid %% tt{1} represents the u direction %% tt{2} represents the v direction nt1 = length(tt{1}); nt2 = length(tt{2}); %% Evaluate along the v direction val = reshape(nurbs.coefs,4*num1,num2); val = bspeval(degree(2),val,nurbs.knots{2},tt{2}); val = reshape(val,[4 num1 nt2]); %% Evaluate along the u direction val = permute(val,[1 3 2]); val = reshape(val,4*nt2,num1); val = bspeval(degree(1),val,nurbs.knots{1},tt{1}); val = reshape(val,[4 nt2 nt1]); val = permute(val,[1 3 2]); w = val(4,:,:); p = val(1:3,:,:); if (foption) p = p./repmat(w,[3 1 1]); end else %% Evaluate at scattered points %% tt(1,:) represents the u direction %% tt(2,:) represents the v direction st = size(tt); if (st(1) ~= 2 && st(2) == 2 && numel(st) == 2) tt = tt'; st = size (tt); end nt = prod(st(2:end)); tt = reshape (tt, [2, nt]); val = reshape(nurbs.coefs,4*num1,num2); val = bspeval(degree(2),val,nurbs.knots{2},tt(2,:)); val = reshape(val,[4 num1 nt]); %% evaluate along the u direction pnts = zeros(4,nt); for v = 1:nt coefs = reshape (val(:,:,v), [4 num1]); pnts(:,v) = bspeval(degree(1),coefs,nurbs.knots{1},tt(1,v)); end w = pnts(4,:); p = pnts(1:3,:); if (foption) p = p./repmat(w,[3, 1]); end if (numel(st) ~= 2) w = reshape (w, [st(2:end)]); p = reshape (p, [3, st(2:end)]); end end end else %% NURBS structure represents a curve %% tt represent a vector of parametric points in the u direction if (iscell (tt) && numel (tt) == 1) tt = cell2mat (tt); end st = size (tt); val = bspeval(nurbs.order-1,nurbs.coefs,nurbs.knots,tt(:)'); w = val(4,:); p = val(1:3,:); if foption p = p./repmat(w,3,1); end if (st(1) ~= 1 || numel(st) ~= 2) w = reshape (w, st); p = reshape (p, [3, st]); end end end %!demo %! srf = nrbtestsrf; %! p = nrbeval(srf,{linspace(0.0,1.0,20) linspace(0.0,1.0,20)}); %! h = surf(squeeze(p(1,:,:)),squeeze(p(2,:,:)),squeeze(p(3,:,:))); %! title('Test surface.'); %! hold off %!test %! knots{1} = [0 0 0 1 1 1]; %! knots{2} = [0 0 0 .5 1 1 1]; %! knots{3} = [0 0 0 0 1 1 1 1]; %! cx = [0 0.5 1]; nx = length(cx); %! cy = [0 0.25 0.75 1]; ny = length(cy); %! cz = [0 1/3 2/3 1]; nz = length(cz); %! coefs(1,:,:,:) = repmat(reshape(cx,nx,1,1),[1 ny nz]); %! coefs(2,:,:,:) = repmat(reshape(cy,1,ny,1),[nx 1 nz]); %! coefs(3,:,:,:) = repmat(reshape(cz,1,1,nz),[nx ny 1]); %! coefs(4,:,:,:) = 1; %! nurbs = nrbmak(coefs, knots); %! x = rand(5,1); y = rand(5,1); z = rand(5,1); %! tt = [x y z]'; %! points = nrbeval(nurbs,tt); %! %! assert(points,tt,1e-10) %! %!test %! knots{1} = [0 0 0 1 1 1]; %! knots{2} = [0 0 0 0 1 1 1 1]; %! knots{3} = [0 0 1 1]; %! cx = [0 0 1]; nx = length(cx); %! cy = [0 0 0 1]; ny = length(cy); %! cz = [0 1]; nz = length(cz); %! coefs(1,:,:,:) = repmat(reshape(cx,nx,1,1),[1 ny nz]); %! coefs(2,:,:,:) = repmat(reshape(cy,1,ny,1),[nx 1 nz]); %! coefs(3,:,:,:) = repmat(reshape(cz,1,1,nz),[nx ny 1]); %! coefs(4,:,:,:) = 1; %! nurbs = nrbmak(coefs, knots); %! x = rand(5,1); y = rand(5,1); z = rand(5,1); %! tt = [x y z]'; %! points = nrbeval(nurbs,tt); %! assert(points,[x.^2 y.^3 z]',1e-10); %! %!test %! knots{1} = [0 0 0 1 1 1]; %! knots{2} = [0 0 0 0 1 1 1 1]; %! knots{3} = [0 0 1 1]; %! cx = [0 0 1]; nx = length(cx); %! cy = [0 0 0 1]; ny = length(cy); %! cz = [0 1]; nz = length(cz); %! coefs(1,:,:,:) = repmat(reshape(cx,nx,1,1),[1 ny nz]); %! coefs(2,:,:,:) = repmat(reshape(cy,1,ny,1),[nx 1 nz]); %! coefs(3,:,:,:) = repmat(reshape(cz,1,1,nz),[nx ny 1]); %! coefs(4,:,:,:) = 1; %! coefs = coefs([2 1 3 4],:,:,:); %! nurbs = nrbmak(coefs, knots); %! x = rand(5,1); y = rand(5,1); z = rand(5,1); %! tt = [x y z]'; %! points = nrbeval(nurbs,tt); %! [y.^3 x.^2 z]'; %! assert(points,[y.^3 x.^2 z]',1e-10); nurbs-1.4.3/inst/PaxHeaders.16248/veccross.m0000644000000000000000000000013214030323741015370 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/veccross.m0000644000175000017500000000343114030323741017470 0ustar00rvazquezrvazquez00000000000000function cross = veccross(vec1,vec2) % % VECCROSS: The cross product of two vectors. % % Calling Sequence: % % cross = veccross(vec1,vec2); % % INPUT: % % vec1 : An array of column vectors represented by a matrix of % vec2 size (dim,nv), where is the dimension of the vector and % nv the number of vectors. % % OUTPUT: % % cross : Array of column vectors, each element is corresponding % to the cross product of the respective components in vec1 % and vec2. % % Description: % % Cross product of two vectors. % % Examples: % % Determine the cross products of: % (2.3,3.4,5.6) and (1.2,4.5,1.2) % (5.1,0.0,2.3) and (2.5,3.2,4.0) % % cross = veccross([2.3 5.1; 3.4 0.0; 5.6 2.3],[1.2 2.5; 4.5 3.2; 1.2 4.0]); % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if size(vec1,1) == 2 % 2D vector cross = zeros(size(vec1)); cross(3,:) = vec1(1,:).*vec2(2,:)-vec1(2,:).*vec2(1,:); else % 3D vector cross = [vec1(2,:).*vec2(3,:)-vec1(3,:).*vec2(2,:); vec1(3,:).*vec2(1,:)-vec1(1,:).*vec2(3,:); vec1(1,:).*vec2(2,:)-vec1(2,:).*vec2(1,:)]; end end nurbs-1.4.3/inst/PaxHeaders.16248/nrbmeasure.m0000644000000000000000000000013014030323741015702 xustar0029 mtime=1617012705.79591932 29 atime=1617012705.79591932 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbmeasure.m0000644000175000017500000000625414030323741020012 0ustar00rvazquezrvazquez00000000000000% NRBMEASURE: Compute the distance between two given points along a NURBS curve. % % Calling Sequence: % % [dist, ddistds, ddistde] = nrbmeasure (nrb) % [dist, ddistds, ddistde] = nrbmeasure (nrb, s, e) % [dist, ddistds, ddistde] = nrbmeasure (nrb, s, e, tol) % % INPUT: % % nrb : a NURBS curve, see nrbmak. % s : starting point in the parametric domain. % e : ending point in the parametric domain. % tol : tolerance for numerical quadrature, to be used in quad. % % OUTPUT: % % dist : distance between the two points along the NURBS curve. % ddistds: derivative of the distance function with respect to the point s. % ddistde: derivative of the distance function with respect to the point e. % % Description: % % Compute the distance between two given points along a NURBS curve, using % quad for numerical integration. The points are given by their coordinates % in the parametric domain. % % Examples: % % Compute the length of a circular arc constructed as a NURBS. % % c = nrbcirc (1, [0 0], 0, pi/2); % s = 0; e = 1; % l = nrbmeasure (c, s, e, 1e-7); % % Copyright (C) 2013 Carlo de Falco % % This program is free software; you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation; either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with Octave; see the file COPYING. If not, see % . function [dist, ddistds, ddistde] = nrbmeasure (nrb, s, e, tol) if (nargin < 4) tol = 1e-6; if (nargin < 3) e = 1; if (nargin < 2) s = 0; end end end nrb.knots = (nrb.knots - nrb.knots(1)) / (nrb.knots(end) - nrb.knots(1)); if (numel (s) > 1 && isscalar (e)) e = e * ones (size(s)); elseif (numel (e) > 1 && isscalar (s)) s = s * ones (size(e)); end ders = nrbderiv (nrb); dist = arrayfun (@(x, y) quad (@(u) len (u, nrb, ders), x, ... y, tol), s, e); if (nargout > 1) ddistds = -len (s, nrb, ders); if (nargout > 2) ddistde = +len (e, nrb, ders); end end end function l = len (u, nrb, ders) [~, d] = nrbdeval (nrb, ders, u); f = d(1, :); g = d(2, :); h = d(3, :); l = sqrt (f.^2 + g.^2 + h.^2); end %!test %! c = nrbcirc (1, [0 0], 0, pi/3); %! l = nrbmeasure(c, 0, 1, 1e-7); %! assert (l, pi/3, 1e-7) %!test %! c = nrbcirc (1, [0 0], 0, pi/2); %! s = zeros (1, 100); e = linspace (0, 1, 100); %! for ii = 1:100 %! l(ii) = nrbmeasure (c, s(ii), e(ii), 1e-7); %! endfor %! xx = nrbeval (c, e); %! theta = atan2 (xx(2,:), xx(1,:)); %! assert (l, theta, 1e-7) %!test %! c = nrbcirc (1, [0 0], 0, pi/2); %! s = 0; e = linspace (0, 1, 100); %! for ii = 1:100 %! l(ii) = nrbmeasure (c, s, e(ii), 1e-7); %! endfor %! l2 = nrbmeasure (c, s, e, 1e-7); %! assert (l, l2, eps) nurbs-1.4.3/inst/PaxHeaders.16248/nrbeval_der_p.m0000644000000000000000000000013214030323741016343 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbeval_der_p.m0000644000175000017500000000744214030323741020451 0ustar00rvazquezrvazquez00000000000000function der = nrbeval_der_p (nrb, i, points) % % NRBEVAL_DER_P: Compute the derivative of a NURBS object at a given point % with respect to the coordinates of the i-th control point. % % Calling Sequence: % % der = nrbeval_der_p (crv, i, u); % der = nrbeval_der_p (srf, i, p); % der = nrbeval_der_p (srf, i, {u v}); % der = nrbeval_der_p (vol, i, p); % der = nrbeval_der_p (vol, i, {u v w}); % % INPUT: % % crv - NURBS curve. % srf - NURBS surface. % vol - NURBS volume. % i - Index of the control point. % u or p(1,:,:) - parametric points along u direction % v or p(2,:,:) - parametric points along v direction % w or p(3,:,:) - parametric points along w direction % % OUTPUT: % % der - Derivative. % size(der) = numel(u) for curves % or numel(u)*numel(v) for surfaces % or numel(u)*numel(v)*numel(w) for volumes % % Copyright (C) 2015 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % [N, id] = nrbbasisfun (points, nrb); der = zeros (1, size(N, 1)); for k = 1:numel (der) [is, loc] = ismember (i, id(k,:)); % id is 1-based if (is) der(k) = N(k,loc); else der(k) = 0; end end end %!test %% 1D %! nrb = nrbkntins (nrbcirc (1, [0 0], 0, pi/2), .5); %! u = 0:.1:.9; %! index = 1:nrb.number; %! e = zeros (numel (u), numel (index), 1); %! for jj = 1:numel (index) %! deltap = .1 * rand (3, 1); %! nrb2 = nrbmodp (nrb, deltap, index(jj)); %! der_ex = nrbeval_der_p (nrb, index(jj), u); %! p2 = nrbeval (nrb2, u); %! p1 = nrbeval (nrb, u); %! der_fd = (p2 - p1) ./ deltap; %! e(:,jj) = sqrt (sum ((repmat (der_ex, 3, 1) - der_fd).^2, 1)); %! end %! assert (max(e(:)) < 1.e-8); %! %!test %% 2D %! crv = nrbline([1 0], [2 0]); %! nrb = nrbtransp (nrbrevolve (crv, [], [0 0 1], pi/2)); %! new_knots = linspace (1/9, 8/9, 8); %! nrb = nrbkntins (nrb, {new_knots, new_knots}); %! u = 0:.1:.9; %! v = u; %! e = zeros (nrb.number(1) * nrb.number(2), numel (u), numel (v)); %! for index = 1:prod(nrb.number) %! deltap = .1 * rand (3, 1); %! nrb2 = nrbmodp (nrb, deltap, index); %! der_ex = nrbeval_der_p (nrb, index, {u v}); %! p2 = nrbeval (nrb2, {u v}); %! p1 = nrbeval (nrb, {u v}); %! der_fd = (p2 - p1) ./ deltap; %! der_ex = reshape (repmat (der_ex, 3, 1), size(der_fd)); %! e(index,:,:) = sqrt (sum ((der_ex - der_fd).^2, 1)); %! end %! assert (max(e(:)) < 1.e-8) %! %!test %% 3D %! crv = nrbline([1 0], [2 0]); %! nrb = nrbtransp (nrbrevolve (crv, [], [0 0 1], pi/2)); %! nrb = nrbextrude (nrb, [0 0 1]); %! u = 0:.1:.9; %! v = u; %! w = u; %! e = zeros (nrb.number(1) * nrb.number(2) * nrb.number(3), numel(u), numel(v), numel(w)); %! for index = 1:prod(nrb.number) %! deltap = .1 * rand (3, 1); %! nrb2 = nrbmodp (nrb, deltap, index); %! der_ex = nrbeval_der_p (nrb, index, {u v w}); %! p2 = nrbeval (nrb2, {u v w}); %! p1 = nrbeval (nrb, {u v w}); %! der_fd = (p2 - p1) ./ deltap; %! der_ex = reshape (repmat (der_ex, 3, 1), size (der_fd)); %! e(index,:,:,:) = sqrt (sum ((der_ex - der_fd).^2, 1)); %! end %! assert (max (e(:)) < 1.e-8); nurbs-1.4.3/inst/PaxHeaders.16248/nrbeval_der_w.m0000644000000000000000000000013214030323741016352 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbeval_der_w.m0000644000175000017500000001132414030323741020452 0ustar00rvazquezrvazquez00000000000000function der = nrbeval_der_w (nrb, i, points) % % NRBEVAL_DER_W: Compute the derivatives of a NURBS object at the point u % with respect to the weight of the i-th control point. % % Calling Sequence: % % der = nrbeval_der_p (crv, i, u); % der = nrbeval_der_p (srf, i, p); % der = nrbeval_der_p (srf, i, {u v}); % der = nrbeval_der_p (vol, i, p); % der = nrbeval_der_p (vol, i, {u v w}); % % INPUT: % % crv - NURBS curve. % srf - NURBS surface. % vol - NURBS volume. % i - Index of the control point. % u or p(1,:,:) - parametric points along u direction % v or p(2,:,:) - parametric points along v direction % w or p(3,:,:) - parametric points along w direction % % OUTPUT: % % der - Derivatives. % size(der) = [3, numel(u)] for curves % or [3, numel(u)*numel(v)] for surfaces % or [3, numel(u)*numel(v)*numel(w)] for volumes % % Copyright (C) 2015 Jacopo Corno % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % if (iscell(points)) npts = prod (cellfun (@numel, points)); else npts = size (points, 2); end der = zeros (3, npts); [evalu, den] = nrbeval (nrb, points); [N, I] = nrbbasisfun (points, nrb); if (iscell (points)) evalu = reshape (evalu, size(evalu, 1), []); den = reshape (den, 1, []); end % if (numel (nrb.number) == 1) % 1D % I = I + 1; % id is 0-based % end [ii, jj, kk] = ind2sub (nrb.number, i); w_i = nrb.coefs(4,ii,jj,kk); P_i = nrb.coefs(1:3,ii,jj,kk) ./ w_i; for ipnt = 1:npts [is, loc] = ismember (i, I(ipnt,:)); if (is) der(:,ipnt) = N(ipnt,loc) ./ w_i .* P_i - evalu(:,ipnt) .* N(ipnt,loc) ./ w_i ./ den(ipnt); end end end %!test % 1D %! nrb = nrbkntins (nrbcirc (1, [0 0], 0, pi/2), .5); %! u = linspace (0, 1, 11); %! delta_w = .01; %! n = nrb.number; %! der_ex = zeros (3, numel (u), n); %! der_fd = zeros (3, numel (u), n); %! for iw = 1:n %! new_w1 = nrb.coefs (4, iw) + delta_w; %! new_w2 = nrb.coefs (4, iw) - delta_w; %! nrb1 = nrbmodw (nrb, new_w1, iw); %! nrb2 = nrbmodw (nrb, new_w2, iw); %! der_ex(:,:,iw) = nrbeval_der_w (nrb, iw, u); %! p2 = nrbeval (nrb2, u); %! p1 = nrbeval (nrb1, u); %! der_fd(:,:,iw) = -(p2 - p1) ./ (2*delta_w); %! end %! error = max (abs (der_ex(:) - der_fd(:))); %! assert (error < 1.e-4) %! %!test %2D %! crv = nrbline([1 0], [2 0]); %! nrb = nrbtransp (nrbrevolve (crv, [], [0 0 1], pi/2)); %! new_knots = linspace (1/9, 8/9, 8); %! nrb = nrbkntins (nrb, {new_knots, new_knots}); %! u = linspace (0, 1, 5); %! v = u; %! delta_w = .01; %! n = nrb.number(1) * nrb.number(2); %! der_ex = zeros (3, numel(u)* numel(v), n); %! der_fd = zeros (3, numel(u)* numel(v), n); %! for iw = 1:prod(nrb.number) %! new_w1 = nrb.coefs (4, iw) + delta_w; %! new_w2 = nrb.coefs (4, iw) - delta_w; %! nrb1 = nrbmodw (nrb, new_w1, iw); %! nrb2 = nrbmodw (nrb, new_w2, iw); %! der_ex(:,:,iw) = nrbeval_der_w (nrb, iw, {u v}); %! p2 = nrbeval (nrb2, {u v}); %! p1 = nrbeval (nrb1, {u v}); %! der_fd(:,:,iw) = reshape (-(p2 - p1) ./ (2*delta_w), 3, []); %! end %! error = max (abs (der_ex(:) - der_fd(:))); %! assert (error < 1.e-5) %! %!test % 3D %! crv = nrbline([1 0], [2 0]); %! nrb = nrbtransp (nrbrevolve (crv, [], [0 0 1], pi/2)); %! nrb = nrbextrude (nrb, [0 0 1]); %! u = 0:.33:.99; %! v = 0:.1:.9; %! w = [.25 .5 .75]; %! delta_w = .01; %! n = nrb.number(1) * nrb.number(2) * nrb.number(3); %! der_ex = zeros (3, numel(u)*numel(v)*numel(w), n); %! der_fd = zeros (3, numel(u)*numel(v)*numel(w), n); %! for iw = 1:prod(nrb.number) %! new_w1 = nrb.coefs (4, iw) + delta_w; %! new_w2 = nrb.coefs (4, iw) - delta_w; %! nrb1 = nrbmodw (nrb, new_w1, iw); %! nrb2 = nrbmodw (nrb, new_w2, iw); %! der_ex(:,:,iw) = nrbeval_der_w (nrb, iw, {u v w}); %! p2 = nrbeval (nrb2, {u v w}); %! p1 = nrbeval (nrb1, {u v w}); %! der_fd(:,:,iw) = reshape (-(p2 - p1) ./ (2*delta_w), 3, []); %! end %! error = max (max (squeeze (max (abs (der_ex - der_fd))))); %! assert (error < 1.e-4) nurbs-1.4.3/inst/PaxHeaders.16248/nrbcirc.m0000644000000000000000000000013214030323741015163 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbcirc.m0000644000175000017500000000573414030323741017273 0ustar00rvazquezrvazquez00000000000000function curve = nrbcirc(radius,center,sang,eang) % % NRBCIRC: Construct a circular arc. % % Calling Sequence: % % crv = nrbcirc() % crv = nrbcirc(radius) % crv = nrbcirc(radius,center) % crv = nrbcirc(radius,center,sang,eang) % % INPUT: % % radius : Radius of the circle, default 1.0 % % center : Center of the circle, default (0,0,0) % % sang : Start angle, default 0 radians (0 degrees) % % eang : End angle, default 2*pi radians (360 degrees) % % OUTPUT: % % crv : NURBS curve for a circular arc. % % Description: % % Constructs NURBS data structure for a circular arc in the x-y plane. If % no rhs arguments are supplied a unit circle with center (0.0,0.0) is % constructed. % % Angles are defined as positive in the anti-clockwise direction. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 1 radius = 1; end if nargin < 2 center = []; end if nargin < 4 sang = 0; eang = 2*pi; end sweep = eang - sang; % sweep angle of arc if sweep < 0 sweep = 2*pi + sweep; end if abs(sweep) <= pi/2 narcs = 1; % number of arc segments knots = [0 0 0 1 1 1]; elseif abs(sweep) <= pi narcs = 2; knots = [0 0 0 0.5 0.5 1 1 1]; elseif abs(sweep) <= 3*pi/2 narcs = 3; knots = [0 0 0 1/3 1/3 2/3 2/3 1 1 1]; else narcs = 4; knots = [0 0 0 0.25 0.25 0.5 0.5 0.75 0.75 1 1 1]; end dsweep = sweep/(2*narcs); % arc segment sweep angle/2 % determine middle control point and weight wm = cos(dsweep); x = radius*wm; y = radius*sin(dsweep); xm = x+y*tan(dsweep); % arc segment control points ctrlpt = [ x wm*xm x; % w*x - coordinate -y 0 y; % w*y - coordinate 0 0 0; % w*z - coordinate 1 wm 1]; % w - coordinate % build up complete arc from rotated segments coefs = zeros(4,2*narcs+1); % nurbs control points of arc xx = vecrotz(sang + dsweep); coefs(:,1:3) = xx*ctrlpt; % rotate to start angle xx = vecrotz(2*dsweep); for n = 2:narcs m = 2*n+[0 1]; coefs(:,m) = xx*coefs(:,m-2); end % vectrans arc if necessary if ~isempty(center) xx = vectrans(center); coefs = xx*coefs; end curve = nrbmak(coefs,knots); end %!demo %! for r = 1:9 %! crv = nrbcirc(r,[],45*pi/180,315*pi/180); %! nrbplot(crv,50); %! hold on; %! end %! hold off; %! axis equal; %! title('NURBS construction of several 2D arcs.'); nurbs-1.4.3/inst/PaxHeaders.16248/bspdegelev.m0000644000000000000000000000013214030323741015661 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/bspdegelev.m0000644000175000017500000005004114030323741017760 0ustar00rvazquezrvazquez00000000000000function [ic,ik] = bspdegelev(d,c,k,t) % BSPDEGELEV: Degree elevate a univariate B-Spline. % % Calling Sequence: % % [ic,ik] = bspdegelev(d,c,k,t) % % INPUT: % % d - Degree of the B-Spline. % c - Control points, matrix of size (dim,nc). % k - Knot sequence, row vector of size nk. % t - Raise the B-Spline degree t times. % % OUTPUT: % % ic - Control points of the new B-Spline. % ik - Knot vector of the new B-Spline. % % Copyright (C) 2000 Mark Spink, 2007 Daniel Claxton % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . [mc,nc] = size(c); % % int bspdegelev(int d, double *c, int mc, int nc, double *k, int nk, % int t, int *nh, double *ic, double *ik) % { % int row,col; % % int ierr = 0; % int i, j, q, s, m, ph, ph2, mpi, mh, r, a, b, cind, oldr, mul; % int n, lbz, rbz, save, tr, kj, first, kind, last, bet, ii; % double inv, ua, ub, numer, den, alf, gam; % double **bezalfs, **bpts, **ebpts, **Nextbpts, *alfs; % % double **ctrl = vec2mat(c, mc, nc); % ic = zeros(mc,nc*(t)); % double **ictrl = vec2mat(ic, mc, nc*(t+1)); % n = nc - 1; % n = nc - 1; % bezalfs = zeros(d+1,d+t+1); % bezalfs = matrix(d+1,d+t+1); bpts = zeros(mc,d+1); % bpts = matrix(mc,d+1); ebpts = zeros(mc,d+t+1); % ebpts = matrix(mc,d+t+1); Nextbpts = zeros(mc,d+1); % Nextbpts = matrix(mc,d+1); alfs = zeros(d,1); % alfs = (double *) mxMalloc(d*sizeof(double)); % m = n + d + 1; % m = n + d + 1; ph = d + t; % ph = d + t; ph2 = floor(ph / 2); % ph2 = ph / 2; % % // compute bezier degree elevation coefficeients bezalfs(1,1) = 1; % bezalfs[0][0] = bezalfs[ph][d] = 1.0; bezalfs(d+1,ph+1) = 1; % for i=1:ph2 % for (i = 1; i <= ph2; i++) { inv = 1/bincoeff(ph,i); % inv = 1.0 / bincoeff(ph,i); mpi = min(d,i); % mpi = min(d,i); % for j=max(0,i-t):mpi % for (j = max(0,i-t); j <= mpi; j++) bezalfs(j+1,i+1) = inv*bincoeff(d,j)*bincoeff(t,i-j); % bezalfs[i][j] = inv * bincoeff(d,j) * bincoeff(t,i-j); end end % } % for i=ph2+1:ph-1 % for (i = ph2+1; i <= ph-1; i++) { mpi = min(d,i); % mpi = min(d, i); for j=max(0,i-t):mpi % for (j = max(0,i-t); j <= mpi; j++) bezalfs(j+1,i+1) = bezalfs(d-j+1,ph-i+1); % bezalfs[i][j] = bezalfs[ph-i][d-j]; end end % } % mh = ph; % mh = ph; kind = ph+1; % kind = ph+1; r = -1; % r = -1; a = d; % a = d; b = d+1; % b = d+1; cind = 1; % cind = 1; ua = k(1); % ua = k[0]; % for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) ic(ii+1,1) = c(ii+1,1); % ictrl[0][ii] = ctrl[0][ii]; end % for i=0:ph % for (i = 0; i <= ph; i++) ik(i+1) = ua; % ik[i] = ua; end % % // initialise first bezier seg for i=0:d % for (i = 0; i <= d; i++) for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) bpts(ii+1,i+1) = c(ii+1,i+1); % bpts[i][ii] = ctrl[i][ii]; end end % % // big loop thru knot vector while b < m % while (b < m) { i = b; % i = b; while b < m && k(b+1) == k(b+2) % while (b < m && k[b] == k[b+1]) b = b + 1; % b++; end % mul = b - i + 1; % mul = b - i + 1; mh = mh + mul + t; % mh += mul + t; ub = k(b+1); % ub = k[b]; oldr = r; % oldr = r; r = d - mul; % r = d - mul; % % // insert knot u(b) r times if oldr > 0 % if (oldr > 0) lbz = floor((oldr+2)/2); % lbz = (oldr+2) / 2; else % else lbz = 1; % lbz = 1; end % if r > 0 % if (r > 0) rbz = ph - floor((r+1)/2); % rbz = ph - (r+1)/2; else % else rbz = ph; % rbz = ph; end % if r > 0 % if (r > 0) { % // insert knot to get bezier segment numer = ub - ua; % numer = ub - ua; for q=d:-1:mul+1 % for (q = d; q > mul; q--) alfs(q-mul) = numer / (k(a+q+1)-ua); % alfs[q-mul-1] = numer / (k[a+q]-ua); end for j=1:r % for (j = 1; j <= r; j++) { save = r - j; % save = r - j; s = mul + j; % s = mul + j; % for q=d:-1:s % for (q = d; q >= s; q--) for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) tmp1 = alfs(q-s+1)*bpts(ii+1,q+1); tmp2 = (1-alfs(q-s+1))*bpts(ii+1,q); bpts(ii+1,q+1) = tmp1 + tmp2; % bpts[q][ii] = alfs[q-s]*bpts[q][ii]+(1.0-alfs[q-s])*bpts[q-1][ii]; end end % for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) Nextbpts(ii+1,save+1) = bpts(ii+1,d+1); % Nextbpts[save][ii] = bpts[d][ii]; end end % } end % } % // end of insert knot % % // degree elevate bezier for i=lbz:ph % for (i = lbz; i <= ph; i++) { for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) ebpts(ii+1,i+1) = 0; % ebpts[i][ii] = 0.0; end mpi = min(d, i); % mpi = min(d, i); for j=max(0,i-t):mpi % for (j = max(0,i-t); j <= mpi; j++) for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) tmp1 = ebpts(ii+1,i+1); tmp2 = bezalfs(j+1,i+1)*bpts(ii+1,j+1); ebpts(ii+1,i+1) = tmp1 + tmp2; % ebpts[i][ii] = ebpts[i][ii] + bezalfs[i][j]*bpts[j][ii]; end end end % } % // end of degree elevating bezier % if oldr > 1 % if (oldr > 1) { % // must remove knot u=k[a] oldr times first = kind - 2; % first = kind - 2; last = kind; % last = kind; den = ub - ua; % den = ub - ua; bet = floor((ub-ik(kind)) / den); % bet = (ub-ik[kind-1]) / den; % % // knot removal loop for tr=1:oldr-1 % for (tr = 1; tr < oldr; tr++) { i = first; % i = first; j = last; % j = last; kj = j - kind + 1; % kj = j - kind + 1; while j-i > tr % while (j - i > tr) { % // loop and compute the new control points % // for one removal step if i < cind % if (i < cind) { alf = (ub-ik(i+1))/(ua-ik(i+1)); % alf = (ub-ik[i])/(ua-ik[i]); for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) tmp1 = alf*ic(ii+1,i+1); tmp2 = (1-alf)*ic(ii+1,i); ic(ii+1,i+1) = tmp1 + tmp2; % ictrl[i][ii] = alf * ictrl[i][ii] + (1.0-alf) * ictrl[i-1][ii]; end end % } if j >= lbz % if (j >= lbz) { if j-tr <= kind-ph+oldr % if (j-tr <= kind-ph+oldr) { gam = (ub-ik(j-tr+1)) / den; % gam = (ub-ik[j-tr]) / den; for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) tmp1 = gam*ebpts(ii+1,kj+1); tmp2 = (1-gam)*ebpts(ii+1,kj+2); ebpts(ii+1,kj+1) = tmp1 + tmp2; % ebpts[kj][ii] = gam*ebpts[kj][ii] + (1.0-gam)*ebpts[kj+1][ii]; end % } else % else { for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) tmp1 = bet*ebpts(ii+1,kj+1); tmp2 = (1-bet)*ebpts(ii+1,kj+2); ebpts(ii+1,kj+1) = tmp1 + tmp2; % ebpts[kj][ii] = bet*ebpts[kj][ii] + (1.0-bet)*ebpts[kj+1][ii]; end end % } end % } i = i + 1; % i++; j = j - 1; % j--; kj = kj - 1; % kj--; end % } % first = first - 1; % first--; last = last + 1; % last++; end % } end % } % // end of removing knot n=k[a] % % // load the knot ua if a ~= d % if (a != d) for i=0:ph-oldr-1 % for (i = 0; i < ph-oldr; i++) { ik(kind+1) = ua; % ik[kind] = ua; kind = kind + 1; % kind++; end end % } % % // load ctrl pts into ic for j=lbz:rbz % for (j = lbz; j <= rbz; j++) { for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) ic(ii+1,cind+1) = ebpts(ii+1,j+1); % ictrl[cind][ii] = ebpts[j][ii]; end cind = cind + 1; % cind++; end % } % if b < m % if (b < m) { % // setup for next pass thru loop for j=0:r-1 % for (j = 0; j < r; j++) for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) bpts(ii+1,j+1) = Nextbpts(ii+1,j+1); % bpts[j][ii] = Nextbpts[j][ii]; end end for j=r:d % for (j = r; j <= d; j++) for ii=0:mc-1 % for (ii = 0; ii < mc; ii++) bpts(ii+1,j+1) = c(ii+1,b-d+j+1); % bpts[j][ii] = ctrl[b-d+j][ii]; end end a = b; % a = b; b = b+1; % b++; ua = ub; % ua = ub; % } else % else % // end knot for i=0:ph % for (i = 0; i <= ph; i++) ik(kind+i+1) = ub; % ik[kind+i] = ub; end end end % } % End big while loop % // end while loop % % *nh = mh - ph - 1; % % freevec2mat(ctrl); % freevec2mat(ictrl); % freematrix(bezalfs); % freematrix(bpts); % freematrix(ebpts); % freematrix(Nextbpts); % mxFree(alfs); % % return(ierr); end % } function b = bincoeff(n,k) % Computes the binomial coefficient. % % ( n ) n! % ( ) = -------- % ( k ) k!(n-k)! % % b = bincoeff(n,k) % % Algorithm from 'Numerical Recipes in C, 2nd Edition' pg215. % double bincoeff(int n, int k) % { b = floor(0.5+exp(factln(n)-factln(k)-factln(n-k))); % return floor(0.5+exp(factln(n)-factln(k)-factln(n-k))); end % } function f = factln(n) % computes ln(n!) if n <= 1, f = 0; return, end f = gammaln(n+1); %log(factorial(n)); endnurbs-1.4.3/inst/PaxHeaders.16248/nrbdegelev.m0000644000000000000000000000013214030323741015656 xustar0030 mtime=1617012705.791919259 30 atime=1617012705.791919259 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbdegelev.m0000644000175000017500000001150414030323741017756 0ustar00rvazquezrvazquez00000000000000function inurbs = nrbdegelev(nurbs, ntimes) % % NRBDEGELEV: Elevate the degree of the NURBS curve, surface or volume. % % Calling Sequence: % % ecrv = nrbdegelev(crv,utimes); % esrf = nrbdegelev(srf,[utimes,vtimes]); % evol = nrbdegelev(vol,[utimes,vtimes,wtimes]); % % INPUT: % % crv : NURBS curve, see nrbmak. % % srf : NURBS surface, see nrbmak. % % vol : NURBS volume, see nrbmak. % % utimes : Increase the degree along U direction utimes. % % vtimes : Increase the degree along V direction vtimes. % % wtimes : Increase the degree along W direction vtimes. % % OUTPUT: % % ecrv : new NURBS structure for a curve with degree elevated. % % esrf : new NURBS structure for a surface with degree elevated. % % evol : new NURBS structure for a volume with degree elevated. % % % Description: % % Degree elevates the NURBS curve or surface. This function uses the % B-Spline function bspdegelev, which interface to an internal 'C' % routine. % % Examples: % % Increase the NURBS surface twice along the V direction. % esrf = nrbdegelev(srf, [0, 2]); % % See also: % % bspdegelev % % Copyright (C) 2000 Mark Spink, 2010 Rafel Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if nargin < 2 error('Input argument must include the NURBS and degree increment.'); end if ~isstruct(nurbs) error('NURBS representation is not structure!'); end if ~strcmp(nurbs.form,'B-NURBS') error('Not a recognised NURBS representation'); end degree = nurbs.order-1; if iscell(nurbs.knots) if size(nurbs.knots,2) == 3 % NURBS represents a volume [dim,num1,num2,num3] = size(nurbs.coefs); % Degree elevate along the w direction if ntimes(3) == 0 coefs = nurbs.coefs; knots{3} = nurbs.knots{3}; else coefs = reshape(nurbs.coefs,4*num1*num2,num3); [coefs,knots{3}] = bspdegelev(degree(3),coefs,nurbs.knots{3},ntimes(3)); num3 = size(coefs,2); coefs = reshape(coefs,[4 num1 num2 num3]); end % Degree elevate along the v direction if ntimes(2) == 0 knots{2} = nurbs.knots{2}; else coefs = permute(coefs,[1 2 4 3]); coefs = reshape(coefs,4*num1*num3,num2); [coefs,knots{2}] = bspdegelev(degree(2),coefs,nurbs.knots{2},ntimes(2)); num2 = size(coefs,2); coefs = reshape(coefs,[4 num1 num3 num2]); coefs = permute(coefs,[1 2 4 3]); end % Degree elevate along the u direction if ntimes(1) == 0 knots{1} = nurbs.knots{1}; else coefs = permute(coefs,[1 3 4 2]); coefs = reshape(coefs,4*num2*num3,num1); [coefs,knots{1}] = bspdegelev(degree(1),coefs,nurbs.knots{1},ntimes(1)); coefs = reshape(coefs,[4 num2 num3 size(coefs,2)]); coefs = permute(coefs,[1 4 2 3]); end elseif size(nurbs.knots,2) == 2 % NURBS represents a surface [dim,num1,num2] = size(nurbs.coefs); % Degree elevate along the v direction if ntimes(2) == 0 coefs = nurbs.coefs; knots{2} = nurbs.knots{2}; else coefs = reshape(nurbs.coefs,4*num1,num2); [coefs,knots{2}] = bspdegelev(degree(2),coefs,nurbs.knots{2},ntimes(2)); num2 = size(coefs,2); coefs = reshape(coefs,[4 num1 num2]); end % Degree elevate along the u direction if ntimes(1) == 0 knots{1} = nurbs.knots{1}; else coefs = permute(coefs,[1 3 2]); coefs = reshape(coefs,4*num2,num1); [coefs,knots{1}] = bspdegelev(degree(1),coefs,nurbs.knots{1},ntimes(1)); coefs = reshape(coefs,[4 num2 size(coefs,2)]); coefs = permute(coefs,[1 3 2]); end end else % NURBS represents a curve if (isempty(ntimes) || ntimes == 0) coefs = nurbs.coefs; knots = nurbs.knots; else [coefs,knots] = bspdegelev(degree,nurbs.coefs,nurbs.knots,ntimes); end end % construct new NURBS inurbs = nrbmak(coefs,knots); end %!demo %! crv = nrbtestcrv; %! plot(crv.coefs(1,:),crv.coefs(2,:),'bo') %! title('Degree elevation along test curve: curve and control polygons.'); %! hold on; %! plot(crv.coefs(1,:),crv.coefs(2,:),'b--'); %! nrbplot(crv,48); %! %! icrv = nrbdegelev(crv, 1); %! %! plot(icrv.coefs(1,:),icrv.coefs(2,:),'ro') %! plot(icrv.coefs(1,:),icrv.coefs(2,:),'r--'); %! %! hold off; nurbs-1.4.3/inst/PaxHeaders.16248/nrbtestsrf.m0000644000000000000000000000013214030323741015735 xustar0030 mtime=1617012705.803919443 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbtestsrf.m0000644000175000017500000000360314030323741020036 0ustar00rvazquezrvazquez00000000000000function srf = nrbtestsrf % NRBTESTSRF: Constructs a simple test surface. % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . % allocate multi-dimensional array of control points pnts = zeros(3,5,5); % define a grid of control points % in this case a regular grid of u,v points % pnts(3,u,v) % pnts(:,:,1) = [ 0.0 3.0 5.0 8.0 10.0; % w*x 0.0 0.0 0.0 0.0 0.0; % w*y 2.0 2.0 7.0 7.0 8.0]; % w*z pnts(:,:,2) = [ 0.0 3.0 5.0 8.0 10.0; 3.0 3.0 3.0 3.0 3.0; 0.0 0.0 5.0 5.0 7.0]; pnts(:,:,3) = [ 0.0 3.0 5.0 8.0 10.0; 5.0 5.0 5.0 5.0 5.0; 0.0 0.0 5.0 5.0 7.0]; pnts(:,:,4) = [ 0.0 3.0 5.0 8.0 10.0; 8.0 8.0 8.0 8.0 8.0; 5.0 5.0 8.0 8.0 10.0]; pnts(:,:,5) = [ 0.0 3.0 5.0 8.0 10.0; 10.0 10.0 10.0 10.0 10.0; 5.0 5.0 8.0 8.0 10.0]; % knots knots{1} = [0 0 0 1/3 2/3 1 1 1]; % knots along u knots{2} = [0 0 0 1/3 2/3 1 1 1]; % knots along v % make and draw nurbs surface srf = nrbmak(pnts,knots); end %!demo %! srf = nrbtestsrf; %! nrbplot(srf,[20 30]) %! title('Test surface') %! hold off nurbs-1.4.3/inst/PaxHeaders.16248/basisfun.m0000644000000000000000000000013214030323741015353 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/basisfun.m0000644000175000017500000000445514030323741017462 0ustar00rvazquezrvazquez00000000000000function B = basisfun (iv, uv, p, U) % BASISFUN: Basis function for B-Spline % % Calling Sequence: % % N = basisfun(iv,uv,p,U) % % INPUT: % % iv - knot span ( from FindSpan() ) % uv - parametric points % p - spline degree % U - knot sequence % % OUTPUT: % % N - Basis functions vector(numel(uv)*(p+1)) % % Adapted from Algorithm A2.2 from 'The NURBS BOOK' pg70. % % See also: % % numbasisfun, basisfunder, findspan % % Copyright (C) 2000 Mark Spink % Copyright (C) 2007 Daniel Claxton % Copyright (C) 2009 Carlo de Falco % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . B = zeros(numel(uv), p+1); for jj = 1:numel(uv) i = iv(jj) + 1; %% findspan uses 0-based numbering u = uv(jj); left = zeros(p+1,1); right = zeros(p+1,1); N(1) = 1; for j=1:p left(j+1) = u - U(i+1-j); right(j+1) = U(i+j) - u; saved = 0; for r=0:j-1 temp = N(r+1)/(right(r+2) + left(j-r+1)); N(r+1) = saved + right(r+2)*temp; saved = left(j-r+1)*temp; end N(j+1) = saved; end B (jj, :) = N; end end %!test %! n = 3; %! U = [0 0 0 1/2 1 1 1]; %! p = 2; %! u = linspace (0, 1, 10); %! s = findspan (n, p, u, U); %! Bref = [1.00000 0.00000 0.00000 %! 0.60494 0.37037 0.02469 %! 0.30864 0.59259 0.09877 %! 0.11111 0.66667 0.22222 %! 0.01235 0.59259 0.39506 %! 0.39506 0.59259 0.01235 %! 0.22222 0.66667 0.11111 %! 0.09877 0.59259 0.30864 %! 0.02469 0.37037 0.60494 %! 0.00000 0.00000 1.00000]; %! B = basisfun (s, u, p, U); %! assert (B, Bref, 1e-5); nurbs-1.4.3/inst/PaxHeaders.16248/vecroty.m0000644000000000000000000000013214030323741015234 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecroty.m0000644000175000017500000000325114030323741017334 0ustar00rvazquezrvazquez00000000000000function ry = vecroty(angle) % % VECROTY: Transformation matrix for a rotation around the y axis. % % Calling Sequence: % % ry = vecroty(angle); % % INPUT: % % angle : rotation angle defined in radians % % OUTPUT: % % ry : (4x4) Transformation matrix. % % % Description: % % Return the (4x4) Transformation matrix for a rotation about the y axis % by the defined angle. % % The matrix is: % % [ cos(angle) 0 sin(angle) 0] % [ 0 1 0 0] % [ -sin(angle) 0 cos(angle) 0] % [ 0 0 0 1] % % Examples: % % Rotate the NURBS line (0.0 0.0 0.0) - (3.0 3.0 3.0) by 45 degrees % around the y-axis % % line = nrbline([0.0 0.0 0.0],[3.0 3.0 3.0]); % trans = vecroty(%pi/4); % rline = nrbtform(line, trans); % % See also: % % nrbtform % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . sn = sin(angle); cn = cos(angle); ry = [cn 0 sn 0; 0 1 0 0; -sn 0 cn 0; 0 0 0 1]; end nurbs-1.4.3/inst/PaxHeaders.16248/nrbplot.m0000644000000000000000000000013214030323741015221 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbplot.m0000644000175000017500000002370614030323741017330 0ustar00rvazquezrvazquez00000000000000function nrbplot (nurbs, subd, varargin) % % NRBPLOT: Plot a NURBS curve or surface, or the boundary of a NURBS volume. % % Calling Sequence: % % nrbplot (nrb, npnts) % nrbplot (nrb, npnts, p, v) % % INPUT: % % nrb : NURBS curve, surface or volume, see nrbmak. % % npnts : Number of evaluation points, for a surface or volume, a row % vector with the number of points along each direction. % % [p,v] : property/value options % % Valid property/value pairs include: % % Property Value/{Default} % ----------------------------------- % light {off} | on % colormap {'copper'} % % Example: % % Plot the test surface with 20 points along the U direction % and 30 along the V direction % % nrbplot(nrbtestsrf, [20 30]) % % Copyright (C) 2000 Mark Spink % Copyright (C) 2010 Carlo de Falco, Rafael Vazquez % Copyright (C) 2012 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . nargs = nargin; if nargs < 2 error ('Need a NURBS to plot and the number of subdivisions!'); elseif rem(nargs+2,2) error ('Param value pairs expected') end if (any (nurbs.order < 2)) warning ('The plot with order smaller than 2 may not be correct') end % Default values light='off'; cmap='summer'; % Recover Param/Value pairs from argument list for i=1:2:nargs-2 Param = varargin{i}; Value = varargin{i+1}; if (~ischar (Param)) error ('Parameter must be a string') elseif size(Param,1)~=1 error ('Parameter must be a non-empty single row string.') end switch lower (Param) case 'light' light = lower (Value); if (~ischar (light)) error ('light must be a string.') elseif ~(strcmp(light,'off') | strcmp(light,'on')) error ('light must be off | on') end case 'colormap' if ischar (Value) cmap = lower(Value); elseif size (Value, 2) ~= 3 error ('colormap must be a string or have exactly three columns.') else cmap=Value; end otherwise error ('Unknown parameter: %s', Param) end end colormap (cmap); % convert the number of subdivisions in number of points subd = subd+1; % plot the curve or surface if (iscell (nurbs.knots)) if (size (nurbs.knots,2) == 2) % plot a NURBS surface knt = nurbs.knots; order = nurbs.order; p = nrbeval (nurbs, {linspace(knt{1}(order(1)),knt{1}(end-order(1)+1),subd(1)) ... linspace(knt{2}(order(2)),knt{2}(end-order(2)+1),subd(2))}); if (strcmp (light,'on')) % light surface surfl (squeeze(p(1,:,:)), squeeze(p(2,:,:)), squeeze(p(3,:,:))); shading interp; else surf (squeeze (p(1,:,:)), squeeze (p(2,:,:)), squeeze (p(3,:,:))); shading faceted; end elseif (size (nurbs.knots,2) == 3) % plot the boundaries of a NURBS volume bnd = nrbextract (nurbs); hold_flag = ishold; nrbplot (bnd(1), subd(2:3), varargin{:}); hold on nrbplot (bnd(2), subd(2:3), varargin{:}); nrbplot (bnd(3), subd([1 3]), varargin{:}); nrbplot (bnd(4), subd([1 3]), varargin{:}); nrbplot (bnd(5), subd(1:2), varargin{:}); nrbplot (bnd(6), subd(1:2), varargin{:}); if (~hold_flag) hold off end else error ('nrbplot: some argument is not correct') end else % plot a NURBS curve order = nurbs.order; p = nrbeval (nurbs, linspace (nurbs.knots(order), nurbs.knots(end-order+1), subd)); if (any (nurbs.coefs(3,:))) % 3D curve plot3 (p(1,:), p(2,:), p(3,:)); grid on; else % 2D curve plot (p(1,:), p(2,:)); end end axis equal; end % plot the control surface % hold on; % mesh(squeeze(pnts(1,:,:)),squeeze(pnts(2,:,:)),squeeze(pnts(3,:,:))); % hold off; %!demo %! crv = nrbtestcrv; %! nrbplot(crv,100) %! title('Test curve') %! hold off %!demo %! coefs = [0.0 7.5 15.0 25.0 35.0 30.0 27.5 30.0; %! 0.0 2.5 0.0 -5.0 5.0 15.0 22.5 30.0]; %! knots = [0.0 0.0 0.0 1/6 1/3 1/2 2/3 5/6 1.0 1.0 1.0]; %! %! geom = [ %! nrbmak(coefs,knots) %! nrbline([30.0 30.0],[20.0 30.0]) %! nrbline([20.0 30.0],[20.0 20.0]) %! nrbcirc(10.0,[10.0 20.0],1.5*pi,0.0) %! nrbline([10.0 10.0],[0.0 10.0]) %! nrbline([0.0 10.0],[0.0 0.0]) %! nrbcirc(5.0,[22.5 7.5]) %! ]; %! %! ng = length(geom); %! for i = 1:ng %! nrbplot(geom(i),500); %! hold on; %! end %! hold off; %! axis equal; %! title('2D Geometry formed by a series of NURBS curves'); %!demo %! sphere = nrbrevolve(nrbcirc(1,[],0.0,pi),[0.0 0.0 0.0],[1.0 0.0 0.0]); %! nrbplot(sphere,[40 40],'light','on'); %! title('Ball and torus - surface construction by revolution'); %! hold on; %! torus = nrbrevolve(nrbcirc(0.2,[0.9 1.0]),[0.0 0.0 0.0],[1.0 0.0 0.0]); %! nrbplot(torus,[40 40],'light','on'); %! hold off %!demo %! knots = {[0 0 0 1/2 1 1 1] [0 0 0 1 1 1]... %! [0 0 0 1/6 2/6 1/2 1/2 4/6 5/6 1 1 1]}; %! %! coefs = [-1.0000 -0.9734 -0.7071 1.4290 1.0000 3.4172 %! 0 2.4172 0 0.0148 -2.0000 -1.9734 %! 0 2.0000 4.9623 9.4508 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -0.8536 0 -0.6036 1.9571 1.2071 3.5000 %! 0.3536 2.5000 0.2500 0.5429 -1.7071 -1.0000 %! 0 2.0000 4.4900 8.5444 3.4142 2.0000 %! 0.8536 1.0000 0.6036 1.0000 0.8536 1.0000 %! -0.3536 -4.0000 -0.2500 -1.2929 1.7071 1.0000 %! 0.8536 0 0.6036 -2.7071 -1.2071 -5.0000 %! 0 2.0000 4.4900 10.0711 3.4142 2.0000 %! 0.8536 1.0000 0.6036 1.0000 0.8536 1.0000 %! 0 -4.0000 0 0.7071 2.0000 5.0000 %! 1.0000 4.0000 0.7071 -0.7071 -1.0000 -5.0000 %! 0 2.0000 4.9623 14.4142 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -2.5000 -4.0000 -1.7678 0.7071 1.0000 5.0000 %! 0 4.0000 0 -0.7071 -3.5000 -5.0000 %! 0 2.0000 6.0418 14.4142 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -2.4379 0 -1.7238 2.7071 1.9527 5.0000 %! 0.9527 4.0000 0.6737 1.2929 -3.4379 -1.0000 %! 0 2.0000 6.6827 10.0711 4.0000 2.0000 %! 1.0000 1.0000 0.7071 1.0000 1.0000 1.0000 %! -0.9734 -1.0000 -0.6883 0.7071 3.4172 1.0000 %! 2.4172 0 1.7092 -1.4142 -1.9734 -2.0000 %! 0 4.0000 6.6827 4.9623 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! 0 -0.8536 0 0.8536 3.5000 1.2071 %! 2.5000 0.3536 1.7678 -1.2071 -1.0000 -1.7071 %! 0 3.4142 6.0418 4.4900 4.0000 0 %! 1.0000 0.8536 0.7071 0.6036 1.0000 0.8536 %! -4.0000 -0.3536 -2.8284 1.2071 1.0000 1.7071 %! 0 0.8536 0 -0.8536 -5.0000 -1.2071 %! 0 3.4142 7.1213 4.4900 4.0000 0 %! 1.0000 0.8536 0.7071 0.6036 1.0000 0.8536 %! -4.0000 0 -2.8284 1.4142 5.0000 2.0000 %! 4.0000 1.0000 2.8284 -0.7071 -5.0000 -1.0000 %! 0 4.0000 10.1924 4.9623 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! -4.0000 -2.5000 -2.8284 0.7071 5.0000 1.0000 %! 4.0000 0 2.8284 -2.4749 -5.0000 -3.5000 %! 0 4.0000 10.1924 6.0418 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! 0 -2.4379 0 1.3808 5.0000 1.9527 %! 4.0000 0.9527 2.8284 -2.4309 -1.0000 -3.4379 %! 0 4.0000 7.1213 6.6827 4.0000 0 %! 1.0000 1.0000 0.7071 0.7071 1.0000 1.0000 %! -1.0000 -0.9734 0.2071 2.4163 1.0000 3.4172 %! 0 2.4172 -1.2071 -1.3954 -2.0000 -1.9734 %! 2.0000 4.0000 7.0178 6.6827 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000 %! -0.8536 0 0.3536 2.4749 1.2071 3.5000 %! 0.3536 2.5000 -0.8536 -0.7071 -1.7071 -1.0000 %! 1.7071 4.0000 6.3498 6.0418 1.7071 0 %! 0.8536 1.0000 0.8536 0.7071 0.8536 1.0000 %! -0.3536 -4.0000 0.8536 0.7071 1.7071 1.0000 %! 0.8536 0 -0.3536 -3.5355 -1.2071 -5.0000 %! 1.7071 4.0000 6.3498 7.1213 1.7071 0 %! 0.8536 1.0000 0.8536 0.7071 0.8536 1.0000 %! 0 -4.0000 1.2071 3.5355 2.0000 5.0000 %! 1.0000 4.0000 -0.2071 -3.5355 -1.0000 -5.0000 %! 2.0000 4.0000 7.0178 10.1924 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000 %! -2.5000 -4.0000 -0.5429 3.5355 1.0000 5.0000 %! 0 4.0000 -1.9571 -3.5355 -3.5000 -5.0000 %! 2.0000 4.0000 8.5444 10.1924 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000 %! -2.4379 0 -0.0355 3.5355 1.9527 5.0000 %! 0.9527 4.0000 -1.4497 -0.7071 -3.4379 -1.0000 %! 2.0000 4.0000 9.4508 7.1213 2.0000 0 %! 1.0000 1.0000 1.0000 0.7071 1.0000 1.0000]; %! coefs = reshape (coefs, 4, 4, 3, 9); %! horseshoe = nrbmak (coefs, knots); %! nrbplot (horseshoe, [6, 6, 50], 'light', 'on'); nurbs-1.4.3/inst/PaxHeaders.16248/vecrotx.m0000644000000000000000000000013214030323741015233 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/vecrotx.m0000644000175000017500000000320114030323741017326 0ustar00rvazquezrvazquez00000000000000function rx = vecrotx(angle) % % VECROTX: Transformation matrix for a rotation around the x axis. % % Calling Sequence: % % rx = vecrotx(angle); % % INPUT: % % angle : rotation angle defined in radians % % OUTPUT: % % rx : (4x4) Transformation matrix. % % % Description: % % Return the (4x4) Transformation matrix for a rotation about the x axis % by the defined angle. % % The matrix is: % % [ 1 0 0 0] % [ 0 cos(angle) -sin(angle) 0] % [ 0 sin(angle) cos(angle) 0] % [ 0 0 0 1] % % Examples: % % Rotate the NURBS line (0.0 0.0 0.0) - (3.0 3.0 3.0) by 45 degrees % around the x-axis % % line = nrbline([0.0 0.0 0.0],[3.0 3.0 3.0]); % trans = vecrotx(%pi/4); % rline = nrbtform(line, trans); % % See also: % % nrbtform % % Copyright (C) 2000 Mark Spink % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . sn = sin(angle); cn = cos(angle); rx = [1 0 0 0; 0 cn -sn 0; 0 sn cn 0; 0 0 0 1]; end nurbs-1.4.3/inst/PaxHeaders.16248/nrbpermute.m0000644000000000000000000000013214030323741015724 xustar0030 mtime=1617012705.799919382 30 atime=1617012705.799919382 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbpermute.m0000644000175000017500000000451714030323741020032 0ustar00rvazquezrvazquez00000000000000function tvol = nrbpermute (vol, ord) % % NRBPERMUTE: Rearrange the directions of a NURBS volume or surface. % % Calling Sequence: % % tvol = nrbpermute(vol,order) % % INPUT: % % vol : NURBS volume or surface, see nrbmak. % order : the order to rearrange the directions of the NURBS entity. % % OUTPUT: % % tvol : NURBS volume or surface with rearranged directions. % % Description: % % Utility function that rearranges the directions of a NURBS volume or % surface. For surfaces, nrbpermute(srf,[2 1]) is the same as % nrbtransp(srf). NURBS curves cannot be rearranged. % % Example: % % nrbpermute (vol, [1 3 2]) % % Copyright (C) 2013 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (~iscell(vol.knots)) error('A NURBS curve cannot be rearranged.'); end tvol = nrbmak (permute (vol.coefs, [1, ord+1]), {vol.knots{ord}}); %!demo %! vol = nrbrevolve (nrb4surf ([1 0], [2 0], [1 1], [2 1]), [0 0 0], [0 1 0], pi/8); %! nrbplot(vol,[5 10 20]); %! title('NURBS volume and the same after reordering the directions') %! hold on %! vol.coefs(1,:,:) = vol.coefs(1,:,:) + 2; %! vol = nrbpermute(vol,[2 3 1]); %! nrbplot(vol,[5 10 20]); %! hold off %!test %! vol = nrbrevolve (nrb4surf ([1 0], [2 0], [1 1], [2 1]), [0 0 0], [0 1 0], pi/8); %! perm1 = [1 3 2]; %! perm2 = [2 1 3]; %! vol2 = nrbpermute (vol, perm1); %! vol3 = nrbpermute (vol, perm2); %! assert (vol.number(perm1), vol2.number) %! assert (vol.order(perm1), vol2.order) %! assert ({vol.knots{perm1}}, vol2.knots) %! assert (permute(vol.coefs, [1, perm1+1]), vol2.coefs) %! assert (vol.number(perm2), vol3.number) %! assert (vol.order(perm2), vol3.order) %! assert ({vol.knots{perm2}}, vol3.knots) %! assert (permute(vol.coefs, [1, perm2+1]), vol3.coefs) nurbs-1.4.3/inst/PaxHeaders.16248/nrbbasisfun.m0000644000000000000000000000013214030323741016055 xustar0030 mtime=1617012705.787919198 30 atime=1617012705.787919198 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/nrbbasisfun.m0000644000175000017500000001662514030323741020166 0ustar00rvazquezrvazquez00000000000000function [B, id] = nrbbasisfun (points, nrb) % NRBBASISFUN: Basis functions for NURBS % % Calling Sequence: % % B = nrbbasisfun (u, crv) % B = nrbbasisfun ({u, v}, srf) % [B, N] = nrbbasisfun ({u, v}, srf) % [B, N] = nrbbasisfun (pts, srf) % [B, N] = nrbbasisfun ({u, v, w}, vol) % [B, N] = nrbbasisfun (pts, vol) % % INPUT: % % u - parametric coordinates along u direction % v - parametric coordinates along v direction % w - parametric coordinates along w direction % pts - array of scattered points in parametric domain, array size: (ndim,num_points) % crv - NURBS curve % srf - NURBS surface % vol - NURBS volume % % If the parametric coordinates are given in a cell-array, the values % are computed in a tensor product set of points % % OUTPUT: % % B - Value of the basis functions at the points % size(B)=[npts, prod(nrb.order)] % % N - Indices of the basis functions that are nonvanishing at each % point. size(N) == size(B) % % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2015 Jacopo Corno % Copyright (C) 2016 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if ( (nargin<2) ... || (nargout>2) ... || (~isstruct(nrb)) ... || (iscell(points) && ~iscell(nrb.knots)) ... || (~iscell(points) && iscell(nrb.knots) && (size(points,1)~=numel(nrb.number))) ... || iscell(points) && numel(points) ~= numel(nrb.number) ... ) error('Incorrect input arguments in nrbbasisfun'); end if (~iscell (nrb.knots)) %% NURBS curve knt = {nrb.knots}; else %% NURBS surface or volume knt = nrb.knots; end ndim = numel (nrb.number); w = reshape (nrb.coefs(4,:), [nrb.number 1]); for idim = 1:ndim if (iscell (points)) pts_dim = points{idim}; else pts_dim = points(idim,:); end sp{idim} = findspan (nrb.number(idim)-1, nrb.order(idim)-1, pts_dim, knt{idim}); N{idim} = basisfun(sp{idim}, pts_dim, nrb.order(idim)-1, knt{idim}); num{idim} = numbasisfun (sp{idim}, pts_dim, nrb.order(idim)-1, knt{idim}) + 1; end if (ndim == 1) id = num{1}; B = reshape (w(num{1}), size(N{1})) .* N{1}; B = bsxfun (@(x,y) x./y, B, sum(B,2)); % B = B ./ sum(B,2); else if (iscell (points)) npts_dim = cellfun (@numel, points); cumnpts = cumprod([1 npts_dim]); npts = prod (npts_dim); val_aux = 1; numaux = 1; cumorder = cumprod ([1 nrb.order]); cumnumber = cumprod ([1 nrb.number]); for idim = 1:ndim val_aux = kron (N{idim}, val_aux); num_dim = reshape (num{idim}, 1, npts_dim(idim), 1, nrb.order(idim)); num_dim = repmat (num_dim, cumnpts(idim), 1, cumorder(idim), 1); num_prev = reshape (numaux, cumnpts(idim), 1, cumorder(idim), 1); num_prev = repmat (num_prev, 1, npts_dim(idim), 1, nrb.order(idim)); numaux = sub2ind ([cumnumber(idim) nrb.number(idim)], num_prev, num_dim); numaux = reshape (numaux, cumnpts(idim+1), cumorder(idim+1)); end B = reshape (val_aux, npts, prod (nrb.order)); id = reshape (numaux, npts, prod (nrb.order)); W = w(id); B = bsxfun (@(x,y) x./y, W.*B, sum (W .* B, 2)); else npts = numel (points(1,:)); B = zeros (npts, prod(nrb.order)); id = zeros (npts, prod(nrb.order)); local_num = cell (ndim, 1); for ipt = 1:npts val_aux = 1; for idim = 1:ndim val_aux = reshape (val_aux.' * N{idim}(ipt,:), 1, []); % val_aux2 = kron (N{idim}(ipt,:), val_aux); local_num{idim} = num{idim}(ipt,:); end [local_num{:}] = ndgrid (local_num{:}); id(ipt,:) = reshape (sub2ind (nrb.number, local_num{:}), 1, size(id, 2)); W = reshape (w(id(ipt,:)), size(val_aux)); val_aux = W .* val_aux; B(ipt,:) = val_aux / sum (val_aux); end end end end %!demo %! U = [0 0 0 0 1 1 1 1]; %! x = [0 1/3 2/3 1] ; %! y = [0 0 0 0]; %! w = [1 1 1 1]; %! nrb = nrbmak ([x;y;y;w], U); %! u = linspace(0, 1, 30); %! B = nrbbasisfun (u, nrb); %! xplot = sum(bsxfun(@(x,y) x.*y, B, x),2); %! plot(xplot, B) %! title('Cubic Bernstein polynomials') %! hold off %!test %! U = [0 0 0 0 1 1 1 1]; %! x = [0 1/3 2/3 1] ; %! y = [0 0 0 0]; %! w = rand(1,4); %! nrb = nrbmak ([x;y;y;w], U); %! u = linspace(0, 1, 30); %! B = nrbbasisfun (u, nrb); %! xplot = sum(bsxfun(@(x,y) x.*y, B, x),2); %! %! yy = y; yy(1) = 1; %! nrb2 = nrbmak ([x.*w;yy;y;w], U); %! aux = nrbeval(nrb2,u); %! %figure, plot(xplot, B(:,1), aux(1,:).', w(1)*aux(2,:).') %! assert(B(:,1), w(1)*aux(2,:).', 1e-6) %! %! yy = y; yy(2) = 1; %! nrb2 = nrbmak ([x.*w;yy;y;w], U); %! aux = nrbeval(nrb2, u); %! %figure, plot(xplot, B(:,2), aux(1,:).', w(2)*aux(2,:).') %! assert(B(:,2), w(2)*aux(2,:).', 1e-6) %! %! yy = y; yy(3) = 1; %! nrb2 = nrbmak ([x.*w;yy;y;w], U); %! aux = nrbeval(nrb2,u); %! %figure, plot(xplot, B(:,3), aux(1,:).', w(3)*aux(2,:).') %! assert(B(:,3), w(3)*aux(2,:).', 1e-6) %! %! yy = y; yy(4) = 1; %! nrb2 = nrbmak ([x.*w;yy;y;w], U); %! aux = nrbeval(nrb2,u); %! %figure, plot(xplot, B(:,4), aux(1,:).', w(4)*aux(2,:).') %! assert(B(:,4), w(4)*aux(2,:).', 1e-6) %!test %! p = 2; q = 3; m = 4; n = 5; %! Lx = 1; Ly = 1; %! nrb = nrb4surf ([0 0], [1 0], [0 1], [1 1]); %! nrb = nrbdegelev (nrb, [p-1, q-1]); %! aux1 = linspace(0,1,m); aux2 = linspace(0,1,n); %! nrb = nrbkntins (nrb, {aux1(2:end-1), aux2(2:end-1)}); %! u = rand (1, 30); v = rand (1, 10); %! u = u - min (u); u = u / max (u); %! v = v - min (v); v = v / max (v); %! [B, N] = nrbbasisfun ({u, v}, nrb); %! assert (sum(B, 2), ones(300, 1), 1e-6) %! assert (all (all (B<=1)), true) %! assert (all (all (B>=0)), true) %! assert (all (all (N>0)), true) %! assert (all (all (N <= prod (nrb.number))), true) %! assert (max (max (N)),prod (nrb.number)) %! assert (min (min (N)),1) %!test %! p1 = 2; p2 = 3; p3 = 2; %! n1 = 4; n2 = 5; n3 = 4; %! Lx = 1; Ly = 1; Lz = 1; %! crv = nrbline([1 0], [2 0]); %! nrb = nrbtransp (nrbrevolve (crv, [], [0 0 1], pi/2)); %! nrb = nrbextrude (nrb, [0 0 1]); %! nrb = nrbdegelev (nrb, [p1-1, p2-2, p3-1]); %! aux1 = linspace(0,1,n1); aux2 = linspace(0,1,n2); aux3 = linspace(0,1,n3); %! nrb = nrbkntins (nrb, {aux1(2:end-1), aux2(2:end-1), aux3(2:end-1)}); %! %! u = rand (1, 12); v = rand (1, 10); w = rand (1, 15); %! u = u - min (u); u = u / max (u); %! v = v - min (v); v = v / max (v); %! w = w - min (w); w = w / max (w); %! [B, N] = nrbbasisfun ({u, v, w}, nrb); %! assert (all(sum(B, 2) - ones(numel(u)*numel(v)*numel(w),1) < 1e-6)) %! assert (all (all (B <= 1)) == true) %! assert (all (all (B >= 0)) == true) %! assert (all (all (N > 0)) == true) %! assert (all (all (N <= prod (nrb.number))) == true) %! assert (max (max (N)) == prod (nrb.number)) %! assert (min (min (N))== 1)nurbs-1.4.3/inst/PaxHeaders.16248/tbasisfun.m0000644000000000000000000000013214030323741015537 xustar0030 mtime=1617012705.807919504 30 atime=1617012705.807919504 30 ctime=1617012705.823919749 nurbs-1.4.3/inst/tbasisfun.m0000644000175000017500000000761114030323741017643 0ustar00rvazquezrvazquez00000000000000function [N, Nder] = tbasisfun (u, p, U) % % TBASISFUN: Compute a B- or T-Spline basis function, and its derivatives, from its local knot vector. % % usage: % % [N, Nder] = tbasisfun (u, p, U) % [N, Nder] = tbasisfun ([u; v], [p q], {U, V}) % [N, Nder] = tbasisfun ([u; v; w], [p q r], {U, V, W}) % % INPUT: % % u or [u; v] : points in parameter space where the basis function is to be % evaluated % % U or {U, V} : local knot vector % % p or [p q] : polynomial degree of the basis function % % OUTPUT: % % N : basis function evaluated at the given parametric points % Nder : basis function gradient evaluated at the given parametric points % % Copyright (C) 2009 Carlo de Falco % Copyright (C) 2012 Rafael Vazquez % % This program is free software: you can redistribute it and/or modify % it under the terms of the GNU General Public License as published by % the Free Software Foundation, either version 3 of the License, or % (at your option) any later version. % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this program. If not, see . if (~ iscell (U)) U = sort (U); if (numel (U) ~= p+2) error ('tbasisfun: knot vector and degree do not correspond') end if (nargout == 1) N = onebasisfun__ (u, p, U); else [N, Nder] = onebasisfunder__ (u, p, U); end elseif (size(U,2) == 2) U{1} = sort(U{1}); U{2} = sort(U{2}); if (numel(U{1}) ~= p(1)+2 || numel(U{2}) ~= p(2)+2) error ('tbasisfun: knot vector and degree do not correspond') end if (nargout == 1) Nu = onebasisfun__ (u(1,:), p(1), U{1}); Nv = onebasisfun__ (u(2,:), p(2), U{2}); N = Nu.*Nv; elseif (nargout == 2) [Nu, Ndu] = onebasisfunder__ (u(1,:), p(1), U{1}); [Nv, Ndv] = onebasisfunder__ (u(2,:), p(2), U{2}); N = Nu.*Nv; Nder(1,:) = Ndu.*Nv; Nder(2,:) = Nu.*Ndv; end elseif (size(U,2) == 3) U{1} = sort(U{1}); U{2} = sort(U{2}); U{3} = sort(U{3}); if (numel(U{1}) ~= p(1)+2 || numel(U{2}) ~= p(2)+2 || numel(U{3}) ~= p(3)+2) error ('tbasisfun: knot vector and degree do not correspond') end if (nargout == 1) Nu = onebasisfun__ (u(1,:), p(1), U{1}); Nv = onebasisfun__ (u(2,:), p(2), U{2}); Nw = onebasisfun__ (u(3,:), p(3), U{3}); N = Nu.*Nv.*Nw; else [Nu, Ndu] = onebasisfunder__ (u(1,:), p(1), U{1}); [Nv, Ndv] = onebasisfunder__ (u(2,:), p(2), U{2}); [Nw, Ndw] = onebasisfunder__ (u(3,:), p(3), U{3}); N = Nu.*Nv.*Nw; Nder(1,:) = Ndu.*Nv.*Nw; Nder(2,:) = Nu.*Ndv.*Nw; Nder(3,:) = Nu.*Nv.*Ndw; end end end %!demo %! U = {[0 0 1/2 1 1], [0 0 0 1 1]}; %! p = [3, 3]; %! [X, Y] = meshgrid (linspace(0, 1, 30)); %! u = [X(:), Y(:)]'; %! N = tbasisfun (u, p, U); %! surf (X, Y, reshape (N, size(X))) %! title('Basis function associated to a local knot vector') %! hold off %!test %! U = [0 1/2 1]; %! p = 1; %! u = [0.3 0.4 0.6 0.7]; %! [N, Nder] = tbasisfun (u, p, U); %! assert (N, [0.6 0.8 0.8 0.6], 1e-12); %! assert (Nder, [2 2 -2 -2], 1e-12); %!test %! U = {[0 1/2 1] [0 1/2 1]}; %! p = [1 1]; %! u = [0.3 0.4 0.6 0.7; 0.3 0.4 0.6 0.7]; %! [N, Nder] = tbasisfun (u, p, U); %! assert (N, [0.36 0.64 0.64 0.36], 1e-12); %! assert (Nder, [1.2 1.6 -1.6 -1.2; 1.2 1.6 -1.6 -1.2], 1e-12); %!test %! U = {[0 1/2 1] [0 1/2 1] [0 1/2 1]}; %! p = [1 1 1]; %! u = [0.4 0.4 0.6 0.6; 0.4 0.4 0.6 0.6; 0.4 0.6 0.4 0.6]; %! [N, Nder] = tbasisfun (u, p, U); %! assert (N, [0.512 0.512 0.512 0.512], 1e-12); %! assert (Nder, [1.28 1.28 -1.28 -1.28; 1.28 1.28 -1.28 -1.28; 1.28 -1.28 1.28 -1.28], 1e-12); nurbs-1.4.3/PaxHeaders.16248/NEWS0000644000000000000000000000013214030323741013105 xustar0030 mtime=1617012705.779919075 30 atime=1617012705.779919075 30 ctime=1617012705.823919749 nurbs-1.4.3/NEWS0000644000175000017500000001102414030323741015202 0ustar00rvazquezrvazquez00000000000000Summary of important user-visible changes for nurbs 1.4.3: ------------------------------------------------------------------- * inst/nrbctrlplot: allow the number of points for the plot as input * inst/nrbkntplot: allow the number of points for the plot as input * inst/nrbmak: added consistency check Summary of important user-visible changes for nurbs 1.4.2: ------------------------------------------------------------------- * remove use of deprecated functionality in oct-files Summary of important user-visible changes for nurbs 1.4.1: ------------------------------------------------------------------- * inst/nrbglue: added new function * inst/nrbinverse: add a way to recognize non-convergence Summary of important user-visible changes for nurbs 1.4.0: ------------------------------------------------------------------- * inst/basiskntins: return the identity when there is no insertion * inst/nrbderiv: 3rd and 4th order derivatives * inst/nrbdeval: 3rd and 4th order derivatives * inst/nrbruled: extended to trivariates * inst/nrbmak: added the possibility to normalize the knot vector * inst/vecnormalize: renamed the old function vecnorm Summary of important user-visible changes for nurbs 1.3.14: ------------------------------------------------------------------- * inst/nrbextract: possibility to extract a list of boundary sides * inst/nrbmultipatch: added tolerance as an optional argument * inst/nrbspheretiling: added new function * inst/nrbspheretile: added new function Summary of important user-visible changes for nurbs 1.3.13: ------------------------------------------------------------------- * inst/aveknt.m: added new function * inst/nrbclamp.: added new function * inst/nrbmodp.m: added new function * inst/nrbmodw.m: added new function * inst/nrbeval_der_p.m: added new function * inst/nrbeval_der_w.m: added new function * inst/nrbsquare.m: added new function * inst/bspinterpcrv.m: added new function * inst/bspinterpsurf.m: added new function * inst/nrbinverse.m: added new function * inst/nrbbasisfun.m: faster version for cell-arrays. Working version for volumes * inst/nrbbasisfunder.m: faster version for cell-arrays. Working version for volumes * inst/nrbnumbasisfun.m: faster version for cell-arrays. Working version for volumes. Now using 0-based indexing for cuves (different from basisfun). Summary of important user-visible changes for nurbs 1.3.12: ------------------------------------------------------------------- * nrbmultipatch: check if two faces match more accurately Summary of important user-visible changes for nurbs 1.3.11: ------------------------------------------------------------------- * nrbextract, nrbmultipatch: generalized for curves * inst/deg2rad, inst/rad2deg: removed functions from the package Summary of important user-visible changes for nurbs 1.3.10: ------------------------------------------------------------------- ** 1.3.10 is being released to allow compatibility with Octave 4.0 * src/low_level_functions.cc(findspan): return an error if a point is outside the knotspan * inst/nrbexport: changed the format of geopdes files * inst/nrbbasisfun: fixed bug to return indices in the 1D case * inst/basiskntins: added new function to compute subdivision coefficients Summary of important user-visible changes for nurbs 1.3.9: ------------------------------------------------------------------- * inst/nrbexport: export multipatch geometries Summary of important user-visible changes for nurbs 1.3.8: ------------------------------------------------------------------- * inst/nrbkntremove.m: added new function * inst/nrbunclamp.m: added new function * inst/nrbmak.m: adapted for unclamped knot vector * inst/nrbplot.m: adapted for unclamped knot vector * inst/nrbkntplot.m: adapted for unclamped knot vector * inst/nrb2iges: added new function * inst/nrbmultipatch: added new function Summary of important user-visible changes for nurbs 1.3.7: ------------------------------------------------------------------- ** 1.3.7 is mainly a maintainance release to distribute small bug fixes that accumulated over time. * inst/nrbpermute.m: added new function * inst/nrbreverse.m: each direction can now be reversed independently * inst/nrbkntplot.m: now works for non-unitary interval * inst/nrbcrvderiveval: code vectorized * inst/nrbplot: fixed bug in affecting plot of trivariate nurbs, now works for non-unitary intervals * inst/nrbctrplot.m: plot the points only once * src/*.cc: avoid use of deprecated array constructors