// Copyright (c) 2006,2007,2009,2010,2011 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: https://github.com/CGAL/cgal/blob/releases/CGAL-4.14.1/Arrangement_on_surface_2/include/CGAL/Surface_sweep_2/Arr_basic_insertion_traits_2.h $
// $Id: Arr_basic_insertion_traits_2.h 7936109 %aI Efi Fogel
// SPDX-License-Identifier: GPL-3.0+
//
//
// Author(s)     : Baruch Zukerman <baruchzu@post.tau.ac.il>
//                 Ron Wein <wein@post.tau.ac.il>
//                 Efi Fogel <efif@post.tau.ac.il>
//                 Eric Berberich <eric@mpi-inf.mpg.de>

#ifndef CGAL_ARR_BASIC_INSERTION_TRAITS_2_H
#define CGAL_ARR_BASIC_INSERTION_TRAITS_2_H

#include <CGAL/license/Arrangement_on_surface_2.h>

/*! \file
 *
 * Defintion of the Arr_basic_insertion_traits_2<Traits,Arrangement> class.
 */

#include <CGAL/Object.h>
#include <CGAL/Arr_tags.h>

#include <list>
#include <iterator>

namespace CGAL {

/*! A basic meta-traits class that stores a halfedge handle with every
 * x-monotone curve, and a vertex handle with each point. This information is
 * used to speed up the aggregated insertion process.
 */
template <typename GeometryTraits_2, typename Arrangement_>
class Arr_basic_insertion_traits_2 {
public:
  typedef GeometryTraits_2                              Geometry_traits_2;
  typedef Arrangement_                                  Arrangement_2;

private:
  typedef Geometry_traits_2                             Gt2;

public:
  typedef typename Arrangement_2::Halfedge_handle       Halfedge_handle;
  typedef typename Arrangement_2::Vertex_handle         Vertex_handle;

  typedef typename Gt2::X_monotone_curve_2     Base_x_monotone_curve_2;
  typedef typename Gt2::Point_2                Base_point_2;
  typedef typename Gt2::Construct_min_vertex_2 Base_construct_min_vertex_2;
  typedef typename Gt2::Construct_max_vertex_2 Base_construct_max_vertex_2;
  typedef typename Gt2::Compare_x_2            Base_compare_x_2;
  typedef typename Gt2::Compare_xy_2           Base_compare_xy_2;
  typedef typename Gt2::Compare_y_at_x_2       Base_compare_y_at_x_2;
  typedef typename Gt2::Compare_y_at_x_right_2 Base_compare_y_at_x_right_2;
  typedef typename Gt2::Equal_2                Base_equal_2;
  typedef typename Gt2::Is_vertical_2          Base_is_vertical_2;

  typedef typename Gt2::Multiplicity           Multiplicity;

  typedef typename Gt2::Has_do_intersect_category
                                               Has_do_intersect_category;


  typedef typename internal::Arr_complete_left_side_category< Gt2>::Category
                                                    Left_side_category;
  typedef typename internal::Arr_complete_bottom_side_category< Gt2>::Category
                                                    Bottom_side_category;
  typedef typename internal::Arr_complete_top_side_category< Gt2>::Category
                                                    Top_side_category;
  typedef typename internal::Arr_complete_right_side_category< Gt2>::Category
                                                    Right_side_category;

  /* Insertion is implemented as surface-sweep visitor. The surface-sweep
   * algorithm never uses Compare_y_at_x_left_2.
   */
  typedef Tag_false                                 Has_left_category;

protected:
  //! The base traits.
  const Gt2* m_base_traits;

public:
  /*! Constructor. */
  Arr_basic_insertion_traits_2(const Gt2& tr) :
    m_base_traits(&tr)
  {}

  /*!
   * Nested extension of the x-monotone curve type.
   */
  class Ex_x_monotone_curve_2 {
  public:
    typedef  Base_x_monotone_curve_2  Base;

  protected:
    Base m_base_xcv;                    // The base x-monotone curve.
    Halfedge_handle m_he_handle;        // The corresponding arrangement edge
                                        // (may be invalid).
    bool m_overlap;                     // Does this curve represent and overlap
                                        // of two other curves.

  public:

    Ex_x_monotone_curve_2():
      m_base_xcv(),
      m_he_handle(),
      m_overlap(false)
    {}

    Ex_x_monotone_curve_2(const Base& xcv):
      m_base_xcv(xcv),
      m_he_handle(),
      m_overlap(false)
    {}

    Ex_x_monotone_curve_2(const Base& xcv, Halfedge_handle he) :
      m_base_xcv(xcv),
      m_he_handle(he),
      m_overlap(false)
    {
      CGAL_precondition(he == Halfedge_handle() ||
                        he->direction() == ARR_RIGHT_TO_LEFT);
    }

    const Base& base() const { return m_base_xcv; }

    Base& base() { return m_base_xcv; }

    operator const Base& () const { return m_base_xcv; }

    Ex_x_monotone_curve_2& operator=(const Base& xcv)
    {
      m_base_xcv = xcv;
      m_he_handle = Halfedge_handle();
      return (*this);
    }

    Halfedge_handle halfedge_handle() const { return m_he_handle; }

    void set_halfedge_handle(Halfedge_handle he)
    {
      CGAL_precondition(he == Halfedge_handle() ||
                        he->direction() == ARR_RIGHT_TO_LEFT);
      m_he_handle = he;
    }

    bool is_overlapping() const { return m_overlap; }

    void set_overlapping() { m_overlap = true; }
  };

  typedef Ex_x_monotone_curve_2                     X_monotone_curve_2;

  // For debugging purposes:
  friend std::ostream& operator<<(std::ostream& os,
                                  const X_monotone_curve_2& xcv)
  {
    os << xcv.base();
    return (os);
  }

  /*!
   * Nested extension of the point type.
   */
  class Ex_point_2 {
  public:
    typedef  Base_point_2            Base;

  protected:
    Base m_base_pt;                     // The base point.
    Vertex_handle m_v;                  // The corresponding arrangement vertex
                                        // (may be invalid).

  public:
    Ex_point_2() :
      m_base_pt(),
      m_v()
    {}

    Ex_point_2(const Base& pt) :
      m_base_pt(pt),
      m_v()
    {}

    Ex_point_2(const Base& pt, Vertex_handle v) :
      m_base_pt(pt),
      m_v(v)
    {}

    const Base& base() const { return m_base_pt; }

    operator const Base& () const { return m_base_pt; }

    Vertex_handle vertex_handle() const { return m_v; }

    void set_vertex_handle(Vertex_handle v) { m_v = v; }
  };

  typedef Ex_point_2                                Point_2;

  // For debugging purposes:
  friend std::ostream& operator<<(std::ostream& os, const Point_2& pt)
  {
    os << pt.base();
    return os;
  }

  /*! A functor that obtains the left endpoint of an x-monotone curve. */
  class Construct_min_vertex_2 {
  protected:
    Base_construct_min_vertex_2 m_base_min_v;
    Base_equal_2 m_base_equal;
    Halfedge_handle invalid_he;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Construct_min_vertex_2(const Base_construct_min_vertex_2& base_min_v,
                           const Base_equal_2& base_equal) :
      m_base_min_v(base_min_v),
      m_base_equal(base_equal),
      invalid_he()
    {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Point_2 operator()(const X_monotone_curve_2& xcv)
    {
      // If there is not halfedge associated with the curve, just return
      // a point with invalid halfedge handle.
      const Base_point_2& base_p = m_base_min_v(xcv.base());

      if (xcv.halfedge_handle() == invalid_he) return (Point_2(base_p));

      // Note that the halfedge associated with the curve is always directed
      // from right to left, so its target is the leftmost vertex.
      // We probably have to associate the point with the target vertex of
      // the halfedge associated with the curve.
      Vertex_handle vh = xcv.halfedge_handle()->target();

      if (! xcv.is_overlapping()) return (Point_2(base_p, vh));

      // In case of an overlapping curve, make sure the curve endpoint equals
      // the point associated with the vertex. If not, we attach an invalid
      // vertex to the extended point.
      if (! vh->is_at_open_boundary() && m_base_equal(base_p, vh->point()))
        return (Point_2(base_p, vh));
      else return (Point_2(base_p));
    }
  };

  /*! Obtain a Construct_min_vertex_2 function object */
  Construct_min_vertex_2 construct_min_vertex_2_object() const
  {
    return (Construct_min_vertex_2
            (m_base_traits->construct_min_vertex_2_object(),
             m_base_traits->equal_2_object()));
  }

  /*! A functor that obtains the right endpoint of an x-monotone curve. */
  class Construct_max_vertex_2 {
  protected:
    Base_construct_max_vertex_2 m_base_max_v;
    Base_equal_2 m_base_equal;
    Halfedge_handle invalid_he;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Construct_max_vertex_2(const Base_construct_max_vertex_2& base_max_v,
                           const Base_equal_2& base_equal) :
      m_base_max_v(base_max_v),
      m_base_equal(base_equal),
      invalid_he()
    {}


    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Point_2 operator()(const X_monotone_curve_2& xcv)
    {
      // If there is not halfedge associated with the curve, just return
      // a point with invalid halfedge handle.
      const Base_point_2& base_p = m_base_max_v(xcv.base());

      if (xcv.halfedge_handle() == invalid_he) return (Point_2(base_p));

      // Note that the halfedge associated with the curve is always directed
      // from right to left, so its source is the rightmost vertex.
      // We probably have to associate the point with the source vertex of
      // the halfedge associated with the curve.
      Vertex_handle vh = xcv.halfedge_handle()->source();

      if (! xcv.is_overlapping()) return (Point_2(base_p, vh));

      // In case of an overlapping curve, make sure the curve endpoint equals
      // the point associated with the vertex. If not, we attach an invalid
      // vertex to the extended point.
      if (! vh->is_at_open_boundary() && m_base_equal(base_p, vh->point()))
        return (Point_2(base_p, vh));
      else return (Point_2(base_p));
    }
  };

  /*! Obtain a Construct_max_vertex_2 function object */
  Construct_max_vertex_2 construct_max_vertex_2_object() const
  {
    return (Construct_max_vertex_2
            (m_base_traits->construct_max_vertex_2_object(),
             m_base_traits->equal_2_object()));
  }

  /*! A functor that compares two points lexigoraphically: by x, then by y. */
  class Compare_xy_2 {
  protected:
    Base_compare_xy_2 m_base_cmp_xy;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_xy_2(const Base_compare_xy_2& base) : m_base_cmp_xy(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Comparison_result operator()(const Point_2& p1, const Point_2& p2) const
    {
      if(p1.vertex_handle() == p2.vertex_handle() &&
         p1.vertex_handle() != Vertex_handle())
        return EQUAL;

      return (m_base_cmp_xy(p1.base(), p2.base()));
    }
  };

  /*! Obtain a Compare_xy_2 function object */
  Compare_xy_2 compare_xy_2_object() const
  { return (Compare_xy_2(m_base_traits->compare_xy_2_object())); }

  /*! A functor that compares the y-coordinates of a point and an
   * x-monotone curve at the point x-coordinate.
   */
  class Compare_y_at_x_2 {
  protected:
    Base_compare_y_at_x_2 m_base_cmp_y_at_x;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_y_at_x_2(const Base_compare_y_at_x_2& base) :
      m_base_cmp_y_at_x(base)
    {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Comparison_result operator()(const Point_2& p,
                                 const X_monotone_curve_2& xcv) const
    { return (m_base_cmp_y_at_x(p.base(), xcv.base())); }
  };

  /*! Obtain a Compare_y_at_x_2 function object */
  Compare_y_at_x_2 compare_y_at_x_2_object() const
  { return (Compare_y_at_x_2(m_base_traits->compare_y_at_x_2_object())); }

  /*! A functor that compares compares the y-coordinates of two x-monotone
   * curves immediately to the right of their intersection point.
   */
  class Compare_y_at_x_right_2 {
  protected:
    Base_compare_y_at_x_right_2 m_base_cmp_y_at_x_right;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_y_at_x_right_2(const Base_compare_y_at_x_right_2& base) :
        m_base_cmp_y_at_x_right(base)
    {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                 const X_monotone_curve_2& xcv2,
                                 const Point_2& p) const
    { return (m_base_cmp_y_at_x_right(xcv1.base(), xcv2.base(), p.base())); }
  };

  /*! Obtain a Compare_y_at_x_right_2 function object */
  Compare_y_at_x_right_2 compare_y_at_x_right_2_object() const
  {
    return (Compare_y_at_x_right_2
	    (m_base_traits->compare_y_at_x_right_2_object()));
  }

  /*! A functor that checks whether two points and two x-monotone curves are
   * identical.
   */
  class Equal_2 {
  protected:
    Base_equal_2 m_base_eq;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Equal_2(const Base_equal_2& base) : m_base_eq(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Check if two curves are the same. */
    bool operator()(const X_monotone_curve_2& xcv1,
                    const X_monotone_curve_2& xcv2) const
    { return (m_base_eq(xcv1.base(), xcv2.base())); }

    /*! Check if the two points are the same. */
    bool operator()(const Point_2& p1, const Point_2& p2) const
    { return (m_base_eq(p1.base(), p2.base())); }
  };

  /*! Obtain a Equal_2 function object */
  Equal_2 equal_2_object() const
  { return (Equal_2(m_base_traits->equal_2_object())); }

  /*! A functor that compares the x-coordinates of two points */
  class Compare_x_2 {
  protected:
    Base_compare_x_2 m_base_cmp_x;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_x_2(const Base_compare_x_2& base) : m_base_cmp_x(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Comparison_result operator()(const Point_2& p1, const Point_2& p2) const
    { return (m_base_cmp_x(p1.base(), p2.base())); }
  };

  /*! Obtain a Compare_x_2 function object */
  Compare_x_2 compare_x_2_object() const
  { return (Compare_x_2(m_base_traits->compare_x_2_object())); }

  /*! A functor that checks whether a given x-monotone curve is vertical. */
  class Is_vertical_2 {
  protected:
    Base_is_vertical_2 m_base_is_vert;

    /*! Constructor.
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Is_vertical_2(const Base_is_vertical_2& base) : m_base_is_vert(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    bool operator()(const X_monotone_curve_2& xcv) const
    { return (m_base_is_vert(xcv.base())); }
  };

  /*! Obtain a Is_vertical_2 function object */
  Is_vertical_2 is_vertical_2_object() const
  { return (Is_vertical_2(m_base_traits->is_vertical_2_object())); }

  // left-right

  /*! A functor that determines whether an endpoint of an x-monotone curve lies
   * on a boundary of the parameter space along the x axis.
   */
  class Parameter_space_in_x_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param tr The base traits class. It must be passed, to handle non
     *           stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Parameter_space_in_x_2(const Gt2* tr) : m_base(tr) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Arr_parameter_space operator()(const X_monotone_curve_2& xcv,
                                   Arr_curve_end ce) const
    { return (m_base->parameter_space_in_x_2_object()(xcv.base(), ce)); }

    Arr_parameter_space operator()(const Point_2& p) const
    { return m_base->parameter_space_in_x_2_object()(p.base()); }

    Arr_parameter_space operator()(const X_monotone_curve_2& xcv) const
    { return m_base->parameter_space_in_x_2_object()(xcv.base()); }
  };

  /*! Obtain a Parameter_space_in_x_2 function object */
  Parameter_space_in_x_2 parameter_space_in_x_2_object() const
  { return Parameter_space_in_x_2(m_base_traits); }

  /*! A function object that determines whether an x-monotone curve or a
   * point coincide with the vertical identification curve.
   */
  class Is_on_x_identification_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param tr The base traits class. It must be passed, to handle non
     *           stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Is_on_x_identification_2(const Gt2* tr) : m_base(tr) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    bool operator()(const Point_2& p) const
    { return m_base->is_on_x_identification_2_object()(p.base()); }

    bool operator()(const X_monotone_curve_2& xcv) const
    { return m_base->is_on_x_identification_2_object()(xcv.base()); }
  };

  /*! Obtain a Is_on_x_identification_2 function object */
  Is_on_x_identification_2 is_on_x_identification_2_object() const
  { return Is_on_x_identification_2(m_base_traits); }

  /*! A functor that compares the y-coordinates of two points on vertical
   * boundaries.
   */
  class Compare_y_on_boundary_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param base The base traits class. It must be passed, to handle non
     *             stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_y_on_boundary_2(const Gt2* base) : m_base(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const Point_2& p1, const Point_2& p2) const
    { return m_base->compare_y_on_boundary_2_object()(p1.base(), p2.base()); }
  };

  /*! Obtain a Compare_y_on_boundary_2 object
   */
  Compare_y_on_boundary_2 compare_y_on_boundary_2_object() const
  { return Compare_y_on_boundary_2(m_base_traits); }

  /*! A functor that compares the y-coordinates of curve ends near the
   * boundary of the parameter space.
   */
  class Compare_y_near_boundary_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param base The base traits class. It must be passed, to handle non
     *             stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_y_near_boundary_2(const Gt2* base) : m_base(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                 const X_monotone_curve_2& xcv2,
                                 Arr_curve_end ce) const
    {
      return m_base->compare_y_near_boundary_2_object()(xcv1.base(),
                                                        xcv2.base(), ce);
    }

  };

  /*! Obtain a Compare_y_near_boundary_2 object
   */
  Compare_y_near_boundary_2 compare_y_near_boundary_2_object() const
  { return Compare_y_near_boundary_2(m_base_traits); }

  // bottom-top

  /*! A functor that determines whether an endpoint of an x-monotone arc lies
   * on a boundary of the parameter space along the y axis.
   */
  class Parameter_space_in_y_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param tr The base traits class. It must be passed, to handle non
     *           stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Parameter_space_in_y_2(const Gt2* tr) : m_base(tr) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    Arr_parameter_space operator()(const X_monotone_curve_2& xcv,
                                   Arr_curve_end ce) const
    { return m_base->parameter_space_in_y_2_object()(xcv.base(), ce); }

    Arr_parameter_space operator()(const Point_2& p) const
    { return m_base->parameter_space_in_y_2_object()(p.base()); }

    Arr_parameter_space operator()(const X_monotone_curve_2& xcv) const
    { return m_base->parameter_space_in_y_2_object()(xcv.base()); }

  };

  /*! Obtain a Parameter_space_in_y_2 function object */
  Parameter_space_in_y_2 parameter_space_in_y_2_object() const
  { return Parameter_space_in_y_2(m_base_traits); }

  /*! A function object that determines whether an x-monotone curve or a
   * point coincide with the horizontal identification curve.
   */
  class Is_on_y_identification_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param tr The base traits class. It must be passed, to handle non
     *           stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Is_on_y_identification_2(const Gt2* tr) : m_base(tr) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    bool operator()(const Point_2& p) const
    { return m_base->is_on_y_identification_2_object()(p.base()); }

    bool operator()(const X_monotone_curve_2 & xcv) const
    { return m_base->is_on_y_identification_2_object()(xcv.base()); }
  };

  /*! Obtain a Is_on_y_identification_2 function object */
  Is_on_y_identification_2 is_on_y_identification_2_object() const
  { return Is_on_y_identification_2(m_base_traits); }

  /*! A functor that compares the x-limits of curve ends on the
   * boundary of the parameter space.
   */
  class Compare_x_at_limit_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param base The base traits class. It must be passed, to handle non
     *             stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_x_at_limit_2(const Gt2* base) : m_base(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const Point_2& p,
                                 const X_monotone_curve_2& xcv,
                                 Arr_curve_end ce) const
    { return m_base->compare_x_at_limit_2_object()(p.base(), xcv.base(), ce); }

    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                 Arr_curve_end ce1,
                                 const X_monotone_curve_2& xcv2,
                                 Arr_curve_end ce2) const
    {
      return m_base->compare_x_at_limit_2_object()(xcv1.base(), ce1,
                                                   xcv2.base(), ce2);
    }
  };

  /*! Obtain a Compare_x_at_limit_2 object
   */
  Compare_x_at_limit_2 compare_x_at_limit_2_object() const
  { return Compare_x_at_limit_2(m_base_traits); }


  /*! A functor that compares the x-coordinates of curve ends near the
   * boundary of the parameter space.
   */
  class Compare_x_near_limit_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param base The base traits class. It must be passed, to handle non
     *             stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_x_near_limit_2(const Gt2* base) : m_base(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                 const X_monotone_curve_2& xcv2,
                                 Arr_curve_end ce) const
    {
      return m_base->compare_x_near_limit_2_object()(xcv1.base(), xcv2.base(),
                                                     ce);
    }

  };

  /*! Obtain a Compare_x_near_limit_2 object
   */
  Compare_x_near_limit_2 compare_x_near_limit_2_object() const
  { return Compare_x_near_limit_2(m_base_traits); }

  /*! A functor that compares the x-coordinates of two points on vertical
   * boundaries.
   */
  class Compare_x_on_boundary_2
  {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param base The base traits class. It must be passed, to handle non
     *             stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_x_on_boundary_2(const Gt2* base) : m_base(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const Point_2& p1, const Point_2& p2) const
    { return m_base->compare_x_on_boundary_2_object()(p1.base(), p2.base()); }

    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const Point_2 & pt,
                                 const X_monotone_curve_2& xcv,
                                 Arr_curve_end ce) const
    {
      return m_base->compare_x_on_boundary_2_object()(pt.base(), xcv.base(), ce);
    }

    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                 Arr_curve_end ce1,
                                 const X_monotone_curve_2& xcv2,
                                 Arr_curve_end ce2) const
    {
      return m_base->compare_x_on_boundary_2_object()(xcv1.base(), ce1,
                                                      xcv2.base(), ce2);
    }
  };

  /*! Obtain a Compare_x_on_boundary_2 object
   */
  Compare_x_on_boundary_2 compare_x_on_boundary_2_object() const
  { return Compare_x_on_boundary_2(m_base_traits); }

  /*! A functor that compares the x-coordinates of curve ends near the
   * boundary of the parameter space.
   */
  class Compare_x_near_boundary_2 {
  protected:
    //! The base traits.
    const Gt2* m_base;

    /*! Constructor.
     * \param base The base traits class. It must be passed, to handle non
     *             stateless traits objects, (which stores data).
     * The constructor is declared private to allow only the functor
     * obtaining function, which is a member of the nesting class,
     * constructing it.
     */
    Compare_x_near_boundary_2(const Gt2* base) : m_base(base) {}

    //! Allow its functor obtaining function calling the private constructor.
    friend class Arr_basic_insertion_traits_2<GeometryTraits_2, Arrangement_>;

  public:
    /*! Use tag dispatching to avoid compilation errors in case the functor
     * is not defined
     */
    Comparison_result operator()(const X_monotone_curve_2& xcv1,
                                 const X_monotone_curve_2& xcv2,
                                 Arr_curve_end ce) const
    {
      return m_base->compare_x_near_boundary_2_object()(xcv1.base(),
                                                        xcv2.base(), ce);
    }
  };

  /*! Obtain a Compare_x_near_boundary_2 object
   */
  Compare_x_near_boundary_2 compare_x_near_boundary_2_object() const
  { return Compare_x_near_boundary_2(m_base_traits); }
};

} //namespace CGAL

#endif
