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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.
+ *
+ */
+
+#ifndef LIB2GEOM_SEEN_BEZIER_H
+#define LIB2GEOM_SEEN_BEZIER_H
+
+#include <algorithm>
+#include <valarray>
+#include <2geom/coord.h>
+#include <2geom/d2.h>
+#include <2geom/math-utils.h>
+
+namespace Geom {
+
+/** @brief Compute the value of a Bernstein-Bezier polynomial.
+ * This method uses a Horner-like fast evaluation scheme.
+ * @param t Time value
+ * @param c_ Pointer to coefficients
+ * @param n Degree of the polynomial (number of coefficients minus one) */
+template <typename T>
+inline T bernstein_value_at(double t, T const *c_, unsigned n) {
+ double u = 1.0 - t;
+ double bc = 1;
+ double tn = 1;
+ T tmp = c_[0]*u;
+ for(unsigned i=1; i<n; i++){
+ tn = tn*t;
+ bc = bc*(n-i+1)/i;
+ tmp = (tmp + tn*bc*c_[i])*u;
+ }
+ return (tmp + tn*t*c_[n]);
+}
+
+/** @brief Perform Casteljau subdivision of a Bezier polynomial.
+ * Given an array of coefficients and a time value, computes two new Bernstein-Bezier basis
+ * polynomials corresponding to the \f$[0, t]\f$ and \f$[t, 1]\f$ intervals of the original one.
+ * @param t Time value
+ * @param v Array of input coordinates
+ * @param left Output polynomial corresponding to \f$[0, t]\f$
+ * @param right Output polynomial corresponding to \f$[t, 1]\f$
+ * @param order Order of the input polynomial, equal to one less the number of coefficients
+ * @return Value of the polynomial at @a t */
+template <typename T>
+inline T casteljau_subdivision(double t, T const *v, T *left, T *right, unsigned order) {
+ // The Horner-like scheme gives very slightly different results, but we need
+ // the result of subdivision to match exactly with Bezier's valueAt function.
+ T val = bernstein_value_at(t, v, order);
+
+ if (!left && !right) {
+ return val;
+ }
+
+ if (!right) {
+ if (left != v) {
+ std::copy(v, v + order + 1, left);
+ }
+ for (std::size_t i = order; i > 0; --i) {
+ for (std::size_t j = i; j <= order; ++j) {
+ left[j] = lerp(t, left[j-1], left[j]);
+ }
+ }
+ left[order] = val;
+ return left[order];
+ }
+
+ if (right != v) {
+ std::copy(v, v + order + 1, right);
+ }
+ for (std::size_t i = 1; i <= order; ++i) {
+ if (left) {
+ left[i-1] = right[0];
+ }
+ for (std::size_t j = i; j > 0; --j) {
+ right[j-1] = lerp(t, right[j-1], right[j]);
+ }
+ }
+ right[0] = val;
+ if (left) {
+ left[order] = right[0];
+ }
+ return right[0];
+}
+
+/**
+ * @brief Polynomial in Bernstein-Bezier basis
+ * @ingroup Fragments
+ */
+class Bezier
+ : boost::arithmetic< Bezier, double
+ , boost::additive< Bezier
+ > >
+{
+private:
+ std::valarray<Coord> c_;
+
+ friend Bezier portion(const Bezier & a, Coord from, Coord to);
+ friend OptInterval bounds_fast(Bezier const & b);
+ friend Bezier derivative(const Bezier & a);
+ friend class Bernstein;
+
+ void
+ find_bezier_roots(std::vector<double> & solutions,
+ double l, double r) const;
+
+protected:
+ Bezier(Coord const c[], unsigned ord)
+ : c_(c, ord+1)
+ {}
+
+public:
+ unsigned order() const { return c_.size()-1;}
+ unsigned degree() const { return order(); }
+ unsigned size() const { return c_.size();}
+
+ Bezier() {}
+ Bezier(const Bezier& b) :c_(b.c_) {}
+ Bezier &operator=(Bezier const &other) {
+ if ( c_.size() != other.c_.size() ) {
+ c_.resize(other.c_.size());
+ }
+ c_ = other.c_;
+ return *this;
+ }
+
+ bool operator==(Bezier const &other) const
+ {
+ if (degree() != other.degree()) {
+ return false;
+ }
+
+ for (size_t i = 0; i < c_.size(); i++) {
+ if (c_[i] != other.c_[i]) {
+ return false;
+ }
+ }
+ return true;
+ }
+
+ bool operator!=(Bezier const &other) const
+ {
+ return !(*this == other);
+ }
+
+ struct Order {
+ unsigned order;
+ explicit Order(Bezier const &b) : order(b.order()) {}
+ explicit Order(unsigned o) : order(o) {}
+ operator unsigned() const { return order; }
+ };
+
+ //Construct an arbitrary order bezier
+ Bezier(Order ord) : c_(0., ord.order+1) {
+ assert(ord.order == order());
+ }
+
+ /// @name Construct Bezier polynomials from their control points
+ /// @{
+ explicit Bezier(Coord c0) : c_(0., 1) {
+ c_[0] = c0;
+ }
+ Bezier(Coord c0, Coord c1) : c_(0., 2) {
+ c_[0] = c0; c_[1] = c1;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2) : c_(0., 3) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3) : c_(0., 4) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4) : c_(0., 5) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3; c_[4] = c4;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4,
+ Coord c5) : c_(0., 6) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3; c_[4] = c4;
+ c_[5] = c5;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4,
+ Coord c5, Coord c6) : c_(0., 7) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3; c_[4] = c4;
+ c_[5] = c5; c_[6] = c6;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4,
+ Coord c5, Coord c6, Coord c7) : c_(0., 8) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3; c_[4] = c4;
+ c_[5] = c5; c_[6] = c6; c_[7] = c7;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4,
+ Coord c5, Coord c6, Coord c7, Coord c8) : c_(0., 9) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3; c_[4] = c4;
+ c_[5] = c5; c_[6] = c6; c_[7] = c7; c_[8] = c8;
+ }
+ Bezier(Coord c0, Coord c1, Coord c2, Coord c3, Coord c4,
+ Coord c5, Coord c6, Coord c7, Coord c8, Coord c9) : c_(0., 10) {
+ c_[0] = c0; c_[1] = c1; c_[2] = c2; c_[3] = c3; c_[4] = c4;
+ c_[5] = c5; c_[6] = c6; c_[7] = c7; c_[8] = c8; c_[9] = c9;
+ }
+
+ template <typename Iter>
+ Bezier(Iter first, Iter last) {
+ c_.resize(std::distance(first, last));
+ for (std::size_t i = 0; first != last; ++first, ++i) {
+ c_[i] = *first;
+ }
+ }
+ Bezier(std::vector<Coord> const &vec)
+ : c_(&vec[0], vec.size())
+ {}
+ /// @}
+
+ void resize (unsigned int n, Coord v = 0) {
+ c_.resize (n, v);
+ }
+ void clear() {
+ c_.resize(0);
+ }
+
+ //IMPL: FragmentConcept
+ typedef Coord output_type;
+ bool isZero(double eps=EPSILON) const {
+ for(unsigned i = 0; i <= order(); i++) {
+ if( ! are_near(c_[i], 0., eps) ) return false;
+ }
+ return true;
+ }
+ bool isConstant(double eps=EPSILON) const {
+ for(unsigned i = 1; i <= order(); i++) {
+ if( ! are_near(c_[i], c_[0], eps) ) return false;
+ }
+ return true;
+ }
+ bool isFinite() const {
+ for(unsigned i = 0; i <= order(); i++) {
+ if(!std::isfinite(c_[i])) return false;
+ }
+ return true;
+ }
+ Coord at0() const { return c_[0]; }
+ Coord &at0() { return c_[0]; }
+ Coord at1() const { return c_[order()]; }
+ Coord &at1() { return c_[order()]; }
+
+ Coord valueAt(double t) const {
+ return bernstein_value_at(t, &c_[0], order());
+ }
+ Coord operator()(double t) const { return valueAt(t); }
+
+ SBasis toSBasis() const;
+
+ Coord &operator[](unsigned ix) { return c_[ix]; }
+ Coord const &operator[](unsigned ix) const { return const_cast<std::valarray<Coord>&>(c_)[ix]; }
+
+ void setCoeff(unsigned ix, double val) { c_[ix] = val; }
+
+ // The size of the returned vector equals n_derivs+1.
+ std::vector<Coord> valueAndDerivatives(Coord t, unsigned n_derivs) const;
+
+ void subdivide(Coord t, Bezier *left, Bezier *right) const;
+ std::pair<Bezier, Bezier> subdivide(Coord t) const;
+
+ std::vector<Coord> roots() const;
+ std::vector<Coord> roots(Interval const &ivl) const;
+
+ Bezier forward_difference(unsigned k) const;
+ Bezier elevate_degree() const;
+ Bezier reduce_degree() const;
+ Bezier elevate_to_degree(unsigned newDegree) const;
+ Bezier deflate() const;
+
+ // basic arithmetic operators
+ Bezier &operator+=(double v) {
+ c_ += v;
+ return *this;
+ }
+ Bezier &operator-=(double v) {
+ c_ -= v;
+ return *this;
+ }
+ Bezier &operator*=(double v) {
+ c_ *= v;
+ return *this;
+ }
+ Bezier &operator/=(double v) {
+ c_ /= v;
+ return *this;
+ }
+ Bezier &operator+=(Bezier const &other);
+ Bezier &operator-=(Bezier const &other);
+
+ /// Unary minus
+ Bezier operator-() const
+ {
+ Bezier result;
+ result.c_ = -c_;
+ return result;
+ }
+};
+
+
+void bezier_to_sbasis (SBasis &sb, Bezier const &bz);
+
+Bezier operator*(Bezier const &f, Bezier const &g);
+inline Bezier multiply(Bezier const &f, Bezier const &g) {
+ Bezier result = f * g;
+ return result;
+}
+
+inline Bezier reverse(const Bezier & a) {
+ Bezier result = Bezier(Bezier::Order(a));
+ for(unsigned i = 0; i <= a.order(); i++)
+ result[i] = a[a.order() - i];
+ return result;
+}
+
+Bezier portion(const Bezier & a, double from, double to);
+
+// XXX Todo: how to handle differing orders
+inline std::vector<Point> bezier_points(const D2<Bezier > & a) {
+ std::vector<Point> result;
+ for(unsigned i = 0; i <= a[0].order(); i++) {
+ Point p;
+ for(unsigned d = 0; d < 2; d++) p[d] = a[d][i];
+ result.push_back(p);
+ }
+ return result;
+}
+
+Bezier derivative(Bezier const &a);
+Bezier integral(Bezier const &a);
+OptInterval bounds_fast(Bezier const &b);
+OptInterval bounds_exact(Bezier const &b);
+OptInterval bounds_local(Bezier const &b, OptInterval const &i);
+
+/// Expand an interval to the image of a Bézier-Bernstein polynomial, assuming it already contains the initial point x0.
+void bezier_expand_to_image(Interval &range, Coord x0, Coord x1, Coord x2);
+void bezier_expand_to_image(Interval &range, Coord x0, Coord x1, Coord x2, Coord x3);
+
+inline std::ostream &operator<< (std::ostream &os, const Bezier & b) {
+ os << "Bezier(";
+ for(unsigned i = 0; i < b.order(); i++) {
+ os << format_coord_nice(b[i]) << ", ";
+ }
+ os << format_coord_nice(b[b.order()]) << ")";
+ return os;
+}
+
+} // namespace Geom
+
+#endif // LIB2GEOM_SEEN_BEZIER_H
+
+/*
+ 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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