// Copyright (c) 2006,2007,2009,2010,2011 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v6.0.1/Arrangement_on_surface_2/include/CGAL/Arr_geometry_traits/Bezier_cache.h $
// $Id: include/CGAL/Arr_geometry_traits/Bezier_cache.h 50cfbde3b84 $
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s)     : Ron Wein     <wein@post.tau.ac.il>

#ifndef CGAL_BEZIER_CACHE_H
#define CGAL_BEZIER_CACHE_H

#include <CGAL/license/Arrangement_on_surface_2.h>


/*! \file
 * Header file for the _Bezier_cache class.
 */

#include <set>
#include <map>
#include <ostream>

namespace CGAL {

/*! \class _Bezier_cache
 * Stores all cached intersection points and vertical tangency points.
 */
template <class Nt_traits_> class _Bezier_cache
{
public:

  typedef Nt_traits_                              Nt_traits;

  typedef typename Nt_traits::Integer             Integer;
  typedef typename Nt_traits::Polynomial          Polynomial;
  typedef typename Nt_traits::Algebraic           Algebraic;

  typedef _Bezier_cache<Nt_traits>                Self;

  /// \name Type definitions for the vertical tangency-point mapping.
  //@{
  typedef size_t                                     Curve_id;
  typedef std::list<Algebraic>                       Vertical_tangency_list;
  typedef
    typename Vertical_tangency_list::const_iterator  Vertical_tangency_iter;
  //@}

  /// \name Type definitions for the intersection-point mapping.
  //@{

  /*! \struct Intersection_point
   * Representation of an intersection point (in both parameter and physical
   * spaces).
   */
  struct Intersection_point
  {
    Algebraic          s;      // The parameter for the first curve.
    Algebraic          t;      // The parameter for the second curve.
    Algebraic          x;      // The x-coordinate.
    Algebraic          y;      // The y-coordinate.

    /*! Constructor. */
    Intersection_point (const Algebraic& _s, const Algebraic& _t,
                          const Algebraic& _x, const Algebraic& _y) :
      s(_s), t(_t),
      x(_x), y(_y)
    {}
  };

  typedef std::pair<Curve_id, Curve_id>              Curve_pair;
  typedef std::pair<Algebraic, Algebraic>            Parameter_pair;
  typedef std::list<Intersection_point>            Intersection_list;
  typedef
    typename Intersection_list::const_iterator       Intersection_iter;

  //@}

private:

  typedef std::map<Curve_id,
                   Vertical_tangency_list>           Vert_tang_map;
  typedef typename Vert_tang_map::value_type         Vert_tang_map_entry;
  typedef typename Vert_tang_map::iterator           Vert_tang_map_iterator;

  /*! \struct Less_curve_pair
   * An auxiliary functor for comparing pair of curve IDs.
   */
  struct Less_curve_pair
  {
    bool operator() (const Curve_pair& cp1, const Curve_pair& cp2) const
    {
      // Compare the pairs of IDs lexicographically.
      return (cp1.first < cp2.first ||
              (cp1.first == cp2.first && cp1.second < cp2.second));
    }
  };

  typedef std::list<Algebraic>                       Parameter_list;
  typedef std::pair<Intersection_list, bool>         Intersection_info;
  typedef std::map<Curve_pair,
                   Intersection_info,
                   Less_curve_pair>                  Intersect_map;
  typedef typename Intersect_map::value_type         Intersect_map_entry;
  typedef typename Intersect_map::iterator           Intersect_map_iterator;

  /*! \struct My_point_2
   * Representation of exact and approximate point coordinates.
   */
  struct My_point_2
  {
    typename Parameter_list::const_iterator  prm_it;
    Algebraic                                x;
    Algebraic                                y;
    double                                   app_x;
    double                                   app_y;

    /*! Default constructor. */
    My_point_2 () :
      app_x (0),
      app_y (0)
    {}

    /*! Constructor. */
    My_point_2 (typename Parameter_list::const_iterator it,
                const Algebraic& _x, const Algebraic& _y) :
      prm_it (it),
      x (_x),
      y (_y),
      app_x (CGAL::to_double(_x)),
      app_y (CGAL::to_double(_y))
    {}

    /*! Get the parameter value. */
    const Algebraic& parameter () const
    {
      return (*prm_it);
    }

    /*! Check if the given two points are equal. */
    bool equals (const My_point_2& other) const
    {
      return (CGAL::compare (x, other.x) == EQUAL &&
              CGAL::compare (y, other.y) == EQUAL);
    }
  };

  /*! \struct Less_distance_iter
   * An auxiliary functor for comparing approximate distances.
   */
  typedef std::list<My_point_2>                      Point_list;
  typedef typename Point_list::iterator              Point_iter;
  typedef std::pair<double, Point_iter>              Distance_iter;

  struct Less_distance_iter
  {
    bool operator() (const Distance_iter& dit1,
                     const Distance_iter& dit2) const
    {
      return (dit1.first < dit2.first);
    }
  };

  // Data members:
  Nt_traits         nt_traits;        /*! Number-type traits. */
  Vert_tang_map     vert_tang_map;    /*! Maps curves to their vertical
                                          tangency parameters. */
  Intersect_map     intersect_map;    /*! Maps curve pairs to their
                                          intersection parameter pairs. */

  // Copy constructor and assignment operator - not supported.
  _Bezier_cache (const Self& );
  Self& operator= (const Self& );

public:

  /*! Constructor. */
  _Bezier_cache ()
  {}

  /*!
   * Get the vertical tangency parameters of a given curve (X(t), Y(t)).
   * \param id The curve ID.
   * \param polyX The coefficients of X(t).
   * \param normX The normalizing factor of X(t).
   * \return A list of parameters 0 < t_1 < ...< t_k < 1 such that X'(t_i) = 0.
   */
  const Vertical_tangency_list&
  get_vertical_tangencies (const Curve_id& id,
                           const Polynomial& polyX, const Integer& normX);

  /*!
   * Get the intersection parameters of two given curve (X_1(s), Y_1(s))
   * and (X_2(t), Y_2(t)).
   * \param id1 The ID of the first curve.
   * \param polyX_1 The coefficients of X_1.
   * \param normX_1 The normalizing factor of X_1.
   * \param polyY_1 The coefficients of Y_1.
   * \param normY_1 The normalizing factor of Y_1.
   * \param id2 The ID of the second curve.
   * \param polyX_2 The coefficients of X_2.
   * \param normX_2 The normalizing factor of X_2.
   * \param polyY_2 The coefficients of Y_2.
   * \param normY_2 The normalizing factor of Y_2.
   * \param do_ovlp Output: Do the two curves overlap.
   * \pre id1 < id2 (swap their order if this is not the case).
   * \return A list of parameter pairs (s_1, t_1), ..., (s_k, t_k) such that
   *         X_1(s_i) = X_2(t_i) and Y_1(s_i) = Y_2(t_i).
   *         Each pair is also associated with the physical point coordinates.
   */
  const Intersection_list&
  get_intersections (const Curve_id& id1,
                     const Polynomial& polyX_1, const Integer& normX_1,
                     const Polynomial& polyY_1, const Integer& normY_1,
                     const Curve_id& id2,
                     const Polynomial& polyX_2, const Integer& normX_2,
                     const Polynomial& polyY_2, const Integer& normY_2,
                     bool& do_ovlp);

  /*!
   * Mark two curves as overlapping.
   * \param id1 The ID of the first curve.
   * \param id2 The ID of the second curve.
   */
  void mark_as_overlapping (const Curve_id& id1,
                            const Curve_id& id2);

private:

  /*!
   * Compute all s-parameter values such that the curve (X_1(s), Y_1(s))
   * intersects with (X_2(t), Y_2(t)) for some t.
   * \param polyX_1 The coefficients of X_1.
   * \param normX_1 The normalizing factor of X_1.
   * \param polyY_1 The coefficients of Y_1.
   * \param normY_1 The normalizing factor of Y_1.
   * \param polyX_2 The coefficients of X_2.
   * \param normX_2 The normalizing factor of X_2.
   * \param polyY_2 The coefficients of Y_2.
   * \param normY_2 The normalizing factor of Y_2.
   * \param s_vals Output: A list of s-values such that (X_1(s), Y_1(s))
   *                       is an intersection point with (X_2(t), Y_2(t)).
   *                       The list is given in ascending order.
   *                       Note that the values are not bounded to [0,1].
   * \return Do the two curves overlap (if they do, s_vals is empty).
   */
  bool _intersection_params (const Polynomial& polyX_1, const Integer& normX_1,
                             const Polynomial& polyY_1, const Integer& normY_1,
                             const Polynomial& polyX_2, const Integer& normX_2,
                             const Polynomial& polyY_2, const Integer& normY_2,
                             Parameter_list& s_vals, bool find_out_of_range=false) const;

  /*!
   * Compute all s-parameter values of the self intersection of (X(s), Y(s))
   * with (X(t), Y(t)) for some t.
   * \param polyX The coefficients of X.
   * \param polyY The coefficients of Y.
   * \param s_vals Output: A list of s-values of the self-intersection points.
   *                       Note that the values are not bounded to [0,1].
   */
  void _self_intersection_params (const Polynomial& polyX,
                                  const Polynomial& polyY,
                                  Parameter_list& s_vals) const;

  /*!
   * Compute the resultant of two bivariate polynomials in x and y with
   * respect to y. The bivariate polynomials are given as vectors of
   * polynomials, where bp1[i] is a coefficient of y^i, which is in turn a
   * polynomial in x.
   * \param bp1 The first bivariate polynomial.
   * \param bp2 The second bivariate polynomial.
   * \return The resultant polynomial (a polynomial in x).
   */
  Polynomial _compute_resultant (const std::vector<Polynomial>& bp1,
                                 const std::vector<Polynomial>& bp2) const;
};

// ---------------------------------------------------------------------------
// Get the vertical tangency parameters of a given curve (X(t), Y(t)).
//
template<class NtTraits>
const typename _Bezier_cache<NtTraits>::Vertical_tangency_list&
_Bezier_cache<NtTraits>::get_vertical_tangencies
        (const Curve_id& id,
         const Polynomial& polyX, const Integer& /* normX */)
{
  // Try to find the curve ID in the map.
  Vert_tang_map_iterator    map_iter = vert_tang_map.find (id);

  if (map_iter != vert_tang_map.end())
  {
    // Found in the map: return the cached parameters' list.
    return (map_iter->second);
  }

  // We need to compute the vertical tangency parameters: find all t-values
  // such that X'(t) = 0, and store them in the cache.
  Vertical_tangency_list&  vert_tang_list = vert_tang_map[id];
  const Polynomial&        polyX_der = nt_traits.derive (polyX);

  nt_traits.compute_polynomial_roots (polyX_der, 0, 1,
                                      std::back_inserter(vert_tang_list));
  return (vert_tang_list);
}

// ---------------------------------------------------------------------------
// Get the intersection parameters of two given curve (X_1(s), Y_1(s))
// and (X_2(t), Y_2(t)).
//
template<class NtTraits>
const typename _Bezier_cache<NtTraits>::Intersection_list&
_Bezier_cache<NtTraits>::get_intersections
        (const Curve_id& id1,
         const Polynomial& polyX_1, const Integer& normX_1,
         const Polynomial& polyY_1, const Integer& normY_1,
         const Curve_id& id2,
         const Polynomial& polyX_2, const Integer& normX_2,
         const Polynomial& polyY_2, const Integer& normY_2,
         bool& do_ovlp)
{
  CGAL_precondition (id1 <= id2);

  // Construct the pair of curve IDs, and try to find it in the map.
  Curve_pair                curve_pair (id1, id2);
  Intersect_map_iterator    map_iter = intersect_map.find (curve_pair);

  if (map_iter != intersect_map.end())
  {
    // Found in the map: return the cached information.
    do_ovlp = map_iter->second.second;
    return (map_iter->second.first);
  }

  // We need to compute the intersection-parameter pairs.
  Intersection_info&      info = intersect_map[curve_pair];

  // Check if we have to compute a self intersection (a special case),
  // or a regular intersection between two curves.
  if (id1 == id2)
  {
    // Compute all parameter values that lead to a self intersection.
    Parameter_list          s_vals;

    _self_intersection_params (polyX_1, polyY_1,
                               s_vals);

    // Match pairs of parameter values that correspond to the same point.
    // Note that we make sure that both parameter pairs are in the range [0, 1]
    // (if not, the self-intersection point is imaginary).
    typename Parameter_list::iterator  s_it;
    typename Parameter_list::iterator  t_it;
    const Algebraic                    one (1);
    const Algebraic&                   denX = nt_traits.convert (normX_1);
    const Algebraic&                   denY = nt_traits.convert (normY_1);
    Point_list                         pts1;

    for (s_it = s_vals.begin(); s_it != s_vals.end(); ++s_it)
    {
      if (CGAL::sign (*s_it) == NEGATIVE)
        continue;

      if (CGAL::compare (*s_it, one) == LARGER)
        break;

      const Algebraic&  x = nt_traits.evaluate_at (polyX_1, *s_it);
      const Algebraic&  y = nt_traits.evaluate_at (polyY_1, *s_it);

      for (t_it = s_it; t_it != s_vals.end(); ++t_it)
      {
        if (CGAL::compare (*t_it, one) == LARGER)
          break;

        if (CGAL::compare (nt_traits.evaluate_at (polyX_1, *t_it),
                           x) == EQUAL &&
            CGAL::compare (nt_traits.evaluate_at (polyY_1, *t_it),
                           y) == EQUAL)
        {
          info.first.push_back (Intersection_point (*s_it, *t_it,
                                                      x / denX, y / denY));
        }
      }
    }

    info.second = false;
    return (info.first);
  }

  // Compute s-values and t-values such that (X_1(s), Y_1(s)) and
  // (X_2(t), Y_2(t)) are the intersection points.
  Parameter_list          s_vals;

  do_ovlp = _intersection_params (polyX_1, normX_1, polyY_1, normY_1,
                                  polyX_2, normX_2, polyY_2, normY_2,
                                  s_vals);

  if (do_ovlp)
  {
    // Update the cache and return an empty list of intersection parameters.
    info.second = true;
    return (info.first);
  }

  Parameter_list          t_vals;

  do_ovlp = _intersection_params (polyX_2, normX_2, polyY_2, normY_2,
                                  polyX_1, normX_1, polyY_1, normY_1,
                                  t_vals, true);

  CGAL_assertion (! do_ovlp);

  // s-values and t-values reported lie in the range [0,1]. We have
  // to pair them together. Note that the numbers of s-values and t-values
  //can be different as a s-value in [0,1] may correspond to a t-values
  //outside this range. To make the pairing correct, we choose the one with
  //less values and look amongst other values the correct one (thus the swapt below).

  // Compute all points on (X_1, Y_1) that match an s-value from the list.
  typename Parameter_list::iterator  s_it;
  const Algebraic&                   denX_1 = nt_traits.convert (normX_1);
  const Algebraic&                   denY_1 = nt_traits.convert (normY_1);
  Point_list                         pts1;

  for (s_it = s_vals.begin(); s_it != s_vals.end(); ++s_it)
  {
    const Algebraic&  x = nt_traits.evaluate_at (polyX_1, *s_it) / denX_1;
    const Algebraic&  y = nt_traits.evaluate_at (polyY_1, *s_it) / denY_1;

    pts1.push_back (My_point_2 (s_it, x, y));
  }

  // Compute all points on (X_2, Y_2) that match a t-value from the list.
  typename Parameter_list::iterator  t_it;
  const Algebraic&                   denX_2 = nt_traits.convert (normX_2);
  const Algebraic&                   denY_2 = nt_traits.convert (normY_2);
  Point_list                         pts2;

  for (t_it = t_vals.begin(); t_it != t_vals.end(); ++t_it)
  {
    const Algebraic&  x = nt_traits.evaluate_at (polyX_2, *t_it) / denX_2;
    const Algebraic&  y = nt_traits.evaluate_at (polyY_2, *t_it) / denY_2;

    pts2.push_back (My_point_2 (t_it, x, y));
  }

  // Go over the points in the pts1 list.
  const bool                x2_simpler = nt_traits.degree(polyX_2) <
                                         nt_traits.degree(polyX_1);
  const bool                y2_simpler = nt_traits.degree(polyY_2) <
                                         nt_traits.degree(polyY_1);
  Point_iter                pit1;
  Point_iter                pit2;
  double                    dx, dy;
  Algebraic                 s, t;
  const Algebraic           one (1);
  unsigned int              k;

  for (pit1 = pts1.begin(); pit1 != pts1.end(); ++pit1)
  {
    // Construct a vector of distances from the current point to all other
    // points in the pts2 list.
    const int                     n_pts2 = static_cast<int>(pts2.size());
    std::vector<Distance_iter>    dist_vec (n_pts2);

    for (k = 0, pit2 = pts2.begin(); pit2 != pts2.end(); k++, ++pit2)
    {
      // Compute the approximate distance between the teo current points.
      dx = pit1->app_x - pit2->app_x;
      dy = pit1->app_y - pit2->app_y;

      dist_vec[k] = Distance_iter (dx*dx + dy*dy, pit2);
    }

    // Sort the vector according to the distances from *pit1.
    std::sort (dist_vec.begin(), dist_vec.end(),
               Less_distance_iter());

    // Go over the vector entries, starting from the most distant from *pit1
    // to the closest and eliminate pairs of points (we expect that
    // eliminating the distant points is done easily). We stop when we find
    // a pait for *pit1 or when we are left with a single point.
    bool                    found = false;
    const Algebraic&        s = pit1->parameter();
    Algebraic               t;

    for (k = n_pts2 - 1; !found && k > 0; k--)
    {
      pit2 = dist_vec[k].second;

      if (pit1->equals (*pit2))
      {
        // Obtain the parameter value, and try to simplify the representation
        // of the intersection point.
        t = pit2->parameter();

        if (x2_simpler)
          pit1->x = pit2->x;
        if (y2_simpler)
          pit1->y = pit2->y;

        // Remove this point from pts2, as we found a match for it.
        pts2.erase (pit2);
        found = true;
      }
    }

    if (! found)
    {
      // We are left with a single point - pair it with *pit1.
      pit2 = dist_vec[0].second;

      // Obtain the parameter value, and try to simplify the representation
      // of the intersection point.
      t = pit2->parameter();

      if (x2_simpler)
        pit1->x = pit2->x;
      if (y2_simpler)
        pit1->y = pit2->y;

      // Remove this point from pts2, as we found a match for it.
      pts2.erase (pit2);
    }

    // Check that  s- and t-values both lie in the legal range of [0,1].
    if(CGAL::sign (s) != NEGATIVE && CGAL::compare (s, one) != LARGER &&
      CGAL::sign (t) != NEGATIVE && CGAL::compare (t, one) != LARGER)
    {
      info.first.push_back(Intersection_point(s, t,pit1->x, pit1->y));
    }
  }

  info.second = false;
  return (info.first);
}

// ---------------------------------------------------------------------------
// Mark two curves as overlapping.
//
template<class NtTraits>
void _Bezier_cache<NtTraits>::mark_as_overlapping (const Curve_id& id1,
                                                   const Curve_id& id2)
{
  CGAL_precondition (id1 < id2);

  // Construct the pair of curve IDs, and try to find it in the map.
  Curve_pair                curve_pair (id1, id2);
  Intersect_map_iterator    map_iter = intersect_map.find (curve_pair);

  if (map_iter != intersect_map.end())
  {
    // Found in the map: Make sure the curves are marked as overlapping.
    CGAL_assertion (map_iter->second.second);
    return;
  }

  // Add a new entry and mark the curves as overlapping.
  Intersection_info&      info = intersect_map[curve_pair];

  info.second = true;
  return;
}

// ---------------------------------------------------------------------------
// Compute all s-parameter values of the intersection of (X_1(s), Y_1(s))
// with (X_2(t), Y_2(t)) for some t.
//
template<class NtTraits>
bool _Bezier_cache<NtTraits>::_intersection_params
        (const Polynomial& polyX_1, const Integer& normX_1,
         const Polynomial& polyY_1, const Integer& normY_1,
         const Polynomial& polyX_2, const Integer& normX_2,
         const Polynomial& polyY_2, const Integer& normY_2,
         Parameter_list& s_vals, bool find_out_of_range) const
{
  // Clear the output parameter list.
  if (! s_vals.empty())
    s_vals.clear();

  // We are looking for the s-values s_0 such that (X_1(s_0), Y_1(s_0)) is
  // an intersection with some point on (X_2(t), Y_2(t)). We therefore have
  // to solve the system of bivariate polynomials in s and t:
  //    I: X_2(t) - X_1(s) = 0
  //   II: Y_2(t) - Y_1(s) = 0
  //
  Integer                  coeff;
  int                      k;

  // Construct the bivariate polynomial that corresponds to Equation I.
  // Note that we represent a bivariate polynomial as a vector of univariate
  // polynomials, whose i'th entry corresponds to the coefficient of t^i,
  // which is in turn a polynomial it s.
  const int                degX_2 = nt_traits.degree (polyX_2);
  std::vector<Polynomial>  coeffsX_st (degX_2 < 0 ? 1 : (degX_2 + 1));

  for (k = degX_2; k >= 0; k--)
  {
    coeff = nt_traits.get_coefficient (polyX_2, k) * normX_1;
    coeffsX_st[k] = nt_traits.construct_polynomial (&coeff, 0);
  }
  coeffsX_st[0] = coeffsX_st[0] - nt_traits.scale (polyX_1, normX_2);

  // Construct the bivariate polynomial that corresponds to Equation II.
  const int                degY_2 = nt_traits.degree (polyY_2);
  std::vector<Polynomial>  coeffsY_st (degY_2 < 0 ? 1 : (degY_2 + 1));

  for (k = degY_2; k >= 0; k--)
  {
    coeff = nt_traits.get_coefficient (polyY_2, k) * normY_1;
    coeffsY_st[k] = nt_traits.construct_polynomial (&coeff, 0);
  }
  coeffsY_st[0] = coeffsY_st[0] - nt_traits.scale (polyY_1, normY_2);

  // Compute the resultant of the two bivariate polynomials and obtain
  // a polynomial in s.
  Polynomial            res = _compute_resultant (coeffsX_st, coeffsY_st);

  if (nt_traits.degree (res) < 0)
  {
    // If the resultant is identiaclly zero, then the two curves overlap.
    return (true);
  }

  // Compute the roots of the resultant polynomial and mark that the curves do
  // not overlap. The roots we are interested in are usually in the interval [0,1].
  if (find_out_of_range)
    nt_traits.compute_polynomial_roots (res,std::back_inserter (s_vals));
  else
    nt_traits.compute_polynomial_roots (res,0,1,std::back_inserter (s_vals));

  return (false);
}

// ---------------------------------------------------------------------------
// Compute all s-parameter values of the self intersection of (X(s), Y(s))
// with (X(t), Y(t)) for some t.
//
template<class NtTraits>
void _Bezier_cache<NtTraits>::_self_intersection_params
        (const Polynomial& polyX,
         const Polynomial& polyY,
         Parameter_list& s_vals) const
{
  // Clear the output parameter list.
  if (! s_vals.empty())
    s_vals.clear();

  // We are looking for the solutions of the following system of bivariate
  // polynomials in s and t:
  //    I: X(t) - X(s) / (t - s) = 0
  //   II: Y(t) - Y(s) / (t - s) = 0
  //
  Integer                 *coeffs;
  int                      i;

  // Construct the bivariate polynomial that corresponds to Equation I.
  // Note that we represent a bivariate polynomial as a vector of univariate
  // polynomials, whose i'th entry corresponds to the coefficient of t^i,
  // which is in turn a polynomial it s.
  const int                degX = nt_traits.degree (polyX);
  CGAL_assertion(degX > 0);
  if (degX <= 0) return; //no self intersection if X is constant
  std::vector<Polynomial>  coeffsX_st (degX);

  coeffs = new Integer [degX];

  for (i = 0; i < degX; i++)
    coeffs[i] = nt_traits.get_coefficient(polyX, i + 1);

  for (i = 0; i < degX; i++)
    coeffsX_st[degX - i - 1] =
        nt_traits.construct_polynomial(coeffs + i, degX - i - 1);

  delete[] coeffs;

  // Construct the bivariate polynomial that corresponds to Equation II.
  const int                degY = nt_traits.degree (polyY);
  CGAL_assertion(degY > 0);
  if (degY <= 0) return; //no self intersection if Y is constant
  std::vector<Polynomial>  coeffsY_st (degY);

  coeffs = new Integer [degY];

  for (i = 0; i < degY; i++)
    coeffs[i] = nt_traits.get_coefficient(polyY, i + 1);

  for (i = 0; i < degY; i++)
    coeffsY_st[degY - i - 1] =
        nt_traits.construct_polynomial(coeffs + i, degY - i - 1);

  delete[] coeffs;

  // Compute the resultant of the two bivariate polynomials and obtain
  // a polynomial in s.
  Polynomial            res = _compute_resultant (coeffsX_st, coeffsY_st);

  if (nt_traits.degree (res) < 0)
      return;

  // Compute the roots of the resultant polynomial.
  nt_traits.compute_polynomial_roots (res, std::back_inserter (s_vals));
  return;
}

// ---------------------------------------------------------------------------
// Compute the resultant of two bivariate polynomials.
//
template<class NtTraits>
typename _Bezier_cache<NtTraits>::Polynomial
_Bezier_cache<NtTraits>::_compute_resultant
        (const std::vector<Polynomial>& bp1,
         const std::vector<Polynomial>& bp2) const
{
  // Create the Sylvester matrix of polynomial coefficients. Also prepare
  // the exp_fact vector, that represents the normalization factor (see
  // below).
  const std::size_t        m = bp1.size() - 1;
  const std::size_t        n = bp2.size() - 1;
  const std::size_t        dim = m + n;
  const Integer    zero = 0;
  const Polynomial zero_poly = nt_traits.construct_polynomial (&zero, 0);
  std::size_t              i, j, k;

  std::vector<std::vector<Polynomial> >  mat (dim);
  std::vector <std::size_t>                      exp_fact (dim);

  for (i = 0; i < dim; i++)
  {
    mat[i].resize (dim);
    exp_fact[i] = 0;

    for (j = 0; j < dim; j++)
      mat[i][j] = zero_poly;
  }

  // Initialize it with copies of the two bivariate polynomials.
  for (i = 0; i < n; i++)
    for (j = 0; j <= m; ++j)
      mat[i][i + j] = bp1[j];

  for (i = 0; i < m; i++)
    for (j = 0; j <= n;++j)
      mat[n + i][i + j] = bp2[j];

  // Perform Gaussian elimination on the Sylvester matrix. The goal is to
  // reach an upper-triangular matrix, whose diagonal elements are mat[0][0]
  // to mat[dim-1][dim-1], such that the determinant of the original matrix
  // is given by:
  //
  //              dim-1
  //             *******
  //              *   *  mat[i][i]
  //              *   *
  //               i=0
  //      ---------------------------------
  //         dim-1
  //        *******            exp_fact[i]
  //         *   *  (mat[i][i])
  //         *   *
  //          i=0
  //
  bool             found_row;
  Polynomial       value;
  int              sign_fact = 1;

  for (i = 0; i < dim; i++)
  {
    // Check if the current diagonal value is a zero polynomial.
    if (nt_traits.degree (mat[i][i]) < 0)
    {
      // If the current diagonal value is a zero polynomial, try to replace
      // the current i'th row with a row with a higher index k, such that
      // mat[k][i] is not a zero polynomial.

      found_row = false;
      for (k = i + 1; k < dim; k++)
      {
        if (nt_traits.degree (mat[k][i]) >= 0)
        {
          found_row = true;
          break;
        }
      }

      if (found_row)
      {
        // Swap the i'th and the k'th rows (note that we start from the i'th
        // column, because the first i entries in every row with index i or
        // higher should be zero by now).
        for (j = i; j < dim; j++)
        {
          value = mat[i][j];
          mat[i][j] = mat[k][j];
          mat[k][j] = value;
        }

        // Swapping two rows should change the sign of the determinant.
        // We therefore swap the sign of the normalization factor.
        sign_fact = -sign_fact;
      }
      else
      {
        // In case we could not find a non-zero value, the matrix is
        // singular and its determinant is a zero polynomial.
        return (mat[i][i]);
      }
    }

    // Zero the whole i'th column of the following rows.
    for (k = i + 1; k < dim; k++)
    {
      if (nt_traits.degree (mat[k][i]) >= 0)
      {
        value = mat[k][i];
        mat[k][i] = zero_poly;

        for (j = i + 1; j < dim; j++)
        {
          mat[k][j] = mat[k][j] * mat[i][i] - mat[i][j] * value;
        }

        // We multiplied the current row by the i'th diagonal entry, thus
        // multiplying the determinant value by it. We therefore increment
        // the exponent of mat[i][i] in the normalization factor.
        exp_fact[i] = exp_fact[i] + 1;
      }
    }
  }

  // Now the determinant is simply the product of all diagonal items,
  // divided by the normalizing factor.
  const Integer    sgn (sign_fact);
  Polynomial       det_factor = nt_traits.construct_polynomial (&sgn, 0);
  Polynomial       diag_prod = mat[dim - 1][dim - 1];

  CGAL_assertion (exp_fact [dim - 1] == 0);
  for (i = 0; i+2 <= dim; ++i)
  {
    // Try to avoid unnecessary multiplications by ignoring the current
    // diagonal item if its exponent in the normalization factor is greater
    // than 0.
    if (exp_fact[i] > 0)
    {
      exp_fact[i] = exp_fact[i] - 1;
    }
    else
    {
      diag_prod *= mat[i][i];
    }

    for (j = 0; j < exp_fact[i]; j++)
      det_factor *= mat[i][i];
  }

  // In case of a trivial normalization factor, just return the product
  // of diagonal elements.
  if (nt_traits.degree(det_factor) == 0)
    return (diag_prod);

  // Divide the product of diagonal elements by the normalization factor
  // and obtain the determinant (note that we should have no remainder).
  Polynomial       det, rem;

  det = nt_traits.divide (diag_prod, det_factor, rem);

  CGAL_assertion (nt_traits.degree(rem) < 0);
  return (det);
}

} //namespace CGAL

#endif
