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* Symmetric Power Basis - Bernstein Basis conversion routines
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* Marco Cecchetti <mrcekets at gmail.com>
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* Nathan Hurst <njh@mail.csse.monash.edu.au>
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* Copyright 2007-2008 authors
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* This library is free software; you can redistribute it and/or
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* modify it either under the terms of the GNU Lesser General Public
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* License version 2.1 as published by the Free Software Foundation
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* (the "LGPL") or, at your option, under the terms of the Mozilla
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* Public License Version 1.1 (the "MPL"). If you do not alter this
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* notice, a recipient may use your version of this file under either
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* the MPL or the LGPL.
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* You should have received a copy of the LGPL along with this library
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* in the file COPYING-LGPL-2.1; if not, write to the Free Software
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* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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* You should have received a copy of the MPL along with this library
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* in the file COPYING-MPL-1.1
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* The contents of this file are subject to the Mozilla Public License
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* Version 1.1 (the "License"); you may not use this file except in
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* compliance with the License. You may obtain a copy of the License at
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* http://www.mozilla.org/MPL/
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* This software is distributed on an "AS IS" basis, WITHOUT WARRANTY
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* OF ANY KIND, either express or implied. See the LGPL or the MPL for
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* the specific language governing rights and limitations.
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#include <2geom/sbasis-to-bezier.h>
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#include <2geom/choose.h>
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#include <2geom/svg-path.h>
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#include <2geom/exception.h>
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* Symmetric Power Basis - Bernstein Basis conversion routines
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* some remark about precision:
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* interval [0,1], subdivisions: 10^3
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* - bezier_to_sbasis : up to degree ~72 precision is at least 10^-5
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* up to degree ~87 precision is at least 10^-3
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* - sbasis_to_bezier : up to order ~63 precision is at least 10^-15
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* precision is at least 10^-14 even beyond order 200
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* interval [-1,1], subdivisions: 10^3
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* - bezier_to_sbasis : up to degree ~21 precision is at least 10^-5
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* up to degree ~24 precision is at least 10^-3
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* - sbasis_to_bezier : up to order ~11 precision is at least 10^-5
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* up to order ~13 precision is at least 10^-3
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* interval [-10,10], subdivisions: 10^3
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* - bezier_to_sbasis : up to degree ~7 precision is at least 10^-5
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* up to degree ~8 precision is at least 10^-3
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* - sbasis_to_bezier : up to order ~3 precision is at least 10^-5
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* up to order ~4 precision is at least 10^-3
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* this implementation is based on the following article:
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* J.Sanchez-Reyes - The Symmetric Analogue of the Polynomial Power Basis
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double binomial(unsigned int n, unsigned int k)
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return choose<double>(n, k);
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int sgn(unsigned int j, unsigned int k)
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// we are sure that j >= k
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return ((j-k) & 1u) ? -1 : 1;
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/** Changes the basis of p to be bernstein.
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\param p the Symmetric basis polynomial
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\returns the Bernstein basis polynomial
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if the degree is even q is the order in the symmetrical power basis,
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if the degree is odd q is the order + 1
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n is always the polynomial degree, i. e. the Bezier order
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sz is the number of bezier handles.
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void sbasis_to_bezier (Bezier & bz, SBasis const& sb, size_t sz)
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if (sb[q-1][0] == sb[q-1][1])
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q = (sz > 2*sb.size()-1) ? sb.size() : (sz+1)/2;
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for (size_t k = 0; k < q; ++k)
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for (size_t j = k; j < n-k; ++j) // j <= n-k-1
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Tjk = binomial(n-2*k-1, j-k);
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bz[j] += (Tjk * sb[k][0]);
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bz[n-j] += (Tjk * sb[k][1]); // n-k <-> [k][1]
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// the resulting coefficients are with respect to the scaled Bernstein
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// basis so we need to divide them by (n, j) binomial coefficient
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for (size_t j = 1; j < n; ++j)
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bz[j] /= binomial(n, j);
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/** Changes the basis of p to be Bernstein.
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\param p the D2 Symmetric basis polynomial
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\returns the D2 Bernstein basis polynomial
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sz is always the polynomial degree, i. e. the Bezier order
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void sbasis_to_bezier (std::vector<Point> & bz, D2<SBasis> const& sb, size_t sz)
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sz = std::max(sb[X].size(), sb[Y].size())*2;
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sbasis_to_bezier(bzx, sb[X], sz);
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sbasis_to_bezier(bzy, sb[Y], sz);
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assert(bzx.size() == bzy.size());
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size_t n = (bzx.size() >= bzy.size()) ? bzx.size() : bzy.size();
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bz.resize(n, Point(0,0));
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for (size_t i = 0; i < bzx.size(); ++i)
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for (size_t i = 0; i < bzy.size(); ++i)
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/** Changes the basis of p to be sbasis.
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\param p the Bernstein basis polynomial
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\returns the Symmetric basis polynomial
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if the degree is even q is the order in the symmetrical power basis,
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if the degree is odd q is the order + 1
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n is always the polynomial degree, i. e. the Bezier order
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void bezier_to_sbasis (SBasis & sb, Bezier const& bz)
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size_t n = bz.order();
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size_t q = (n+1) / 2;
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size_t even = (n & 1u) ? 0 : 1;
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sb.resize(q + even, Linear(0, 0));
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for (size_t k = 0; k < q; ++k)
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for (size_t j = k; j < q; ++j)
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Tjk = sgn(j, k) * binomial(n-j-k, j-k) * binomial(n, k);
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sb[j][0] += (Tjk * bz[k]);
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sb[j][1] += (Tjk * bz[n-k]); // n-j <-> [j][1]
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for (size_t j = k+1; j < q; ++j)
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Tjk = sgn(j, k) * binomial(n-j-k-1, j-k-1) * binomial(n, k);
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sb[j][0] += (Tjk * bz[n-k]);
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sb[j][1] += (Tjk * bz[k]); // n-j <-> [j][1]
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for (size_t k = 0; k < q; ++k)
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Tjk = sgn(q,k) * binomial(n, k);
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sb[q][0] += (Tjk * (bz[k] + bz[n-k]));
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sb[q][0] += (binomial(n, q) * bz[q]);
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/** Changes the basis of d2 p to be sbasis.
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\param p the d2 Bernstein basis polynomial
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\returns the d2 Symmetric basis polynomial
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if the degree is even q is the order in the symmetrical power basis,
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if the degree is odd q is the order + 1
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n is always the polynomial degree, i. e. the Bezier order
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void bezier_to_sbasis (D2<SBasis> & sb, std::vector<Point> const& bz)
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size_t n = bz.size() - 1;
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size_t q = (n+1) / 2;
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size_t even = (n & 1u) ? 0 : 1;
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sb[X].resize(q + even, Linear(0, 0));
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sb[Y].resize(q + even, Linear(0, 0));
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for (size_t k = 0; k < q; ++k)
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for (size_t j = k; j < q; ++j)
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Tjk = sgn(j, k) * binomial(n-j-k, j-k) * binomial(n, k);
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sb[X][j][0] += (Tjk * bz[k][X]);
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sb[X][j][1] += (Tjk * bz[n-k][X]);
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sb[Y][j][0] += (Tjk * bz[k][Y]);
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sb[Y][j][1] += (Tjk * bz[n-k][Y]);
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for (size_t j = k+1; j < q; ++j)
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Tjk = sgn(j, k) * binomial(n-j-k-1, j-k-1) * binomial(n, k);
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sb[X][j][0] += (Tjk * bz[n-k][X]);
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sb[X][j][1] += (Tjk * bz[k][X]);
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sb[Y][j][0] += (Tjk * bz[n-k][Y]);
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sb[Y][j][1] += (Tjk * bz[k][Y]);
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for (size_t k = 0; k < q; ++k)
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Tjk = sgn(q,k) * binomial(n, k);
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sb[X][q][0] += (Tjk * (bz[k][X] + bz[n-k][X]));
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sb[Y][q][0] += (Tjk * (bz[k][Y] + bz[n-k][Y]));
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sb[X][q][0] += (binomial(n, q) * bz[q][X]);
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sb[X][q][1] = sb[X][q][0];
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sb[Y][q][0] += (binomial(n, q) * bz[q][Y]);
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sb[Y][q][1] = sb[Y][q][0];
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sb[X][0][0] = bz[0][X];
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sb[X][0][1] = bz[n][X];
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sb[Y][0][0] = bz[0][Y];
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sb[Y][0][1] = bz[n][Y];
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} // end namespace Geom
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* This version works by inverting a reasonable upper bound on the error term after subdividing the
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* curve at $a$. We keep biting off pieces until there is no more curve left.
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* Derivation: The tail of the power series is $a_ks^k + a_{k+1}s^{k+1} + \ldots = e$. A
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* subdivision at $a$ results in a tail error of $e*A^k, A = (1-a)a$. Let this be the desired
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* tolerance tol $= e*A^k$ and invert getting $A = e^{1/k}$ and $a = 1/2 - \sqrt{1/4 - A}$
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subpath_from_sbasis_incremental(Geom::OldPathSetBuilder &pb, D2<SBasis> B, double tol, bool initial) {
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const unsigned k = 2; // cubic bezier
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double te = B.tail_error(k);
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assert(B[0].IS_FINITE());
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assert(B[1].IS_FINITE());
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//std::cout << "tol = " << tol << std::endl;
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double A = std::sqrt(tol/te); // pow(te, 1./k)
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A = std::min(A, 0.25);
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a = 0.5 - std::sqrt(0.25 - A); // quadratic formula
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if(a > 1) a = 1; // clamp to the end of the segment
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//std::cout << "te = " << te << std::endl;
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//std::cout << "A = " << A << "; a=" << a << std::endl;
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D2<SBasis> Bs = compose(B, Linear(0, a));
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assert(Bs.tail_error(k));
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std::vector<Geom::Point> bez = sbasis_to_bezier(Bs, 2);
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reverse(bez.begin(), bez.end());
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pb.start_subpath(bez[0]);
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pb.push_cubic(bez[1], bez[2], bez[3]);
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// move to next piece of curve
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B = compose(B, Linear(a, 1));
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te = B.tail_error(k);
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/** Make a path from a d2 sbasis.
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\param p the d2 Symmetric basis polynomial
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If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
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void build_from_sbasis(Geom::PathBuilder &pb, D2<SBasis> const &B, double tol, bool only_cubicbeziers) {
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THROW_EXCEPTION("assertion failed: B.isFinite()");
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if(tail_error(B, 2) < tol || sbasis_size(B) == 2) { // nearly cubic enough
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if( !only_cubicbeziers && (sbasis_size(B) <= 1) ) {
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std::vector<Geom::Point> bez;
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sbasis_to_bezier(bez, B, 4);
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pb.curveTo(bez[1], bez[2], bez[3]);
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build_from_sbasis(pb, compose(B, Linear(0, 0.5)), tol, only_cubicbeziers);
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build_from_sbasis(pb, compose(B, Linear(0.5, 1)), tol, only_cubicbeziers);
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/** Make a path from a d2 sbasis.
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\param p the d2 Symmetric basis polynomial
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If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
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path_from_sbasis(D2<SBasis> const &B, double tol, bool only_cubicbeziers) {
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build_from_sbasis(pb, B, tol, only_cubicbeziers);
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return pb.peek().front();
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/** Make a path from a d2 sbasis.
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\param p the d2 Symmetric basis polynomial
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If only_cubicbeziers is true, the resulting path may only contain CubicBezier curves.
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TODO: some of this logic should be lifted into svg-path
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std::vector<Geom::Path>
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path_from_piecewise(Geom::Piecewise<Geom::D2<Geom::SBasis> > const &B, double tol, bool only_cubicbeziers) {
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Geom::PathBuilder pb;
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if(B.size() == 0) return pb.peek();
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Geom::Point start = B[0].at0();
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for(unsigned i = 0; ; i++) {
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if(i+1 == B.size() || !are_near(B[i+1].at0(), B[i].at1(), tol)) {
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//start of a new path
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if(are_near(start, B[i].at1()) && sbasis_size(B[i]) <= 1) {
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//last line seg already there (because of .closePath())
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build_from_sbasis(pb, B[i], tol, only_cubicbeziers);
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if(are_near(start, B[i].at1())) {
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//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.
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if(i+1 >= B.size()) break;
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start = B[i+1].at0();
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build_from_sbasis(pb, B[i], tol, only_cubicbeziers);
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c-file-style:"stroustrup"
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c-file-offsets:((innamespace . 0)(inline-open . 0)(case-label . +))
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// vim: filetype=cpp:expandtab:shiftwidth=4:tabstop=8:softtabstop=4:encoding=utf-8:textwidth=99 :
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inkscape-0.47pre1/src/2geom/sbasis-to-bezier.cpp
