path: root/include/2geom/bezier.h

/**
 * @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 :