// Copyright (c) 2006,2007,2008,2009,2010,2011 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.6.2/Arrangement_on_surface_2/include/CGAL/Arr_geometry_traits/Conic_arc_2.h $
// $Id: Conic_arc_2.h ccdb5a0b54a 2023-02-15T14:36:44+02:00 Efi Fogel
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s): Ron Wein  <wein@post.tau.ac.il>
//            Efi Fogel <efifogel@gmail.com>

#ifndef CGAL_CONIC_ARC_2_H
#define CGAL_CONIC_ARC_2_H

#include <CGAL/license/Arrangement_on_surface_2.h>

/*! \file
 * Header file for the Conic_arc_2<Int_kernel, Alg_kernel, Nt_traits> class.
 */

#include <ostream>

#include <CGAL/Arr_geometry_traits/Conic_point_2.h>

// The following is included for the benefit of deprecated functions.
#include <CGAL/Arr_geometry_traits/Conic_intersections_2.h>
#include <CGAL/Bbox_2.h>

namespace CGAL {

/*! Representation of a conic arc -- a bounded segment that lies on a conic
 * curve, the loci of all points satisfying the equation:
 *   r*x^2 + s*y^2 + t*xy + u*x + v*y +w = 0
 *
 * The class is templated with three parameters:
 * Rat_kernel A kernel that provides the input objects or coefficients.
 *            Rat_kernel::FT should be an integral or a rational type.
 * Alg_kernel A geometric kernel, where Alg_kernel::FT is the number type
 *            for the coordinates of arrangement vertices, which are algebraic
 *            numbers of degree up to 4 (preferably it is CORE::Expr).
 * Nt_traits A traits class for performing various operations on the integer,
 *           rational and algebraic types.
 */

template <typename RatKernel, typename AlgKernel, typename NtTraits>
class Conic_arc_2 {
public:
  typedef RatKernel                                        Rat_kernel;
  typedef AlgKernel                                        Alg_kernel;
  typedef NtTraits                                         Nt_traits;

  typedef Conic_arc_2<Rat_kernel, Alg_kernel, Nt_traits>   Self;

  typedef typename Rat_kernel::FT                          Rational;
  typedef typename Rat_kernel::Point_2                     Rat_point_2;
  typedef typename Rat_kernel::Segment_2                   Rat_segment_2;
  typedef typename Rat_kernel::Circle_2                    Rat_circle_2;

  typedef typename Nt_traits::Integer                      Integer;

  typedef typename Alg_kernel::FT                          Algebraic;
  typedef typename Alg_kernel::Point_2                     Alg_point_2;
  typedef Conic_point_2<Alg_kernel>                        Point_2;

  /*! \struct
   * For arcs whose base is a hyperbola we store the axis (a*x + b*y + c = 0)
   * which separates the two bracnes of the hyperbola. We also store the side
   * (NEGATIVE or POSITIVE) that the arc occupies.
   * In case of line segments connecting two algebraic endpoints, we use this
   * structure two store the coefficients of the line supporting this segment.
   * In this case we set the side field to be ZERO.
   */
  struct Extra_data {
    Algebraic a;
    Algebraic b;
    Algebraic c;
    Sign side;
  };

  // Bit masks for the m_info field.
  enum {
    IS_VALID = 0,
    IS_FULL_CONIC,
    LAST_INFO,
  };

protected:
  Integer m_r;          //
  Integer m_s;          // The coefficients of the supporting conic curve:
  Integer m_t;          //
  Integer m_u;          //
  Integer m_v;          //   r*x^2 + s*y^2 + t*xy + u*x + v*y + w = 0
  Integer m_w;          //

  Orientation m_orient;         // The orientation of the conic.
  int m_info;                   // does the arc represent a full conic curve.
  Point_2 m_source;             // the source of the arc (if not a full curve).
  Point_2 m_target;             // the target of the arc (if not a full curve).
  Extra_data* m_extra_data;     // The extra data stored with the arc
                                // (may be nullptr).

public:
  /// \name Deprecated Constructions.
  //@{

  /*!Construct a conic arc which is the full conic:
   *   C: r*x^2 + s*y^2 + t*xy + u*x + v*y + w = 0
   * \pre The conic C must be an ellipse (so 4rs - t^2 > 0).
   */
  CGAL_DEPRECATED Conic_arc_2(const Rational& r, const Rational& s,
                              const Rational& t, const Rational& u,
                              const Rational& v, const Rational& w) :
    m_info(0),
    m_extra_data(nullptr)
  {
    // Ensure that the given curve is an ellipse (4rs - t^2 is positive).
    CGAL_precondition(CGAL::sign(4*r*s - t*t) == POSITIVE);

    // Set the arc to be the full conic (and compute the orientation).
    Rational rat_coeffs[6];
    rat_coeffs[0] = r;
    rat_coeffs[1] = s;
    rat_coeffs[2] = t;
    rat_coeffs[3] = u;
    rat_coeffs[4] = v;
    rat_coeffs[5] = w;
    _set_full(rat_coeffs, true);
  }

  /*! Construct a conic arc which lies on the conic:
   *   C: r*x^2 + s*y^2 + t*xy + u*x + v*y + w = 0
   * \param orient The orientation of the arc (clockwise or counterclockwise).
   * \param source The source point.
   * \param target The target point.
   * \pre The source and the target must be on the conic boundary and must
   * not be the same.
   */
  CGAL_DEPRECATED
  Conic_arc_2(const Rational& r, const Rational& s, const Rational& t,
              const Rational& u, const Rational& v, const Rational& w,
              const Orientation& orient,
              const Point_2& source, const Point_2& target) :
    m_orient(orient),
    m_source(source),
    m_target(target),
    m_extra_data(nullptr)
  {
    // Make sure that the source and the target are not the same.
    CGAL_precondition(Alg_kernel().compare_xy_2_object()(source, target) !=
                      EQUAL);

    // Set the arc properties (no need to compute the orientation).
    Rational rat_coeffs[6] = {r, s, t, u, v, w};
    _set(rat_coeffs);
  }

  /*! Construct a segment conic arc from two endpoints.
   * \param source the source point with rational coordinates.
   */
  CGAL_DEPRECATED Conic_arc_2(const Point_2& source, const Point_2& target) :
    m_orient(COLLINEAR),
    m_info(0),
    m_source(source),
    m_target(target),
    m_extra_data(nullptr)
  {
    set_flag(IS_VALID);
    CGAL_precondition(Alg_kernel().compare_xy_2_object()(source, target) !=
                      EQUAL);

    // Compose the equation of the underlying line.
    const Algebraic x1 = source.x();
    const Algebraic y1 = source.y();
    const Algebraic x2 = target.x();
    const Algebraic y2 = target.y();

    // The supporting line is A*x + B*y + C = 0, where:
    //  A = y2 - y1,    B = x1 - x2,    C = x2*y1 - x1*y2
    // We use the extra data field to store the equation of this line.
    m_extra_data = new Extra_data;
    m_extra_data->a = y2 - y1;
    m_extra_data->b = x1 - x2;
    m_extra_data->c = x2*y1 - x1*y2;
    m_extra_data->side = ZERO;
  }

  /*! Construct a conic arc from the given line segment.
   * \param seg The line segment with rational endpoints.
   */
  CGAL_DEPRECATED Conic_arc_2 (const Rat_segment_2& seg) :
    m_orient(COLLINEAR),
    m_info(0),
    m_extra_data(nullptr)
  {
    // Set the source and target.
    Rat_kernel ker;
    Rat_point_2 source = ker.construct_vertex_2_object()(seg, 0);
    Rat_point_2 target = ker.construct_vertex_2_object()(seg, 1);
    Rational x1 = source.x();
    Rational y1 = source.y();
    Rational x2 = target.x();
    Rational y2 = target.y();
    Nt_traits nt_traits;

    m_source = Point_2(nt_traits.convert (x1), nt_traits.convert(y1));
    m_target = Point_2(nt_traits.convert (x2), nt_traits.convert(y2));

    // Make sure that the source and the target are not the same.
    CGAL_precondition (Alg_kernel().compare_xy_2_object()(m_source, m_target)
                       != EQUAL);

    // The supporting conic is r=s=t=0, and u*x + v*y + w = 0 should hold
    // for both the source (x1,y1) and the target (x2, y2).
    const Rational zero (0);
    const Rational one (1);
    Rational rat_coeffs[6];

    rat_coeffs[0] = zero;
    rat_coeffs[1] = zero;
    rat_coeffs[2] = zero;

    if (CGAL::compare(x1, x2) == EQUAL) {
      // The supporting conic is a vertical line, of the form x = CONST.
      rat_coeffs[3] = one;
      rat_coeffs[4] = zero;
      rat_coeffs[5] = -x1;
    }
    else {
      // The supporting line is A*x + B*y + C = 0, where:
      //
      //  A = y2 - y1,    B = x1 - x2,    C = x2*y1 - x1*y2
      //
      rat_coeffs[3] = y2 - y1;
      rat_coeffs[4] = x1 - x2;
      rat_coeffs[5] = x2*y1 - x1*y2;
    }

    // Set the arc properties (no need to compute the orientation).
    _set(rat_coeffs);
  }

  /*! Set a circular arc that corresponds to a full circle.
   * \param circ The circle (with rational center and rational squared radius).
   */
  CGAL_DEPRECATED Conic_arc_2(const Rat_circle_2& circ) :
    m_orient(CLOCKWISE),
    m_info(0),
    m_extra_data(nullptr)
  {
    // Get the circle properties.
    Rat_kernel ker;
    Rat_point_2 center = ker.construct_center_2_object()(circ);
    Rational x0 = center.x();
    Rational y0 = center.y();
    Rational R_sqr = ker.compute_squared_radius_2_object()(circ);

    // Produce the corresponding conic: if the circle center is (x0,y0)
    // and its squared radius is R^2, that its equation is:
    //   x^2 + y^2 - 2*x0*x - 2*y0*y + (x0^2 + y0^2 - R^2) = 0
    // Note that this equation describes a curve with a negative (clockwise)
    // orientation.
    const Rational zero(0);
    const Rational one (1);
    const Rational minus_two(-2);
    Rational rat_coeffs[6];

    rat_coeffs[0] = one;
    rat_coeffs[1] = one;
    rat_coeffs[2] = zero;
    rat_coeffs[3] = minus_two*x0;
    rat_coeffs[4] = minus_two*y0;
    rat_coeffs[5] = x0*x0 + y0*y0 - R_sqr;

    // Set the arc to be the full conic (no need to compute the orientation).
    _set_full(rat_coeffs, false);
  }

  /*! Set a circular arc that lies on the given circle:
   *   C: (x - x0)^2 + (y - y0)^2 = R^2
   * \param orient The orientation of the circle.
   * \param source The source point.
   * \param target The target point.
   * \pre The source and the target must be on the conic boundary and must
   *      not be the same.
   */
  CGAL_DEPRECATED
  Conic_arc_2(const Rat_circle_2& circ, const Orientation& orient,
              const Point_2& source, const Point_2& target) :
    m_orient(orient),
    m_info(0),
    m_source(source),
    m_target(target),
    m_extra_data(nullptr)
  {
    // Make sure that the source and the target are not the same.
    CGAL_precondition(Alg_kernel().compare_xy_2_object()(source, target) !=
                      EQUAL);
    CGAL_precondition(orient != COLLINEAR);

    // Get the circle properties.
    Rat_kernel ker;
    Rat_point_2 center = ker.construct_center_2_object()(circ);
    Rational x0 = center.x();
    Rational y0 = center.y();
    Rational R_sqr = ker.compute_squared_radius_2_object()(circ);

    // Produce the corresponding conic: if the circle center is (x0,y0)
    // and it squared radius is R^2, that its equation is:
    //   x^2 + y^2 - 2*x0*x - 2*y0*y + (x0^2 + y0^2 - R^2) = 0
    // Since this equation describes a curve with a negative (clockwise)
    // orientation, we multiply it by -1 if nece_Conic_arc_2 ssary to obtain a
    // positive (counterclockwise) orientation.
    const Rational zero (0);
    Rational rat_coeffs[6];

    if (m_orient == COUNTERCLOCKWISE) {
      const Rational minus_one(-1);
      const Rational two(2);

      rat_coeffs[0] = minus_one;
      rat_coeffs[1] = minus_one;
      rat_coeffs[2] = zero;
      rat_coeffs[3] = two*x0;
      rat_coeffs[4] = two*y0;
      rat_coeffs[5] = R_sqr - x0*x0 - y0*y0;
    }
    else {
      const Rational one(1);
      const Rational minus_two(-2);

      rat_coeffs[0] = one;
      rat_coeffs[1] = one;
      rat_coeffs[2] = zero;
      rat_coeffs[3] = minus_two*x0;
      rat_coeffs[4] = minus_two*y0;
      rat_coeffs[5] = x0*x0 + y0*y0 - R_sqr;
    }

    // Set the arc properties (no need to compute the orientation).
    _set(rat_coeffs);
  }

  /*! Construct a circular arc from the given three points.
   * \param p1 The arc source.
   * \param p2 A point in the interior of the arc.
   * \param p3 The arc target.
   * \pre The three points must not be collinear.
   */
  CGAL_DEPRECATED Conic_arc_2(const Rat_point_2& p1, const Rat_point_2& p2,
                              const Rat_point_2& p3) :
    m_info(0),
    m_extra_data(nullptr)
  {
    // Set the source and target.
    Rational x1 = p1.x();
    Rational y1 = p1.y();
    Rational x2 = p2.x();
    Rational y2 = p2.y();
    Rational x3 = p3.x();
    Rational y3 = p3.y();
    Nt_traits nt_traits;

    m_source = Point_2(nt_traits.convert (x1), nt_traits.convert(y1));
    m_target = Point_2(nt_traits.convert (x3), nt_traits.convert(y3));

    // Make sure that the source and the target are not the same.
    CGAL_precondition(Alg_kernel().compare_xy_2_object()(m_source, m_target) !=
                      EQUAL);

    // Compute the lines: A1*x + B1*y + C1 = 0,
    //               and: A2*x + B2*y + C2 = 0,
    // where:
    const Rational two  = 2;

    const Rational A1 = two*(x1 - x2);
    const Rational B1 = two*(y1 - y2);
    const Rational C1 = y2*y2 - y1*y1 + x2*x2 - x1*x1;

    const Rational A2 = two*(x2 - x3);
    const Rational B2 = two*(y2 - y3);
    const Rational C2 = y3*y3 - y2*y2 + x3*x3 - x2*x2;

    // Compute the coordinates of the intersection point between the
    // two lines, given by (Nx / D, Ny / D), where:
    const Rational Nx = B1*C2 - B2*C1;
    const Rational Ny = A2*C1 - A1*C2;
    const Rational D = A1*B2 - A2*B1;

    // Make sure the three points are not collinear.
    const bool points_collinear = (CGAL::sign(D) == ZERO);

    if (points_collinear) {
      reset_flags();            // inavlid arc
      return;
    }

    // The equation of the underlying circle is given by:
    Rational rat_coeffs[6];
    rat_coeffs[0] = D*D;
    rat_coeffs[1] = D*D;
    rat_coeffs[2] = 0;
    rat_coeffs[3] = -two*D*Nx;
    rat_coeffs[4] = -two*D*Ny;
    rat_coeffs[5] =
      Nx*Nx + Ny*Ny - ((D*x2 - Nx)*(D*x2 - Nx) + (D*y2 - Ny)*(D*y2 - Ny));

    // Determine the orientation: If the mid-point forms a left-turn with
    // the source and the target points, the orientation is positive (going
    // counterclockwise).
    // Otherwise, it is negative (going clockwise).
    Alg_kernel ker;
    typename Alg_kernel::Orientation_2 orient_f = ker.orientation_2_object();

    Point_2 p_mid = Point_2(nt_traits.convert(x2), nt_traits.convert(y2));

    if (orient_f(m_source, p_mid, m_target) == LEFT_TURN)
      m_orient = COUNTERCLOCKWISE;
    else
      m_orient = CLOCKWISE;

    // Set the arc properties (no need to compute the orientation).
    _set(rat_coeffs);
  }

  /*! Construct a conic arc from the given five points, specified by the
   * points p1, p2, p3, p4 and p5.
   * \param p1 The source point of the given arc.
   * \param p2,p3,p4 Points lying on the conic arc, between p1 and p5.
   * \param p5 The target point of the given arc.
   * \pre No three points are collinear.
   */
  CGAL_DEPRECATED Conic_arc_2(const Rat_point_2& p1, const Rat_point_2& p2,
                              const Rat_point_2& p3, const Rat_point_2& p4,
                              const Rat_point_2& p5) :
    m_info(0),
    m_extra_data(nullptr)
  {
    // Make sure that no three points are collinear.
    Rat_kernel ker;
    typename Rat_kernel::Orientation_2 orient_f = ker.orientation_2_object();
    const bool point_collinear =
      (orient_f (p1, p2, p3) == COLLINEAR ||
       orient_f (p1, p2, p4) == COLLINEAR ||
       orient_f (p1, p2, p5) == COLLINEAR ||
       orient_f (p1, p3, p4) == COLLINEAR ||
       orient_f (p1, p3, p5) == COLLINEAR ||
       orient_f (p1, p4, p5) == COLLINEAR ||
       orient_f (p2, p3, p4) == COLLINEAR ||
       orient_f (p2, p3, p5) == COLLINEAR ||
       orient_f (p2, p4, p5) == COLLINEAR ||
       orient_f (p3, p4, p5) == COLLINEAR);

    if (point_collinear) {
      reset_flags();            // inavlid arc
      return;
    }

    // Set the source and target.
    Rational x1 = p1.x();
    Rational y1 = p1.y();
    Rational x5 = p5.x();
    Rational y5 = p5.y();
    Nt_traits nt_traits;

    m_source = Point_2(nt_traits.convert (x1), nt_traits.convert(y1));
    m_target = Point_2(nt_traits.convert (x5), nt_traits.convert(y5));

    // Set a conic curve that passes through the five given point.
    typename Rat_kernel::Conic_2 temp_conic;
    Rational rat_coeffs [6];

    temp_conic.set(p1, p2, p3, p4, p5);

    // Get the conic coefficients.
    rat_coeffs[0] = temp_conic.r();
    rat_coeffs[1] = temp_conic.s();
    rat_coeffs[2] = temp_conic.t();
    rat_coeffs[3] = temp_conic.u();
    rat_coeffs[4] = temp_conic.v();
    rat_coeffs[5] = temp_conic.w();

    // Determine the orientation: If one of the midpoints forms a left-turn
    // with the source and the target points, the orientation is positive
    // (going counterclockwise).
    // Otherwise, it is negative (going clockwise).
    const Orientation turn = orient_f(p1, p2, p5);

    if (turn == LEFT_TURN) {
      m_orient = COUNTERCLOCKWISE;
      CGAL_precondition(orient_f(p1, p3, p5) == LEFT_TURN &&
                        orient_f(p1, p4, p5) == LEFT_TURN);
    }
    else {
      m_orient = CLOCKWISE;
      CGAL_precondition(orient_f(p1, p3, p5) != LEFT_TURN &&
                        orient_f(p1, p4, p5) != LEFT_TURN);
    }

    // Set the arc properties (no need to compute the orientation).
    _set(rat_coeffs);

    // Make sure that all midpoints are strictly between the
    // source and the target.
    Point_2 mp2 = Point_2(nt_traits.convert(p2.x()), nt_traits.convert(p2.y()));
    Point_2 mp3 = Point_2(nt_traits.convert(p3.x()), nt_traits.convert(p3.y()));
    Point_2 mp4 = Point_2(nt_traits.convert(p4.x()), nt_traits.convert(p4.y()));

    if (! _is_strictly_between_endpoints(mp2) ||
        ! _is_strictly_between_endpoints(mp3) ||
        ! _is_strictly_between_endpoints(mp4))
    {
      reset_flags();            // inavlid arc
    }
  }

  /*! Construct a conic arc which lies on the conic:
   *   C: r*x^2 + s*y^2 + t*xy + u*x + v*y + w = 0
   * The source and the target are specified by the intersection of the
   * conic with:
   *   C_1: r_1*x^2 + s_1*y^2 + t_1*xy + u_1*x + v_1*y + w_1 = 0
   *   C_2: r_2*x^2 + s_2*y^2 + t_2*xy + u_2*x + v_2*y + w_2 = 0
   * The user must also specify the source and the target with approximated
   * coordinates. The actual intersection points that best fits the source
   * (or the target) will be selected.
   */
  CGAL_DEPRECATED
  Conic_arc_2(const Rational& r, const Rational& s, const Rational& t,
              const Rational& u, const Rational& v, const Rational& w,
              const Orientation& orient,
              const Point_2& app_source,
              const Rational& r_1, const Rational& s_1, const Rational& t_1,
              const Rational& u_1, const Rational& v_1, const Rational& w_1,
              const Point_2& app_target,
              const Rational& r_2, const Rational& s_2, const Rational& t_2,
              const Rational& u_2, const Rational& v_2, const Rational& w_2):
    m_orient(orient),
    m_info(0),
    m_extra_data(nullptr)
  {
    // Create the integer coefficients of the base conic.
    Nt_traits nt_traits;
    Integer base_coeffs[6];
    int deg_base;
    Rational rat_coeffs [6] = {r, s, t, u, v, w};
    nt_traits.convert_coefficients(rat_coeffs, rat_coeffs + 6, base_coeffs);

    if (CGAL::sign (base_coeffs[0]) == ZERO &&
        CGAL::sign (base_coeffs[1]) == ZERO &&
        CGAL::sign (base_coeffs[2]) == ZERO) {
      deg_base = 1;
    }
    else {
      deg_base = 2;
    }

    // Compute the endpoints.
    Rational aux_rat_coeffs [6];
    Integer aux_coeffs[6];
    int  deg_aux;
    Algebraic xs[4];
    int n_xs;
    Algebraic ys[4];
    int n_ys;
    int i, j;
    Algebraic val;
    bool found;
    double dx, dy;
    double curr_dist;
    double min_dist = -1;
    int k;

    for (k = 1; k <= 2; ++k) {
      // Get the integer coefficients of the k'th auxiliary conic curve.
      aux_rat_coeffs[0] = (k == 1) ? r_1 : r_2;
      aux_rat_coeffs[1] = (k == 1) ? s_1 : s_2;
      aux_rat_coeffs[2] = (k == 1) ? t_1 : t_2;
      aux_rat_coeffs[3] = (k == 1) ? u_1 : u_2;
      aux_rat_coeffs[4] = (k == 1) ? v_1 : v_2;
      aux_rat_coeffs[5] = (k == 1) ? w_1 : w_2;

      nt_traits.convert_coefficients(aux_rat_coeffs, aux_rat_coeffs + 6,
                                     aux_coeffs);

      if (CGAL::sign (aux_coeffs[0]) == ZERO &&
          CGAL::sign (aux_coeffs[1]) == ZERO &&
          CGAL::sign (aux_coeffs[2]) == ZERO) {
        deg_aux = 1;
      }
      else {
        deg_aux = 2;
      }

      // Compute the x- and y-coordinates of intersection points of the base
      // conic and the k'th auxiliary conic.
      n_xs = compute_resultant_roots(nt_traits,
                                     base_coeffs[0], base_coeffs[1],
                                     base_coeffs[2],
                                     base_coeffs[3], base_coeffs[4],
                                     base_coeffs[5],
                                     deg_base,
                                     aux_coeffs[0], aux_coeffs[1],
                                     aux_coeffs[2],
                                     aux_coeffs[3], aux_coeffs[4],
                                     aux_coeffs[5],
                                     deg_aux,
                                     xs);

      n_ys = compute_resultant_roots(nt_traits,
                                     base_coeffs[1], base_coeffs[0],
                                     base_coeffs[2],
                                     base_coeffs[4], base_coeffs[3],
                                     base_coeffs[5],
                                     deg_base,
                                     aux_coeffs[1], aux_coeffs[0],
                                     aux_coeffs[2],
                                     aux_coeffs[4], aux_coeffs[3],
                                     aux_coeffs[5],
                                     deg_aux,
                                     ys);

      // Find the intersection point which is nearest the given approximation
      // and set it as the endpoint.
      found = false;
      for (i = 0; i < n_xs; ++i) {
        for (j = 0; j < n_ys; ++j) {
          // Check if the point (xs[i], ys[j]) lies on both conics.
          val = nt_traits.convert(base_coeffs[0]) * xs[i]*xs[i] +
                nt_traits.convert(base_coeffs[1]) * ys[j]*ys[j] +
                nt_traits.convert(base_coeffs[2]) * xs[i]*ys[j] +
                nt_traits.convert(base_coeffs[3]) * xs[i] +
                nt_traits.convert(base_coeffs[4]) * ys[j] +
                nt_traits.convert(base_coeffs[5]);

          if (CGAL::sign (val) != ZERO) continue;

          val = nt_traits.convert(aux_coeffs[0]) * xs[i]*xs[i] +
                nt_traits.convert(aux_coeffs[1]) * ys[j]*ys[j] +
                nt_traits.convert(aux_coeffs[2]) * xs[i]*ys[j] +
                nt_traits.convert(aux_coeffs[3]) * xs[i] +
                nt_traits.convert(aux_coeffs[4]) * ys[j] +
                nt_traits.convert(aux_coeffs[5]);

          if (CGAL::sign (val) == ZERO) {
            // Compute the distance of (xs[i], ys[j]) from the approximated
            // endpoint.
            if (k == 1) {
              dx = CGAL::to_double (xs[i] - app_source.x());
              dy = CGAL::to_double (ys[j] - app_source.y());
            }
            else {
              dx = CGAL::to_double (xs[i] - app_target.x());
              dy = CGAL::to_double (ys[j] - app_target.y());
            }

            curr_dist = dx*dx + dy*dy;

            // Update the endpoint if (xs[i], ys[j]) is the nearest pair so
            // far.
            if (! found || curr_dist < min_dist) {
              if (k == 1) m_source = Point_2 (xs[i], ys[j]);
              else m_target = Point_2 (xs[i], ys[j]);
              min_dist = curr_dist;
              found = true;
            }
          }
        }
      }

      if (! found) {
        reset_flags();          // inavlid arc
        return;
      }
    }

    // Make sure that the source and the target are not the same.
    if (Alg_kernel().compare_xy_2_object()(m_source, m_target) == EQUAL) {
      reset_flags();            // inavlid arc
      return;
    }

    // Set the arc properties (no need to compute the orientation).
    _set(rat_coeffs);
  }

  /*! Get a bounding box for the conic arc.
   * \return The bounding box.
   */
  CGAL_DEPRECATED Bbox_2 bbox() const {
    CGAL_precondition (is_valid());

    double x_min(0), y_min(0);
    double x_max(0), y_max(0);

    if (is_full_conic()) {
      // In case of a full conic (an ellipse or a circle), compute the
      // horizontal and vertical tangency points and use them to bound the arc.
      Point_2 tan_ps[2];
      CGAL_assertion_code(int n_tan_ps);

      CGAL_assertion_code(n_tan_ps = vertical_tangency_points(tan_ps));
      CGAL_assertion(n_tan_ps == 2);

      if (CGAL::to_double(tan_ps[0].x()) < CGAL::to_double(tan_ps[1].x())) {
        x_min = CGAL::to_double(tan_ps[0].x());
        x_max = CGAL::to_double(tan_ps[1].x());
      }
      else {
        x_min = CGAL::to_double(tan_ps[1].x());
        x_max = CGAL::to_double(tan_ps[0].x());
      }

      CGAL_assertion_code(n_tan_ps = horizontal_tangency_points(tan_ps));
      CGAL_assertion(n_tan_ps == 2);

      if (CGAL::to_double(tan_ps[0].y()) < CGAL::to_double(tan_ps[1].y())) {
        y_min = CGAL::to_double(tan_ps[0].y());
        y_max = CGAL::to_double(tan_ps[1].y());
      }
      else {
        y_min = CGAL::to_double(tan_ps[1].y());
        y_max = CGAL::to_double(tan_ps[0].y());
      }
    }
    else {
      // Use the source and target to initialize the exterme points.
      bool   source_left =
        CGAL::to_double(m_source.x()) < CGAL::to_double(m_target.x());
      x_min = source_left ?
        CGAL::to_double(m_source.x()) : CGAL::to_double(m_target.x());
      x_max = source_left ?
        CGAL::to_double(m_target.x()) : CGAL::to_double(m_source.x());

      bool source_down =
        CGAL::to_double(m_source.y()) < CGAL::to_double(m_target.y());
      y_min = source_down ?
        CGAL::to_double(m_source.y()) : CGAL::to_double(m_target.y());
      y_max = source_down ?
        CGAL::to_double(m_target.y()) : CGAL::to_double(m_source.y());

      // Go over the vertical tangency points and try to update the x-points.
      Alg_point_2 tan_ps[2];
      int n_tan_ps;
      int i;

      n_tan_ps = vertical_tangency_points(tan_ps);
      for (i = 0; i < n_tan_ps; ++i) {
        if (CGAL::to_double(tan_ps[i].x()) < x_min)
          x_min = CGAL::to_double(tan_ps[i].x());
        if (CGAL::to_double(tan_ps[i].x()) > x_max)
          x_max = CGAL::to_double(tan_ps[i].x());
      }

      // Go over the horizontal tangency points and try to update the y-points.
      n_tan_ps = horizontal_tangency_points(tan_ps);
      for (i = 0; i < n_tan_ps; ++i) {
        if (CGAL::to_double(tan_ps[i].y()) < y_min)
          y_min = CGAL::to_double(tan_ps[i].y());
        if (CGAL::to_double(tan_ps[i].y()) > y_max)
          y_max = CGAL::to_double(tan_ps[i].y());
      }
    }

    // Return the resulting bounding box.
    return Bbox_2(x_min, y_min, x_max, y_max);
  }

  //@}

protected:
  /*! Calculate the vertical tangency points of the arc.
   * \param vpts The vertical tangency points.
   * \pre The vpts vector should be allocated at the size of 2.
   * \return The number of vertical tangency points.
   */
  CGAL_DEPRECATED int vertical_tangency_points(Alg_point_2* vpts) const {
    // No vertical tangency points for line segments:
    if (m_orient == COLLINEAR) return 0;

    // Calculate the vertical tangency points of the supporting conic.
    Alg_point_2 ps[2];
    int n = _conic_vertical_tangency_points(ps);

    // Return only the points that are contained in the arc interior.
    int m(0);

    for (int i = 0; i < n; ++i) {
      if (is_full_conic() || _is_strictly_between_endpoints(ps[i])) {
        vpts[m] = ps[i];
        ++m;
      }
    }

    // Return the number of vertical tangency points found.
    CGAL_assertion(m <= 2);
    return m;
  }

  /*! Calculate the horizontal tangency points of the arc.
   * \param hpts The horizontal tangency points.
   * \pre The hpts vector should be allocated at the size of 2.
   * \return The number of horizontal tangency points.
   */
  CGAL_DEPRECATED int horizontal_tangency_points(Alg_point_2* hpts) const {
    // No horizontal tangency points for line segments:
    if (m_orient == COLLINEAR) return 0;

    // Calculate the horizontal tangency points of the conic.
    Alg_point_2 ps[2];
    int n = _conic_horizontal_tangency_points(ps);

    // Return only the points that are contained in the arc interior.
    int m = 0;

    for (int i = 0; i < n; ++i) {
      if (is_full_conic() || _is_strictly_between_endpoints(ps[i])) {
        hpts[m] = ps[i];
        ++m;
      }
    }

    // Return the number of horizontal tangency points found.
    CGAL_assertion(m <= 2);
    return m;
  }

  /*! Check whether the given point is strictly between the source and the
   * target (but not any of them).
   * The point is assumed to be on the conic's boundary.
   * \param p The query point.
   * \return (true) if the point is strictly between the two endpoints,
   *         (false) if it is not.
   */
  CGAL_DEPRECATED bool _is_strictly_between_endpoints(const Point_2& p) const {
    // In case this is a full conic, any point on its boundary is between
    // its end points.
    if (is_full_conic()) return true;

    // Check if we have extra data available.
    if (m_extra_data != nullptr) {
      if (m_extra_data->side != ZERO) {
        // In case of a hyperbolic arc, make sure the point is located on the
        // same branch as the arc.
        if (sign_of_extra_data(p.x(), p.y()) != m_extra_data->side)
          return false;
      }
      else {
        // In case we have a segment of a line pair, make sure that p really
        // satisfies the equation of the line.
        if (sign_of_extra_data(p.x(), p.y()) != ZERO) return false;
      }
    }

    // Act according to the conic degree.
    Alg_kernel ker;

    if (m_orient == COLLINEAR) {
      Comparison_result res1;
      Comparison_result res2;

      if (ker.compare_x_2_object()(m_source, m_target) == EQUAL) {
        // In case of a vertical segment - just check whether the y coordinate
        // of p is between those of the source's and of the target's.
        res1 = ker.compare_y_2_object()(p, m_source);
        res2 = ker.compare_y_2_object()(p, m_target);
      }
      else {
        // Otherwise, since the segment is x-monotone, just check whether the
        // x coordinate of p is between those of the source's and of the
        // target's.
        res1 = ker.compare_x_2_object()(p, m_source);
        res2 = ker.compare_x_2_object()(p, m_target);
      }

      // If p is not in the (open) x-range (or y-range) of the segment, it
      // cannot be contained in the segment.
      if (res1 == EQUAL || res2 == EQUAL || res1 == res2) return false;

      // Perform an orientation test: This is crucial for segment of line
      // pairs, as we want to make sure that p lies on the same line as the
      // source and the target.
      return (ker.orientation_2_object()(m_source, p, m_target) == COLLINEAR);
    }
    else {
      // In case of a conic of degree 2, make a decision based on the conic's
      // orientation and whether (source,p,target) is a right or a left turn.
      if (m_orient == COUNTERCLOCKWISE)
        return (ker.orientation_2_object()(m_source, p, m_target) == LEFT_TURN);
      else
        return (ker.orientation_2_object()(m_source, p, m_target) == RIGHT_TURN);
    }
  }

  /*! Find the vertical tangency points of the undelying conic.
   * \param ps The output points of vertical tangency.
   *           This area must be allocated at the size of 2.
   * \return The number of vertical tangency points.
   */
  CGAL_DEPRECATED
  int _conic_vertical_tangency_points(Alg_point_2* ps) const {
    const auto& cv = *this;
    Nt_traits nt_traits;

    // In case the base conic is of degree 1 (and not 2), the arc has no
    // vertical tangency points.
    if (CGAL::sign(cv.s()) == ZERO) return 0;

    // We are interested in the x coordinates where the quadratic equation:
    //  s*y^2 + (t*x + v)*y + (r*x^2 + u*x + w) = 0
    // has a single solution (obviously if s = 0, there are no such points).
    // We therefore demand that the discriminant of this equation is zero:
    //  (t*x + v)^2 - 4*s*(r*x^2 + u*x + w) = 0
    const Integer two(2);
    const Integer four(4);
    Algebraic xs[2];

    auto r = cv.r();
    auto s = cv.s();
    auto t = cv.t();
    auto u = cv.u();
    auto v = cv.v();
    auto w = cv.w();
    Algebraic* xs_end = nt_traits.solve_quadratic_equation(t*t - four*r*s,
                                                           two*t*v - four*s*u,
                                                           v*v - four*s*w,
                                                           xs);
    auto n_xs = static_cast<int>(xs_end - xs);

    // Find the y-coordinates of the vertical tangency points.
    Algebraic ys[2];
    Algebraic* ys_end;
    int n_ys;

    if (CGAL::sign(cv.t()) == ZERO) {
      // The two vertical tangency points have the same y coordinate:
      ys[0] = nt_traits.convert(-v) / nt_traits.convert(two*s);
      n_ys = 1;
    }
    else {
      ys_end = nt_traits.solve_quadratic_equation(four*r*s*s - s*t*t,
                                                  four*r*s*v - two*s*t*u,
                                                  r*v*v - t*u*v +
                                                  t*t*w,
                                                  ys);
      n_ys = static_cast<int>(ys_end - ys);
    }

    // Pair the x and y coordinates and obtain the vertical tangency points.
    int n(0);

    for (int i = 0; i < n_xs; ++i) {
      if (n_ys == 1) {
        ps[n++] = Point_2(xs[i], ys[0]);
      }
      else {
        for (int j = 0; j < n_ys; ++j) {
          if (CGAL::compare(nt_traits.convert(two*s) * ys[j],
                            -(nt_traits.convert(t) * xs[i] +
                              nt_traits.convert(v))) == EQUAL)
          {
            ps[n++] = Point_2(xs[i], ys[j]);
            break;
          }
        }
      }
    }

    CGAL_assertion(n <= 2);
    return n;
  }

  /*! Find the horizontal tangency points of the undelying conic.
   * \param ps The output points of horizontal tangency.
   *           This area must be allocated at the size of 2.
   * \return The number of horizontal tangency points.
   */
  CGAL_DEPRECATED
  size_t _conic_horizontal_tangency_points(Alg_point_2* ps) const {
    const auto& cv = *this;
    Nt_traits nt_traits;

    const Integer zero(0);

    // In case the base conic is of degree 1 (and not 2), the arc has no
    // vertical tangency points.
    if (CGAL::sign(cv.r()) == ZERO) return 0;

    // We are interested in the y coordinates were the quadratic equation:
    //  r*x^2 + (t*y + u)*x + (s*y^2 + v*y + w) = 0
    // has a single solution (obviously if r = 0, there are no such points).
    // We therefore demand that the discriminant of this equation is zero:
    //  (t*y + u)^2 - 4*r*(s*y^2 + v*y + w) = 0
    const Integer two(2);
    const Integer four(4);
    Algebraic ys[2];

    auto r = cv.r();
    auto s = cv.s();
    auto t = cv.t();
    auto u = cv.u();
    auto v = cv.v();
    auto w = cv.w();
    Algebraic* ys_end = nt_traits.solve_quadratic_equation(t*t - four*r*s,
                                                           two*t*u - four*r*v,
                                                           u*u - four*r*w,
                                                           ys);
    auto n = static_cast<int>(ys_end - ys);

    // Compute the x coordinates and construct the horizontal tangency points.
    for (int i = 0; i < n; ++i) {
      // Having computed y, x is the single solution to the quadratic equation
      // above, and since its discriminant is 0, x is simply given by:
      Algebraic x = -(nt_traits.convert(t)*ys[i] + nt_traits.convert(u)) /
        nt_traits.convert(two*r);
      ps[i] = Point_2(x, ys[i]);
    }

    CGAL_assertion(n <= 2);
    return n;
  }

  /*! Set the properties of a conic arc that is really a full curve
   * (that is, an ellipse).
   * \param rat_coeffs A vector of size 6, storing the rational coefficients
   *                   of x^2, y^2, xy, x, y and the free coefficient resp.
   * \param comp_orient Should we compute the orientation of the given curve.
   */
  CGAL_DEPRECATED
  void _set_full(const Rational* rat_coeffs, const bool& comp_orient) {
    // Convert the coefficients vector to an equivalent vector of integer
    // coefficients.
    Nt_traits nt_traits;
    Integer int_coeffs[6];

    nt_traits.convert_coefficients(rat_coeffs, rat_coeffs + 6, int_coeffs);

    // Check the orientation of conic curve, and negate the conic coefficients
    // if its given orientation.
    typename Rat_kernel::Conic_2 temp_conic(rat_coeffs[0], rat_coeffs[1],
                                            rat_coeffs[2], rat_coeffs[3],
                                            rat_coeffs[4], rat_coeffs[5]);
    const Orientation temp_orient = temp_conic.orientation();

    if (comp_orient) set_orientation(temp_orient);

    if (orientation() == temp_orient) {
      m_r = int_coeffs[0];
      m_s = int_coeffs[1];
      m_t = int_coeffs[2];
      m_u = int_coeffs[3];
      m_v = int_coeffs[4];
      m_w = int_coeffs[5];
    }
    else {
      m_r = -int_coeffs[0];
      m_s = -int_coeffs[1];
      m_t = -int_coeffs[2];
      m_u = -int_coeffs[3];
      m_v = -int_coeffs[4];
      m_w = -int_coeffs[5];
    }

    // Make sure the conic is a non-degenerate ellipse:
    // The coefficients should satisfy (4rs - t^2) > 0.
    const bool is_ellipse = (CGAL::sign(4*m_r*m_s - m_t*m_t) == POSITIVE);
    CGAL_assertion(is_ellipse);

    // Mark that this arc is a full conic curve.
    if (is_ellipse) {
      set_flag(IS_VALID);
      set_flag(IS_FULL_CONIC);
    }
    else reset_flags();         // inavlid arc
  }

  /*! Check whether the given point lies on the supporting conic of the arc.
   * \param p The query point.
   * \return (true) if p lies on the supporting conic; (false) otherwise.
   */
  CGAL_DEPRECATED bool _is_on_supporting_conic(const Point_2& p) const {
    // Check whether p satisfies the conic equation.
    // The point must satisfy: r*x^2 + s*y^2 + t*xy + u*x + v*y + w = 0.
    Nt_traits nt_traits;
    const Algebraic val =
      (nt_traits.convert(m_r)*p.x() +
       nt_traits.convert(m_t)*p.y() +
       nt_traits.convert(m_u)) * p.x() +
      (nt_traits.convert(m_s)*p.y() +
       nt_traits.convert(m_v)) * p.y() +
      nt_traits.convert(m_w);

    return (CGAL::sign(val) == ZERO);
  }

  /*! Build the data for hyperbolic arc, contaning the characterization of the
   * hyperbolic branch the arc is placed on.
   */
  CGAL_DEPRECATED void _build_hyperbolic_arc_data() {
    // Let phi be the rotation angle of the conic from its canonic form.
    // We can write:
    //
    //                          t
    //  sin(2*phi) = -----------------------
    //                sqrt((r - s)^2 + t^2)
    //
    //                        r - s
    //  cos(2*phi) = -----------------------
    //                sqrt((r - s)^2 + t^2)
    //
    Nt_traits nt_traits;
    const int or_fact = (m_orient == CLOCKWISE) ? -1 : 1;
    const Algebraic r = nt_traits.convert(or_fact * m_r);
    const Algebraic s = nt_traits.convert(or_fact * m_s);
    const Algebraic t = nt_traits.convert(or_fact * m_t);
    const Algebraic cos_2phi = (r - s) / nt_traits.sqrt((r-s)*(r-s) + t*t);
    const Algebraic zero = 0;
    const Algebraic one = 1;
    const Algebraic two = 2;
    Algebraic sin_phi;
    Algebraic cos_phi;

    // Calculate sin(phi) and cos(phi) according to the half-angle formulae:
    //
    //  sin(phi)^2 = 0.5 * (1 - cos(2*phi))
    //  cos(phi)^2 = 0.5 * (1 + cos(2*phi))
    Sign sign_t = CGAL::sign(t);

    if (sign_t == ZERO) {
      // sin(2*phi) == 0, so phi = 0 or phi = PI/2
      if (CGAL::sign(cos_2phi) == POSITIVE) {
        // phi = 0.
        sin_phi = zero;
        cos_phi = one;
      }
      else {
        // phi = PI/2.
        sin_phi = one;
        cos_phi = zero;
      }
    }
    else if (sign_t == POSITIVE) {
      // sin(2*phi) > 0 so 0 < phi < PI/2.
      sin_phi = nt_traits.sqrt((one + cos_2phi) / two);
      cos_phi = nt_traits.sqrt((one - cos_2phi) / two);
    }
    else
    {
      // sin(2*phi) < 0 so PI/2 < phi < PI.
      sin_phi = nt_traits.sqrt((one + cos_2phi) / two);
      cos_phi = -nt_traits.sqrt((one - cos_2phi) / two);
    }

    // Calculate the center (x0, y0) of the conic, given by the formulae:
    //
    //        t*v - 2*s*u                t*u - 2*r*v
    //  x0 = -------------   ,     y0 = -------------
    //        4*r*s - t^2                4*r*s - t^2
    //
    // The denominator (4*r*s - t^2) must be negative for hyperbolas.
    const Algebraic u = nt_traits.convert(or_fact * m_u);
    const Algebraic v = nt_traits.convert(or_fact * m_v);
    const Algebraic det = 4*r*s - t*t;
    Algebraic x0, y0;

    CGAL_assertion (CGAL::sign(det) == NEGATIVE);

    x0 = (t*v - two*s*u) / det;
    y0 = (t*u - two*r*v) / det;

    // The axis separating the two branches of the hyperbola is now given by:
    //
    //  cos(phi)*x + sin(phi)*y - (cos(phi)*x0 + sin(phi)*y0) = 0
    //
    // We store the equation of this line in the extra data structure and also
    // the sign (side of half-plane) our arc occupies with respect to the line.
    m_extra_data = new Extra_data;

    m_extra_data->a = cos_phi;
    m_extra_data->b = sin_phi;
    m_extra_data->c = - (cos_phi*x0 + sin_phi*y0);

    // Make sure that the two endpoints are located on the same branch
    // of the hyperbola.
    m_extra_data->side = sign_of_extra_data(m_source.x(), m_source.y());

    CGAL_assertion(m_extra_data->side != ZERO);
    CGAL_assertion(m_extra_data->side ==
                   sign_of_extra_data(m_target.x(), m_target.y()));
  }

  /*! Check whether the given point is between the source and the target.
   * The point is assumed to be on the conic's boundary.
   * \param p The query point.
   * \return (true) if the point is between the two endpoints,
   *         (false) if it is not.
   */
  CGAL_DEPRECATED bool _is_between_endpoints(const Point_2& p) const {
    CGAL_precondition(! is_full_conic());

    // Check if p is one of the endpoints.
    Alg_kernel ker;
    if (ker.equal_2_object()(p, m_source) ||
        ker.equal_2_object()(p, m_target))
      return true;
    else return _is_strictly_between_endpoints(p);
  }

  /*! Find the y coordinates of the underlying conic at a given x coordinate.
   * \param x The x coordinate.
   * \param ys The output y coordinates.
   * \pre The vector ys must be allocated at the size of 2.
   * \return The number of y coordinates computed (either 0, 1 or 2).
   */
  CGAL_DEPRECATED
  int _conic_get_y_coordinates(const Algebraic& x, Algebraic* ys) const {
    // Solve the quadratic equation for a given x and find the y values:
    //  s*y^2 + (t*x + v)*y + (r*x^2 + u*x + w) = 0
    Nt_traits nt_traits;
    Algebraic A = nt_traits.convert(m_s);
    Algebraic B = nt_traits.convert(m_t)*x + nt_traits.convert(m_v);
    Algebraic C = (nt_traits.convert(m_r)*x +
                   nt_traits.convert(m_u))*x + nt_traits.convert(m_w);
    return (_solve_quadratic_equation(A, B, C, ys[0], ys[1]));
  }

  /*! Find all points on the arc with a given x-coordinate.
   * \param p A placeholder for the x-coordinate.
   * \param ps The point on the arc at x(p).
   * \pre The vector ps should be allocated at the size of 2.
   * \return The number of points found.
   */
  CGAL_DEPRECATED int _points_at_x(const Point_2& p, Point_2* ps) const {
    // Get the y coordinates of the points on the conic.
    Algebraic ys[2];
    int n = _conic_get_y_coordinates(p.x(), ys);

    // Find all the points that are contained in the arc.
    int m(0);

    for (int i = 0; i < n; ++i) {
      ps[m] = Point_2 (p.x(), ys[i]);
      if (is_full_conic() || _is_between_endpoints(ps[m])) ++m;
    }

    // Return the number of points on the arc.
    CGAL_assertion(m <= 2);
    return m;
  }

  /*! Solve the given quadratic equation: Ax^2 + B*x + C = 0.
   * \param x_minus The root obtained from taking -sqrt(discriminant).
   * \param x_plus The root obtained from taking -sqrt(discriminant).
   * \return The number of disticnt solutions to the equation.
   */
  CGAL_DEPRECATED int _solve_quadratic_equation(const Algebraic& A,
                                                const Algebraic& B,
                                                const Algebraic& C,
                                                Algebraic& x_minus,
                                                Algebraic& x_plus) const {
    // Check if we actually have a linear equation.
    if (CGAL::sign(A) == ZERO) {
      if (CGAL::sign(B) == ZERO) return (0);
      x_minus = x_plus = -C / B;
      return 1;
    }

    // Compute the discriminant and act according to its sign.
    const Algebraic disc = B*B - 4*A*C;
    Sign sign_disc = CGAL::sign(disc);

    if (sign_disc == NEGATIVE) {
      // No real-valued solutions:
      return 0;
    }
    else if (sign_disc == ZERO) {
      // One distinct solution:
      x_minus = x_plus = -B / (2*A);
      return 1;
    }

    // Compute the two distinct solutions:
    Algebraic _2A = 2*A;
    Nt_traits nt_traits;
    Algebraic sqrt_disc = nt_traits.sqrt (disc);

    x_minus = -(B + sqrt_disc) / _2A;
    x_plus = (sqrt_disc - B) / _2A;
    return 2;
  }

  /*! Find the x coordinates of the underlying conic at a given y coordinate.
   * \param y The y coordinate.
   * \param xs The output x coordinates.
   * \pre The vector xs must be allocated at the size of 2.
   * \return The number of x coordinates computed (either 0, 1 or 2).
   */
  CGAL_DEPRECATED
  int _conic_get_x_coordinates(const Algebraic& y, Algebraic* xs) const {
    // Solve the quadratic equation for a given y and find the x values:
    //  r*x^2 + (t*y + u)*x + (s*y^2 + v*y + w) = 0
    Nt_traits nt_traits;
    Algebraic A = nt_traits.convert(m_r);
    Algebraic B = nt_traits.convert(m_t)*y + nt_traits.convert(m_u);
    Algebraic C = (nt_traits.convert(m_s)*y +
                   nt_traits.convert(m_v))*y + nt_traits.convert(m_w);

    return (_solve_quadratic_equation(A, B, C, xs[0], xs[1]));
  }

  /*! Find all points on the arc with a given y-coordinate.
   * \param p A placeholder for the y-coordinate.
   * \param ps The point on the arc at x(p).
   * \pre The vector ps should be allocated at the size of 2.
   * \return The number of points found.
   */
  CGAL_DEPRECATED int _points_at_y(const Point_2& p, Point_2* ps) const {
    // Get the y coordinates of the points on the conic.
    Algebraic xs[2];
    int n = _conic_get_x_coordinates(p.y(), xs);

    // Find all the points that are contained in the arc.
    int m(0);

    for (int i = 0; i < n; ++i) {
      ps[m] = Point_2 (xs[i], p.y());
      if (is_full_conic() || _is_between_endpoints(ps[m])) ++m;
    }

    // Return the number of points on the arc.
    CGAL_assertion(m <= 2);
    return m;
  }

  /*! Set the properties of a conic arc (for the usage of the constructors).
   * \param rat_coeffs A vector of size 6, storing the rational coefficients
   *                   of x^2, y^2, xy, x, y and the free coefficient resp.
   */
  CGAL_DEPRECATED void _set(const Rational* rat_coeffs) {
    set_flag(IS_VALID);
    Nt_traits nt_traits;

    // Convert the coefficients vector to an equivalent vector of integer
    // coefficients.
    Integer int_coeffs[6];
    nt_traits.convert_coefficients(rat_coeffs, rat_coeffs + 6, int_coeffs);

    // Check the orientation of conic curve, and negate the conic coefficients
    // if its given orientation.
    typename Rat_kernel::Conic_2 temp_conic(rat_coeffs[0], rat_coeffs[1],
                                            rat_coeffs[2], rat_coeffs[3],
                                            rat_coeffs[4], rat_coeffs[5]);

    if (orientation() == temp_conic.orientation()) {
      m_r = int_coeffs[0];
      m_s = int_coeffs[1];
      m_t = int_coeffs[2];
      m_u = int_coeffs[3];
      m_v = int_coeffs[4];
      m_w = int_coeffs[5];
    }
    else {
      m_r = -int_coeffs[0];
      m_s = -int_coeffs[1];
      m_t = -int_coeffs[2];
      m_u = -int_coeffs[3];
      m_v = -int_coeffs[4];
      m_w = -int_coeffs[5];
    }

    // Make sure both endpoint lie on the supporting conic.
    if (! _is_on_supporting_conic(m_source) ||
        ! _is_on_supporting_conic(m_target))
    {
      reset_flags();    // inavlid arc
      return;
    }

    // Check whether we have a degree 2 curve.
    if ((CGAL::sign(m_r) != ZERO) || (CGAL::sign(m_s) != ZERO) ||
        (CGAL::sign(m_t) != ZERO))
    {
      if (orientation() == COLLINEAR) {
        // Make sure the midpoint is on the line pair (thus making sure that
        // the two points are not taken from different lines).
        Alg_kernel ker;
        auto ctr_mid_point = ker.construct_midpoint_2_object();
        Point_2 p_mid = ctr_mid_point(m_source, m_target);
        // if (! is_on_supporting_conic(arc, p_mid))
        if (CGAL::sign((nt_traits.convert(m_r)*p_mid.x() +
                        nt_traits.convert(m_t)*p_mid.y() +
                        nt_traits.convert(m_u)) * p_mid.x() +
                       (nt_traits.convert(m_s)*p_mid.y() +
                        nt_traits.convert(m_v)) * p_mid.y() +
                       nt_traits.convert(m_w)) != ZERO)
        {
          reset_flags();        // inavlid arc
          return;
        }

        // We have a segment of a line pair with rational coefficients.
        // Compose the equation of the underlying line
        // (with algebraic coefficients).
        const Algebraic& x1 = m_source.x();
        const Algebraic& y1 = m_source.y();
        const Algebraic& x2 = m_target.x();
        const Algebraic& y2 = m_target.y();

        // The supporting line is A*x + B*y + C = 0, where:
        //
        //  A = y2 - y1,    B = x1 - x2,    C = x2*y1 - x1*y2
        //
        // We use the extra dat field to store the equation of this line.
        m_extra_data = new Extra_data;
        m_extra_data->a = y2 - y1;
        m_extra_data->b = x1 - x2;
        m_extra_data->c = x2*y1 - x1*y2;
        m_extra_data->side = ZERO;
      }
      else {
        // The sign of (4rs - t^2) detetmines the conic type:
        // - if it is possitive, the conic is an ellipse,
        // - if it is negative, the conic is a hyperbola,
        // - if it is zero, the conic is a parabola.
        CGAL::Sign sign_conic = CGAL::sign(4*m_r*m_s - m_t*m_t);

        // Build the extra hyperbolic data if necessary
        if (sign_conic == NEGATIVE) _build_hyperbolic_arc_data();

        if (sign_conic != POSITIVE) {
          // In case of a non-degenerate parabola or a hyperbola, make sure
          // the arc is not infinite.
          Alg_kernel ker;
          Point_2 p_mid = ker.construct_midpoint_2_object()(m_source, m_target);
          Point_2 ps[2];

          bool finite_at_x = (_points_at_x(p_mid, ps) > 0);
          bool finite_at_y = (_points_at_y(p_mid, ps) > 0);

          if (! finite_at_x && ! finite_at_y) {
            reset_flags();      // inavlid arc
            return;
          }
        }
      }
    }


    set_flag(IS_VALID);                 // arc is valid
    reset_flag(IS_FULL_CONIC);          // not a full conic
  }

public:
  /// \name Construction and destruction functions.
  //@{

  /*! Default constructor.
   */
  Conic_arc_2() :
    m_r(0), m_s(0), m_t(0), m_u(0), m_v(0), m_w(0),
    m_orient(COLLINEAR),
    m_info(0),
    m_extra_data(nullptr)
  {}

  /*! Copy constructor.
   * \param arc The copied arc.
   */
  Conic_arc_2(const Self& arc) :
    m_r(arc.m_r), m_s(arc.m_s), m_t(arc.m_t),
    m_u(arc.m_u), m_v(arc.m_v), m_w(arc.m_w),
    m_orient(arc.m_orient),
    m_info(arc.m_info),
    m_source(arc.m_source),
    m_target(arc.m_target)
  {
    m_extra_data = (arc.m_extra_data != nullptr) ?
      new Extra_data(*(arc.m_extra_data)) : nullptr;
  }

  /*! Destructor.
   */
  virtual ~Conic_arc_2() { if (m_extra_data != nullptr) delete m_extra_data; }

  /*! Assignment operator.
   * \param arc The copied arc.
   */
  const Self& operator=(const Self& arc) {
    if (this == &arc) return (*this);

    // Free any existing data.
    if (m_extra_data != nullptr) delete m_extra_data;

    // Copy the arc's attributes.
    m_r = arc.m_r;
    m_s = arc.m_s;
    m_t = arc.m_t;
    m_u = arc.m_u;
    m_v = arc.m_v;
    m_w = arc.m_w;

    m_orient = arc.m_orient;
    m_info = arc.m_info;
    m_source = arc.m_source;
    m_target = arc.m_target;

    // Duplicate the extra data, if necessary.
    m_extra_data = (arc.m_extra_data != nullptr) ?
      new Extra_data(*(arc.m_extra_data)) : nullptr;

    return (*this);
  }
  //@}

  /// \name Get the arc properties.
  //@{

  /*! Determine wheather the arc is valid.
   */
  bool is_valid() const { return test_flag(IS_VALID); }

  /*! Determine whether the arc represents a full conic curve.
   */
  bool is_full_conic() const { return test_flag(IS_FULL_CONIC); }

  /*! Obtain the coefficients of the underlying conic.
   */
  const Integer& r() const { return (m_r); }
  const Integer& s() const { return (m_s); }
  const Integer& t() const { return (m_t); }
  const Integer& u() const { return (m_u); }
  const Integer& v() const { return (m_v); }
  const Integer& w() const { return (m_w); }

  /*! Obtain the arc's source.
   * \return The source point.
   * \pre The arc does not represent a full conic curve.
   */
  const Point_2& source() const {
    CGAL_precondition(! is_full_conic());
    return m_source;
  }

  /*! Obtain the non const reference to the curve source point.
   */
  Point_2& source() {
    CGAL_precondition(! is_full_conic());
    return (this->m_source);
  }

  /*! Obtain the arc's target.
   * \return The target point.
   * \pre The arc does not represent a full conic curve.
   */
  const Point_2& target() const {
    CGAL_precondition(! is_full_conic());
    return m_target;
  }

  /*! Obtain the non const reference to the curve source point.
   */
  Point_2& target() {
    CGAL_precondition(! is_full_conic());
    return this->m_target;
  }

  /*! Obtain the orientation of the arc.
   * \return The orientation.
   */
  Orientation orientation() const { return m_orient; }

  /*! Obtain the extra data.
   */
  const Extra_data* extra_data() const { return m_extra_data; }
  //@}

protected:
  template <typename, typename, typename> friend class Arr_conic_traits_2;

  /// \name Flag manipulation functions.
  //@{
  template <typename T>
  static constexpr auto flag_mask(const T flag) { return 0x1 << flag; }

  void reset_flags() { m_info = 0; }

  template <typename T>
  void set_flag(const T flag) { m_info |= flag_mask(flag); }

  template <typename T>
  void reset_flag(const T flag) { m_info &= ~flag_mask(flag); }

  template <typename T>
  void flip_flag(const T flag) { m_info ^= flag_mask(flag);}

  template <typename T>
  bool test_flag(const T flag) const
  { return (m_info & flag_mask(flag)); }
  //@}

public:
  /// \name Modifying functions (setters);
  // only friends have the priviledge to use.
  //@{

  /*! Set the source point of the conic arc.
   * \param ps The new source point.
   */
  void set_source(const Point_2& ps) { m_source = ps; }

  /*! Set the target point of the conic arc.
   * \param pt The new source point.
   */
  void set_target(const Point_2& pt) { m_target = pt; }

  /*! Set the coefficients.
   */
  void set_coefficients(Integer r, Integer s, Integer t,
                        Integer u, Integer v, Integer w) {
    m_r = r;
    m_s = s;
    m_t = t;
    m_u = u;
    m_v = v;
    m_w = w;
  }

  /*! Set the orientation.
   */
  void set_orientation(Orientation orient) { m_orient = orient; }

  /*! Set the endpoints.
   */
  void set_endpoints(const Point_2& source, const Point_2& target) {
    m_source = source;
    m_target = target;
  }

  /*! Set the extra data field.
   */
  void set_extra_data(Extra_data* extra_data) { m_extra_data = extra_data; }

  /*! Set the extra data field.
   */
  void set_extra_data(const Algebraic& a, const Algebraic& b,
                      const Algebraic& c, Sign side)
  {
    m_extra_data = new Extra_data;
    m_extra_data->a = a;
    m_extra_data->b = b;
    m_extra_data->c = c;
    m_extra_data->side = side;
  }

  /*! Update the extra data field.
   */
  void update_extra_data() {
    const Algebraic x1 = source().x();
    const Algebraic y1 = source().y();
    const Algebraic x2 = target().x();
    const Algebraic y2 = target().y();

    // The supporting line is A*x + B*y + C = 0, where:
    //  A = y2 - y1, B = x1 - x2, C = x2*y1 - x1*y2
    // We use the extra data field to store the equation of this line.
    m_extra_data = new Extra_data;
    m_extra_data->a = y2 - y1;
    m_extra_data->b = x1 - x2;
    m_extra_data->c = x2*y1 - x1*y2;
    m_extra_data->side = ZERO;
  }
  //@}

public:
  /// \name Auxiliary functions.
  //@{

  /*! Evaluate the sign of (a*x + b*y + c) stored with the extra data field
   * at a given point.
   * \param px The x-coordinate of query point.
   * \param py The y-coordinate of query point.
   * \return The sign of (a*x + b*y + c).
   */
  Sign sign_of_extra_data(const Algebraic& px, const Algebraic& py) const {
    CGAL_assertion(m_extra_data != nullptr);
    if (m_extra_data == nullptr) return ZERO;
    Algebraic val = m_extra_data->a*px + m_extra_data->b*py + m_extra_data->c;
    return CGAL::sign(val);
  }

  //@}
};

/*! Exporter for conic arcs.
 */
template <typename Rat_kernel, typename Alg_kernel, typename Nt_traits>
std::ostream&
operator<< (std::ostream& os,
            const Conic_arc_2<Rat_kernel, Alg_kernel, Nt_traits>& arc)
{
  os << "{" << CGAL::to_double(arc.r()) << "*x^2 + "
     << CGAL::to_double(arc.s()) << "*y^2 + "
     << CGAL::to_double(arc.t()) << "*xy + "
     << CGAL::to_double(arc.u()) << "*x + "
     << CGAL::to_double(arc.v()) << "*y + "
     << CGAL::to_double(arc.w()) << "}";

  if (arc.is_full_conic()) os << " - Full curve";
  else {
    os << " : (" << CGAL::to_double(arc.source().x()) << ","
       << CGAL::to_double(arc.source().y()) << ") ";

    if (arc.orientation() == CLOCKWISE) os << "--cw-->";
    else if (arc.orientation() == COUNTERCLOCKWISE) os << "--ccw-->";
    else os << "--l-->";

    os << " (" << CGAL::to_double(arc.target().x()) << ","
       << CGAL::to_double(arc.target().y()) << ")";
  }

  return os;
}

} //namespace CGAL

#endif
