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Diffstat (limited to 'src/2geom/sbasis-to-bezier.cpp')
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diff --git a/src/2geom/sbasis-to-bezier.cpp b/src/2geom/sbasis-to-bezier.cpp new file mode 100644 index 0000000..5580956 --- /dev/null +++ b/src/2geom/sbasis-to-bezier.cpp @@ -0,0 +1,584 @@ +/* + * Symmetric Power Basis - Bernstein Basis conversion routines + * + * Authors: + * Marco Cecchetti <mrcekets at gmail.com> + * Nathan Hurst <njh@mail.csse.monash.edu.au> + * + * Copyright 2007-2008 authors + * + * This library is free software; you can redistribute it and/or + * modify it either under the terms of the GNU Lesser General Public + * License version 2.1 as published by the Free Software Foundation + * (the "LGPL") or, at your option, under the terms of the Mozilla + * Public License Version 1.1 (the "MPL"). If you do not alter this + * notice, a recipient may use your version of this file under either + * the MPL or the LGPL. + * + * You should have received a copy of the LGPL along with this library + * in the file COPYING-LGPL-2.1; if not, write to the Free Software + * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * You should have received a copy of the MPL along with this library + * in the file COPYING-MPL-1.1 + * + * The contents of this file are subject to the Mozilla Public License + * Version 1.1 (the "License"); you may not use this file except in + * compliance with the License. You may obtain a copy of the License at + * http://www.mozilla.org/MPL/ + * + * This software is distributed on an "AS IS" basis, WITHOUT WARRANTY + * OF ANY KIND, either express or implied. See the LGPL or the MPL for + * the specific language governing rights and limitations. + */ + + +#include <2geom/sbasis-to-bezier.h> +#include <2geom/d2.h> +#include <2geom/choose.h> +#include <2geom/path-sink.h> +#include <2geom/exception.h> +#include <2geom/convex-hull.h> + +#include <iostream> + + + + +namespace Geom +{ + +/* + * Symmetric Power Basis - Bernstein Basis conversion routines + * + * some remark about precision: + * interval [0,1], subdivisions: 10^3 + * - bezier_to_sbasis : up to degree ~72 precision is at least 10^-5 + * up to degree ~87 precision is at least 10^-3 + * - sbasis_to_bezier : up to order ~63 precision is at least 10^-15 + * precision is at least 10^-14 even beyond order 200 + * + * interval [-1,1], subdivisions: 10^3 + * - bezier_to_sbasis : up to degree ~21 precision is at least 10^-5 + * up to degree ~24 precision is at least 10^-3 + * - sbasis_to_bezier : up to order ~11 precision is at least 10^-5 + * up to order ~13 precision is at least 10^-3 + * + * interval [-10,10], subdivisions: 10^3 + * - bezier_to_sbasis : up to degree ~7 precision is at least 10^-5 + * up to degree ~8 precision is at least 10^-3 + * - sbasis_to_bezier : up to order ~3 precision is at least 10^-5 + * up to order ~4 precision is at least 10^-3 + * + * references: + * this implementation is based on the following article: + * J.Sanchez-Reyes - The Symmetric Analogue of the Polynomial Power Basis + */ + +/** Changes the basis of p to be bernstein. + \param p the Symmetric basis polynomial + \returns the Bernstein basis polynomial + + if the degree is even q is the order in the symmetrical power basis, + if the degree is odd q is the order + 1 + n is always the polynomial degree, i. e. the Bezier order + sz is the number of bezier handles. +*/ +void sbasis_to_bezier (Bezier & bz, SBasis const& sb, size_t sz) +{ + assert(sb.size() > 0); + + size_t q, n; + bool even; + if (sz == 0) + { + q = sb.size(); + if (sb[q-1][0] == sb[q-1][1]) + { + even = true; + --q; + n = 2*q; + } + else + { + even = false; + n = 2*q-1; + } + } + else + { + q = (sz > 2*sb.size()-1) ? sb.size() : (sz+1)/2; + n = sz-1; + even = false; + } + bz.clear(); + bz.resize(n+1); + for (size_t k = 0; k < q; ++k) + { + int Tjk = 1; + for (size_t j = k; j < n-k; ++j) // j <= n-k-1 + { + bz[j] += (Tjk * sb[k][0]); + bz[n-j] += (Tjk * sb[k][1]); // n-k <-> [k][1] + // assert(Tjk == binomial(n-2*k-1, j-k)); + binomial_increment_k(Tjk, n-2*k-1, j-k); + } + } + if (even) + { + bz[q] += sb[q][0]; + } + // the resulting coefficients are with respect to the scaled Bernstein + // basis so we need to divide them by (n, j) binomial coefficient + int bcj = n; + for (size_t j = 1; j < n; ++j) + { + bz[j] /= bcj; + // assert(bcj == binomial(n, j)); + binomial_increment_k(bcj, n, j); + } + bz[0] = sb[0][0]; + bz[n] = sb[0][1]; +} + +void sbasis_to_bezier(D2<Bezier> &bz, D2<SBasis> const &sb, size_t sz) +{ + if (sz == 0) { + sz = std::max(sb[X].size(), sb[Y].size())*2; + } + sbasis_to_bezier(bz[X], sb[X], sz); + sbasis_to_bezier(bz[Y], sb[Y], sz); +} + +/** Changes the basis of p to be Bernstein. + \param p the D2 Symmetric basis polynomial + \returns the D2 Bernstein basis polynomial + + sz is always the polynomial degree, i. e. the Bezier order +*/ +void sbasis_to_bezier (std::vector<Point> & bz, D2<SBasis> const& sb, size_t sz) +{ + D2<Bezier> bez; + sbasis_to_bezier(bez, sb, sz); + bz = bezier_points(bez); +} + +/** Changes the basis of p to be Bernstein. + \param p the D2 Symmetric basis polynomial + \returns the D2 Bernstein basis cubic polynomial + +Bezier is always cubic. +For general asymmetric case, fit the SBasis function value at midpoint +For parallel, symmetric case, find the point of closest approach to the midpoint +For parallel, anti-symmetric case, fit the SBasis slope at midpoint +*/ +void sbasis_to_cubic_bezier (std::vector<Point> & bz, D2<SBasis> const& sb) +{ + double delx[2], dely[2]; + double xprime[2], yprime[2]; + double midx = 0; + double midy = 0; + double midx_0, midy_0; + double numer[2], numer_0[2]; + double denom; + double div; + + if ((sb[X].size() == 0) || (sb[Y].size() == 0)) { + THROW_RANGEERROR("size of sb is too small"); + } + + sbasis_to_bezier(bz, sb, 4); // zeroth-order estimate + if ((sb[X].size() < 3) && (sb[Y].size() < 3)) + return; // cubic bezier estimate is exact + Geom::ConvexHull bezhull(bz); + +// calculate first derivatives of x and y wrt t + + for (int i = 0; i < 2; ++i) { + xprime[i] = sb[X][0][1] - sb[X][0][0]; + yprime[i] = sb[Y][0][1] - sb[Y][0][0]; + } + if (sb[X].size() > 1) { + xprime[0] += sb[X][1][0]; + xprime[1] -= sb[X][1][1]; + } + if (sb[Y].size() > 1) { + yprime[0] += sb[Y][1][0]; + yprime[1] -= sb[Y][1][1]; + } + +// calculate midpoint at t = 0.5 + + div = 2; + for (auto i : sb[X]) { + midx += (i[0] + i[1])/div; + div *= 4; + } + + div = 2; + for (auto i : sb[Y]) { + midy += (i[0] + i[1])/div; + div *= 4; + } + +// is midpoint in hull: if not, the solution will be ill-conditioned, LP Bug 1428683 + + if (!bezhull.contains(Geom::Point(midx, midy))) + return; + +// calculate Bezier control arms + + midx = 8*midx - 4*bz[0][X] - 4*bz[3][X]; // re-define relative to center + midy = 8*midy - 4*bz[0][Y] - 4*bz[3][Y]; + midx_0 = sb[X].size() > 1 ? sb[X][1][0] + sb[X][1][1] : 0; // zeroth order estimate + midy_0 = sb[Y].size() > 1 ? sb[Y][1][0] + sb[Y][1][1] : 0; + + if ((std::abs(xprime[0]) < EPSILON) && (std::abs(yprime[0]) < EPSILON) + && ((std::abs(xprime[1]) > EPSILON) || (std::abs(yprime[1]) > EPSILON))) { // degenerate handle at 0 : use distance of closest approach + numer[0] = midx*xprime[1] + midy*yprime[1]; + denom = 3.0*(xprime[1]*xprime[1] + yprime[1]*yprime[1]); + delx[0] = 0; + dely[0] = 0; + delx[1] = -xprime[1]*numer[0]/denom; + dely[1] = -yprime[1]*numer[0]/denom; + } else if ((std::abs(xprime[1]) < EPSILON) && (std::abs(yprime[1]) < EPSILON) + && ((std::abs(xprime[0]) > EPSILON) || (std::abs(yprime[0]) > EPSILON))) { // degenerate handle at 1 : ditto + numer[1] = midx*xprime[0] + midy*yprime[0]; + denom = 3.0*(xprime[0]*xprime[0] + yprime[0]*yprime[0]); + delx[0] = xprime[0]*numer[1]/denom; + dely[0] = yprime[0]*numer[1]/denom; + delx[1] = 0; + dely[1] = 0; + } else if (std::abs(xprime[1]*yprime[0] - yprime[1]*xprime[0]) > // general case : fit mid fxn value + 0.002 * std::abs(xprime[1]*xprime[0] + yprime[1]*yprime[0])) { // approx. 0.1 degree of angle + double test1 = (bz[1][Y] - bz[0][Y])*(bz[3][X] - bz[0][X]) - (bz[1][X] - bz[0][X])*(bz[3][Y] - bz[0][Y]); + double test2 = (bz[2][Y] - bz[0][Y])*(bz[3][X] - bz[0][X]) - (bz[2][X] - bz[0][X])*(bz[3][Y] - bz[0][Y]); + if (test1*test2 < 0) // reject anti-symmetric case, LP Bug 1428267 & Bug 1428683 + return; + denom = 3.0*(xprime[1]*yprime[0] - yprime[1]*xprime[0]); + for (int i = 0; i < 2; ++i) { + numer_0[i] = xprime[1 - i]*midy_0 - yprime[1 - i]*midx_0; + numer[i] = xprime[1 - i]*midy - yprime[1 - i]*midx; + delx[i] = xprime[i]*numer[i]/denom; + dely[i] = yprime[i]*numer[i]/denom; + if (numer_0[i]*numer[i] < 0) // check for reversal of direction, LP Bug 1544680 + return; + } + if (std::abs((numer[0] - numer_0[0])*numer_0[1]) > 10.0*std::abs((numer[1] - numer_0[1])*numer_0[0]) // check for asymmetry + || std::abs((numer[1] - numer_0[1])*numer_0[0]) > 10.0*std::abs((numer[0] - numer_0[0])*numer_0[1])) + return; + } else if ((xprime[0]*xprime[1] < 0) || (yprime[0]*yprime[1] < 0)) { // symmetric case : use distance of closest approach + numer[0] = midx*xprime[0] + midy*yprime[0]; + denom = 6.0*(xprime[0]*xprime[0] + yprime[0]*yprime[0]); + delx[0] = xprime[0]*numer[0]/denom; + dely[0] = yprime[0]*numer[0]/denom; + delx[1] = -delx[0]; + dely[1] = -dely[0]; + } else { // anti-symmetric case : fit mid slope + // calculate slope at t = 0.5 + midx = 0; + div = 1; + for (auto i : sb[X]) { + midx += (i[1] - i[0])/div; + div *= 4; + } + midy = 0; + div = 1; + for (auto i : sb[Y]) { + midy += (i[1] - i[0])/div; + div *= 4; + } + if (midx*yprime[0] != midy*xprime[0]) { + denom = midx*yprime[0] - midy*xprime[0]; + numer[0] = midx*(bz[3][Y] - bz[0][Y]) - midy*(bz[3][X] - bz[0][X]); + for (int i = 0; i < 2; ++i) { + delx[i] = xprime[0]*numer[0]/denom; + dely[i] = yprime[0]*numer[0]/denom; + } + } else { // linear case + for (int i = 0; i < 2; ++i) { + delx[i] = (bz[3][X] - bz[0][X])/3; + dely[i] = (bz[3][Y] - bz[0][Y])/3; + } + } + } + bz[1][X] = bz[0][X] + delx[0]; + bz[1][Y] = bz[0][Y] + dely[0]; + bz[2][X] = bz[3][X] - delx[1]; + bz[2][Y] = bz[3][Y] - dely[1]; +} + +/** Changes the basis of p to be sbasis. + \param p the Bernstein basis polynomial + \returns the Symmetric basis polynomial + + if the degree is even q is the order in the symmetrical power basis, + if the degree is odd q is the order + 1 + n is always the polynomial degree, i. e. the Bezier order +*/ +void bezier_to_sbasis (SBasis & sb, Bezier const& bz) +{ + size_t n = bz.order(); + size_t q = (n+1) / 2; + size_t even = (n & 1u) ? 0 : 1; + sb.clear(); + sb.resize(q + even, Linear(0, 0)); + int nck = 1; + for (size_t k = 0; k < q; ++k) + { + int Tjk = nck; + for (size_t j = k; j < q; ++j) + { + sb[j][0] += (Tjk * bz[k]); + sb[j][1] += (Tjk * bz[n-k]); // n-j <-> [j][1] + // assert(Tjk == sgn(j, k) * binomial(n-j-k, j-k) * binomial(n, k)); + binomial_increment_k(Tjk, n-j-k, j-k); + binomial_decrement_n(Tjk, n-j-k, j-k+1); + Tjk = -Tjk; + } + Tjk = -nck; + for (size_t j = k+1; j < q; ++j) + { + sb[j][0] += (Tjk * bz[n-k]); + sb[j][1] += (Tjk * bz[k]); // n-j <-> [j][1] + // assert(Tjk == sgn(j, k) * binomial(n-j-k-1, j-k-1) * binomial(n, k)); + binomial_increment_k(Tjk, n-j-k-1, j-k-1); + binomial_decrement_n(Tjk, n-j-k-1, j-k); + Tjk = -Tjk; + } + // assert(nck == binomial(n, k)); + binomial_increment_k(nck, n, k); + } + if (even) + { + int Tjk = q & 1 ? -1 : 1; + for (size_t k = 0; k < q; ++k) + { + sb[q][0] += (Tjk * (bz[k] + bz[n-k])); + // assert(Tjk == sgn(q,k) * binomial(n, k)); + binomial_increment_k(Tjk, n, k); + Tjk = -Tjk; + } + // assert(Tjk == binomial(n, q)); + sb[q][0] += Tjk * bz[q]; + sb[q][1] = sb[q][0]; + } + sb[0][0] = bz[0]; + sb[0][1] = bz[n]; +} + +/** Changes the basis of d2 p to be sbasis. + \param p the d2 Bernstein basis polynomial + \returns the d2 Symmetric basis polynomial + + if the degree is even q is the order in the symmetrical power basis, + if the degree is odd q is the order + 1 + n is always the polynomial degree, i. e. the Bezier order +*/ +void bezier_to_sbasis (D2<SBasis> & sb, std::vector<Point> const& bz) +{ + size_t n = bz.size() - 1; + size_t q = (n+1) / 2; + size_t even = (n & 1u) ? 0 : 1; + sb[X].clear(); + sb[Y].clear(); + sb[X].resize(q + even, Linear(0, 0)); + sb[Y].resize(q + even, Linear(0, 0)); + int nck = 1; + for (size_t k = 0; k < q; ++k) + { + int Tjk = nck; + for (size_t j = k; j < q; ++j) + { + sb[X][j][0] += (Tjk * bz[k][X]); + sb[X][j][1] += (Tjk * bz[n-k][X]); + sb[Y][j][0] += (Tjk * bz[k][Y]); + sb[Y][j][1] += (Tjk * bz[n-k][Y]); + // assert(Tjk == sgn(j, k) * binomial(n-j-k, j-k) * binomial(n, k)); + binomial_increment_k(Tjk, n-j-k, j-k); + binomial_decrement_n(Tjk, n-j-k, j-k+1); + Tjk = -Tjk; + } + Tjk = -nck; + for (size_t j = k+1; j < q; ++j) + { + sb[X][j][0] += (Tjk * bz[n-k][X]); + sb[X][j][1] += (Tjk * bz[k][X]); + sb[Y][j][0] += (Tjk * bz[n-k][Y]); + sb[Y][j][1] += (Tjk * bz[k][Y]); + // assert(Tjk == sgn(j, k) * binomial(n-j-k-1, j-k-1) * binomial(n, k)); + binomial_increment_k(Tjk, n-j-k-1, j-k-1); + binomial_decrement_n(Tjk, n-j-k-1, j-k); + Tjk = -Tjk; + } + // assert(nck == binomial(n, k)); + binomial_increment_k(nck, n, k); + } + if (even) + { + int Tjk = q & 1 ? -1 : 1; + for (size_t k = 0; k < q; ++k) + { + sb[X][q][0] += (Tjk * (bz[k][X] + bz[n-k][X])); + sb[Y][q][0] += (Tjk * (bz[k][Y] + bz[n-k][Y])); + // assert(Tjk == sgn(q,k) * binomial(n, k)); + binomial_increment_k(Tjk, n, k); + Tjk = -Tjk; + } + // assert(Tjk == binomial(n, q)); + sb[X][q][0] += Tjk * bz[q][X]; + sb[X][q][1] = sb[X][q][0]; + sb[Y][q][0] += Tjk * bz[q][Y]; + sb[Y][q][1] = sb[Y][q][0]; + } + sb[X][0][0] = bz[0][X]; + sb[X][0][1] = bz[n][X]; + sb[Y][0][0] = bz[0][Y]; + sb[Y][0][1] = bz[n][Y]; +} + +} // namespace Geom + +#if 0 +/* +* This version works by inverting a reasonable upper bound on the error term after subdividing the +* curve at $a$. We keep biting off pieces until there is no more curve left. +* +* Derivation: The tail of the power series is $a_ks^k + a_{k+1}s^{k+1} + \ldots = e$. A +* subdivision at $a$ results in a tail error of $e*A^k, A = (1-a)a$. Let this be the desired +* tolerance tol $= e*A^k$ and invert getting $A = e^{1/k}$ and $a = 1/2 - \sqrt{1/4 - A}$ +*/ +void +subpath_from_sbasis_incremental(Geom::OldPathSetBuilder &pb, D2<SBasis> B, double tol, bool initial) { + const unsigned k = 2; // cubic bezier + double te = B.tail_error(k); + assert(B[0].std::isfinite()); + assert(B[1].std::isfinite()); + + //std::cout << "tol = " << tol << std::endl; + while(1) { + double A = std::sqrt(tol/te); // pow(te, 1./k) + double a = A; + if(A < 1) { + A = std::min(A, 0.25); + a = 0.5 - std::sqrt(0.25 - A); // quadratic formula + if(a > 1) a = 1; // clamp to the end of the segment + } else + a = 1; + assert(a > 0); + //std::cout << "te = " << te << std::endl; + //std::cout << "A = " << A << "; a=" << a << std::endl; + D2<SBasis> Bs = compose(B, Linear(0, a)); + assert(Bs.tail_error(k)); + std::vector<Geom::Point> bez = sbasis_to_bezier(Bs, 2); + reverse(bez.begin(), bez.end()); + if (initial) { + pb.start_subpath(bez[0]); + initial = false; + } + pb.push_cubic(bez[1], bez[2], bez[3]); + +// move to next piece of curve + if(a >= 1) break; + B = compose(B, Linear(a, 1)); + te = B.tail_error(k); + } +} + +#endif + +namespace Geom{ + +/** Make a path from a d2 sbasis. + \param p the d2 Symmetric basis polynomial + \returns a Path + + If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves. +*/ +void build_from_sbasis(Geom::PathBuilder &pb, D2<SBasis> const &B, double tol, bool only_cubicbeziers) { + if (!B.isFinite()) { + THROW_EXCEPTION("assertion failed: B.isFinite()"); + } + if(tail_error(B, 3) < tol || sbasis_size(B) == 2) { // nearly cubic enough + if( !only_cubicbeziers && (sbasis_size(B) <= 1) ) { + pb.lineTo(B.at1()); + } else { + std::vector<Geom::Point> bez; +// sbasis_to_bezier(bez, B, 4); + sbasis_to_cubic_bezier(bez, B); + pb.curveTo(bez[1], bez[2], bez[3]); + } + } else { + build_from_sbasis(pb, compose(B, Linear(0, 0.5)), tol, only_cubicbeziers); + build_from_sbasis(pb, compose(B, Linear(0.5, 1)), tol, only_cubicbeziers); + } +} + +/** Make a path from a d2 sbasis. + \param p the d2 Symmetric basis polynomial + \returns a Path + + If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves. +*/ +Path +path_from_sbasis(D2<SBasis> const &B, double tol, bool only_cubicbeziers) { + PathBuilder pb; + pb.moveTo(B.at0()); + build_from_sbasis(pb, B, tol, only_cubicbeziers); + pb.flush(); + return pb.peek().front(); +} + +/** Make a path from a d2 sbasis. + \param p the d2 Symmetric basis polynomial + \returns a Path + + If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves. + TODO: some of this logic should be lifted into svg-path +*/ +PathVector +path_from_piecewise(Geom::Piecewise<Geom::D2<Geom::SBasis> > const &B, double tol, bool only_cubicbeziers) { + Geom::PathBuilder pb; + if(B.size() == 0) return pb.peek(); + Geom::Point start = B[0].at0(); + pb.moveTo(start); + for(unsigned i = 0; ; i++) { + if ( (i+1 == B.size()) + || !are_near(B[i+1].at0(), B[i].at1(), tol) ) + { + //start of a new path + if (are_near(start, B[i].at1()) && sbasis_size(B[i]) <= 1) { + pb.closePath(); + //last line seg already there (because of .closePath()) + goto no_add; + } + build_from_sbasis(pb, B[i], tol, only_cubicbeziers); + if (are_near(start, B[i].at1())) { + //it's closed, the last closing segment was not a straight line so it needed to be added, but still make it closed here with degenerate straight line. + pb.closePath(); + } + no_add: + if (i+1 >= B.size()) { + break; + } + start = B[i+1].at0(); + pb.moveTo(start); + } else { + build_from_sbasis(pb, B[i], tol, only_cubicbeziers); + } + } + pb.flush(); + return pb.peek(); +} + +} + +/* + Local Variables: + mode:c++ + c-file-style:"stroustrup" + c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +)) + indent-tabs-mode:nil + fill-column:99 + End: +*/ +// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:fileencoding=utf-8:textwidth=99 : |