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+/**
+ * @file
+ * @brief Bernstein-Bezier polynomial
+ *//*
+ * Authors:
+ * MenTaLguY <mental@rydia.net>
+ * Michael Sloan <mgsloan@gmail.com>
+ * Nathan Hurst <njh@njhurst.com>
+ * Krzysztof Kosiński <tweenk.pl@gmail.com>
+ *
+ * Copyright 2007-2015 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/bezier.h>
+#include <2geom/solver.h>
+#include <2geom/concepts.h>
+#include <2geom/choose.h>
+
+namespace Geom {
+
+std::vector<Coord> Bezier::valueAndDerivatives(Coord t, unsigned n_derivs) const {
+ /* This is inelegant, as it uses several extra stores. I think there might be a way to
+ * evaluate roughly in situ. */
+
+ // initialize return vector with zeroes, such that we only need to replace the non-zero derivs
+ std::vector<Coord> val_n_der(n_derivs + 1, Coord(0.0));
+
+ // initialize temp storage variables
+ std::valarray<Coord> d_(order()+1);
+ for(unsigned i = 0; i < size(); i++) {
+ d_[i] = c_[i];
+ }
+
+ unsigned nn = n_derivs + 1;
+ if(n_derivs > order()) {
+ nn = order()+1; // only calculate the non zero derivs
+ }
+ for(unsigned di = 0; di < nn; di++) {
+ //val_n_der[di] = (casteljau_subdivision(t, &d_[0], NULL, NULL, order() - di));
+ val_n_der[di] = bernstein_value_at(t, &d_[0], order() - di);
+ for(unsigned i = 0; i < order() - di; i++) {
+ d_[i] = (order()-di)*(d_[i+1] - d_[i]);
+ }
+ }
+
+ return val_n_der;
+}
+
+void Bezier::subdivide(Coord t, Bezier *left, Bezier *right) const
+{
+ if (left) {
+ left->c_.resize(size());
+ if (right) {
+ right->c_.resize(size());
+ casteljau_subdivision<double>(t, &const_cast<std::valarray<Coord>&>(c_)[0],
+ &left->c_[0], &right->c_[0], order());
+ } else {
+ casteljau_subdivision<double>(t, &const_cast<std::valarray<Coord>&>(c_)[0],
+ &left->c_[0], NULL, order());
+ }
+ } else if (right) {
+ right->c_.resize(size());
+ casteljau_subdivision<double>(t, &const_cast<std::valarray<Coord>&>(c_)[0],
+ NULL, &right->c_[0], order());
+ }
+}
+
+std::pair<Bezier, Bezier> Bezier::subdivide(Coord t) const
+{
+ std::pair<Bezier, Bezier> ret;
+ subdivide(t, &ret.first, &ret.second);
+ return ret;
+}
+
+std::vector<Coord> Bezier::roots() const
+{
+ std::vector<Coord> solutions;
+ find_bezier_roots(solutions, 0, 1);
+ std::sort(solutions.begin(), solutions.end());
+ return solutions;
+}
+
+std::vector<Coord> Bezier::roots(Interval const &ivl) const
+{
+ std::vector<Coord> solutions;
+ find_bernstein_roots(&const_cast<std::valarray<Coord>&>(c_)[0], order(), solutions, 0, ivl.min(), ivl.max());
+ std::sort(solutions.begin(), solutions.end());
+ return solutions;
+}
+
+Bezier Bezier::forward_difference(unsigned k) const
+{
+ Bezier fd(Order(order() - k));
+ int n = fd.size();
+
+ for (int i = 0; i < n; i++) {
+ fd[i] = 0;
+ int b = (i & 1) ? -1 : 1; // b = (-1)^j binomial(n, j - i)
+ for (int j = i; j < n; j++) {
+ fd[i] += c_[j] * b;
+ binomial_increment_k(b, n, j - i);
+ b = -b;
+ }
+ }
+ return fd;
+}
+
+Bezier Bezier::elevate_degree() const
+{
+ Bezier ed(Order(order()+1));
+ unsigned n = size();
+ ed[0] = c_[0];
+ ed[n] = c_[n-1];
+ for(unsigned i = 1; i < n; i++) {
+ ed[i] = (i*c_[i-1] + (n - i)*c_[i])/(n);
+ }
+ return ed;
+}
+
+Bezier Bezier::reduce_degree() const
+{
+ if(order() == 0) return *this;
+ Bezier ed(Order(order()-1));
+ unsigned n = size();
+ ed[0] = c_[0];
+ ed[n-1] = c_[n]; // ensure exact endpoints
+ unsigned middle = n/2;
+ for(unsigned i = 1; i < middle; i++) {
+ ed[i] = (n*c_[i] - i*ed[i-1])/(n-i);
+ }
+ for(unsigned i = n-1; i >= middle; i--) {
+ ed[i] = (n*c_[i] - i*ed[n-i])/(i);
+ }
+ return ed;
+}
+
+Bezier Bezier::elevate_to_degree(unsigned newDegree) const
+{
+ Bezier ed = *this;
+ for(unsigned i = degree(); i < newDegree; i++) {
+ ed = ed.elevate_degree();
+ }
+ return ed;
+}
+
+Bezier Bezier::deflate() const
+{
+ if(order() == 0) return *this;
+ unsigned n = order();
+ Bezier b(Order(n-1));
+ for(unsigned i = 0; i < n; i++) {
+ b[i] = (n*c_[i+1])/(i+1);
+ }
+ return b;
+}
+
+SBasis Bezier::toSBasis() const
+{
+ SBasis sb;
+ bezier_to_sbasis(sb, (*this));
+ return sb;
+ //return bezier_to_sbasis(&c_[0], order());
+}
+
+Bezier &Bezier::operator+=(Bezier const &other)
+{
+ if (c_.size() > other.size()) {
+ c_ += other.elevate_to_degree(degree()).c_;
+ } else if (c_.size() < other.size()) {
+ *this = elevate_to_degree(other.degree());
+ c_ += other.c_;
+ } else {
+ c_ += other.c_;
+ }
+ return *this;
+}
+
+Bezier &Bezier::operator-=(Bezier const &other)
+{
+ if (c_.size() > other.size()) {
+ c_ -= other.elevate_to_degree(degree()).c_;
+ } else if (c_.size() < other.size()) {
+ *this = elevate_to_degree(other.degree());
+ c_ -= other.c_;
+ } else {
+ c_ -= other.c_;
+ }
+ return *this;
+}
+
+Bezier operator*(Bezier const &f, Bezier const &g)
+{
+ int m = f.order();
+ int n = g.order();
+ Bezier h(Bezier::Order(m+n));
+ // h_k = sum_(i+j=k) (m i)f_i (n j)g_j / (m+n k)
+
+ int mci = 1;
+ for (int i = 0; i <= m; i++) {
+ double const fi = mci * f[i];
+
+ int ncj = 1;
+ for (int j = 0; j <= n; j++) {
+ h[i + j] += fi * ncj * g[j];
+ binomial_increment_k(ncj, n, j);
+ }
+
+ binomial_increment_k(mci, m, i);
+ }
+
+ int mnck = 1;
+ for (int k = 0; k <= m + n; k++) {
+ h[k] /= mnck;
+ binomial_increment_k(mnck, m + n, k);
+ }
+
+ return h;
+}
+
+Bezier portion(Bezier const &a, double from, double to)
+{
+ Bezier ret(a);
+
+ bool reverse_result = false;
+ if (from > to) {
+ std::swap(from, to);
+ reverse_result = true;
+ }
+
+ do {
+ if (from == 0) {
+ if (to == 1) {
+ break;
+ }
+ casteljau_subdivision<double>(to, &ret.c_[0], &ret.c_[0], NULL, ret.order());
+ break;
+ }
+ casteljau_subdivision<double>(from, &ret.c_[0], NULL, &ret.c_[0], ret.order());
+ if (to == 1) break;
+ casteljau_subdivision<double>((to - from) / (1 - from), &ret.c_[0], &ret.c_[0], NULL, ret.order());
+ // to protect against numerical inaccuracy in the above expression, we manually set
+ // the last coefficient to a value evaluated directly from the original polynomial
+ ret.c_[ret.order()] = a.valueAt(to);
+ } while(0);
+
+ if (reverse_result) {
+ std::reverse(&ret.c_[0], &ret.c_[0] + ret.c_.size());
+ }
+ return ret;
+}
+
+Bezier derivative(Bezier const &a)
+{
+ //if(a.order() == 1) return Bezier(0.0);
+ if(a.order() == 1) return Bezier(a.c_[1]-a.c_[0]);
+ Bezier der(Bezier::Order(a.order()-1));
+
+ for(unsigned i = 0; i < a.order(); i++) {
+ der.c_[i] = a.order()*(a.c_[i+1] - a.c_[i]);
+ }
+ return der;
+}
+
+Bezier integral(Bezier const &a)
+{
+ Bezier inte(Bezier::Order(a.order()+1));
+
+ inte[0] = 0;
+ for(unsigned i = 0; i < inte.order(); i++) {
+ inte[i+1] = inte[i] + a[i]/(inte.order());
+ }
+ return inte;
+}
+
+OptInterval bounds_fast(Bezier const &b)
+{
+ OptInterval ret = Interval::from_array(&const_cast<Bezier&>(b).c_[0], b.size());
+ return ret;
+}
+
+OptInterval bounds_exact(Bezier const &b)
+{
+ OptInterval ret(b.at0(), b.at1());
+ std::vector<Coord> r = derivative(b).roots();
+ for (double i : r) {
+ ret->expandTo(b.valueAt(i));
+ }
+ return ret;
+}
+
+OptInterval bounds_local(Bezier const &b, OptInterval const &i)
+{
+ //return bounds_local(b.toSBasis(), i);
+ if (i) {
+ return bounds_fast(portion(b, i->min(), i->max()));
+ } else {
+ return OptInterval();
+ }
+}
+
+/*
+ * The general bézier of degree n is
+ *
+ * p(t) = sum_{i = 0...n} binomial(n, i) t^i (1 - t)^(n - i) x[i]
+ *
+ * It can be written explicitly as a polynomial in t as
+ *
+ * p(t) = sum_{i = 0...n} binomial(n, i) t^i [ sum_{j = 0...i} binomial(i, j) (-1)^(i - j) x[j] ]
+ *
+ * Its derivative is
+ *
+ * p'(t) = n sum_{i = 1...n} binomial(n - 1, i - 1) t^(i - 1) [ sum_{j = 0...i} binomial(i, j) (-1)^(i - j) x[j] ]
+ *
+ * This is used by the various specialisations below as an optimisation for low degree n <= 3.
+ * In the remaining cases, the generic implementation is used which resorts to iteration.
+ */
+
+void bezier_expand_to_image(Interval &range, Coord x0, Coord x1, Coord x2)
+{
+ range.expandTo(x2);
+
+ if (range.contains(x1)) {
+ // The interval contains all control points, and therefore the entire curve.
+ return;
+ }
+
+ // p'(t) / 2 = at + b
+ auto a = (x2 - x1) - (x1 - x0);
+ auto b = x1 - x0;
+
+ // t = -b / a
+ if (std::abs(a) > EPSILON) {
+ auto t = -b / a;
+ if (t > 0.0 && t < 1.0) {
+ auto s = 1.0 - t;
+ auto x = s * s * x0 + 2 * s * t * x1 + t * t * x2;
+ range.expandTo(x);
+ }
+ }
+}
+
+void bezier_expand_to_image(Interval &range, Coord x0, Coord x1, Coord x2, Coord x3)
+{
+ range.expandTo(x3);
+
+ if (range.contains(x1) && range.contains(x2)) {
+ // The interval contains all control points, and therefore the entire curve.
+ return;
+ }
+
+ // p'(t) / 3 = at^2 + 2bt + c
+ auto a = (x3 - x0) - 3 * (x2 - x1);
+ auto b = (x2 - x1) - (x1 - x0);
+ auto c = x1 - x0;
+
+ auto expand = [&] (Coord t) {
+ if (t > 0.0 && t < 1.0) {
+ auto s = 1.0 - t;
+ auto x = s * s * s * x0 + 3 * s * s * t * x1 + 3 * t * t * s * x2 + t * t * t * x3;
+ range.expandTo(x);
+ }
+ };
+
+ // t = (-b ± sqrt(b^2 - ac)) / a
+ if (std::abs(a) < EPSILON) {
+ if (std::abs(b) > EPSILON) {
+ expand(-c / (2 * b));
+ }
+ } else {
+ auto d2 = b * b - a * c;
+ if (d2 >= 0.0) {
+ auto bsign = b >= 0.0 ? 1 : -1;
+ auto tmp = -(b + bsign * std::sqrt(d2));
+ expand(tmp / a);
+ expand(c / tmp); // Using Vieta's formula: product of roots == c/a
+ }
+ }
+}
+
+} // namespace Geom
+
+/*
+ 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 :
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