/**
* @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 <boost/optional.hpp>
#include <2geom/choose.h>
#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;
}
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(!IS_FINITE(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);
};
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);
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;
}
}
#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:
*/
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