#include <Rcpp.h>
// Copyright (c) 2003,2004,2005,2006  INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.6.1/Segment_Delaunay_graph_2/include/CGAL/Segment_Delaunay_graph_2.h $
// $Id: Segment_Delaunay_graph_2.h 23e56a8 2024-01-11T08:43:48+00:00 Andreas Fabri
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s)     : Menelaos Karavelas <mkaravel@iacm.forth.gr>



#ifndef CGAL_SEGMENT_DELAUNAY_GRAPH_2_H
#define CGAL_SEGMENT_DELAUNAY_GRAPH_2_H

#include <CGAL/license/Segment_Delaunay_graph_2.h>

#include <CGAL/disable_warnings.h>

#include <iostream>
#include <vector>
#include <list>
#include <set>
#include <stack>
#include <map>
#include <algorithm>
#include <boost/tuple/tuple.hpp>

#include <CGAL/Segment_Delaunay_graph_2/basic.h>

#include <CGAL/Triangulation_2.h>
#include <CGAL/Segment_Delaunay_graph_storage_traits_2.h>
#include <CGAL/Segment_Delaunay_graph_vertex_base_2.h>
#include <CGAL/Segment_Delaunay_graph_face_base_2.h>
#include <CGAL/Triangulation_data_structure_2.h>

#include <CGAL/Segment_Delaunay_graph_2/in_place_edge_list.h>
#include <CGAL/TDS_2/internal/edge_list.h>
#include <CGAL/Segment_Delaunay_graph_2/Traits_wrapper_2.h>
#include <CGAL/Segment_Delaunay_graph_2/Constructions_C2.h>

#include <CGAL/utility.h>
#include <CGAL/tss.h>

#include <CGAL/spatial_sort.h>
#include <CGAL/Spatial_sort_traits_adapter_2.h>

#include <CGAL/assertions.h>

#include <CGAL/boost/iterator/counting_iterator.hpp>
#include <CGAL/boost/iterator/transform_iterator.hpp>

/*
  Conventions:
  ------------
  1. we treat segments as open; the endpoints are separate objects
  2. a segment of length zero is treated as a point
  3. a point is deleted only if it has no segment adjacent to it
  4. when a segment is deleted it's endpoints are not deleted
  5. the user can force the deletion of endpoints; this is only done
     if condition 3 is met.
  6. when objects are written to a stream we distinguish between
     points and segments; points start by a 'p' and segments by an 's'.
*/


namespace CGAL {

namespace SegmentDelaunayGraph_2 {

namespace Internal {

  template<typename Edge, typename LTag> struct Which_list;

  // use the in-place edge list
  template<typename E>
  struct Which_list<E,Tag_true>
  {
    typedef E                           Edge;
    typedef In_place_edge_list_for_sdg<Edge>    List;
  };

  // do not use the in-place edge list
  template<typename E>
  struct Which_list<E,Tag_false>
  {
    typedef E                                 Edge;
    // change the following to Tag_false in order to use
    // CGAL's Unique_hash_map
    typedef Tag_true                          Use_stl_map_tag;
    typedef Edge_list<Edge,Use_stl_map_tag>   List;
  };


  template < class Node >
  struct Project_site_2 {
    typedef Node                   argument_type;
    typedef typename Node::Site_2  Site;
    typedef Site                   result_type;

    Site operator()(const Node& x) const {
      return x.site();
    }

  };

  template < class Node, class Site_t >
  struct Project_input_to_site_2 {
    typedef Node                   argument_type;
    typedef Site_t                 Site;
    typedef Site                   result_type;

    Site operator()(const Node& x) const {
      if ( boost::tuples::get<2>(x) /*x.third*/ ) { // it is a point
        return Site::construct_site_2( *boost::tuples::get<0>(x) );
      }
      return Site::construct_site_2
        (*boost::tuples::get<0>(x), *boost::tuples::get<1>(x));
    }

  };

  template<typename T, typename U>
  struct Check_type_equality_for_info
  {
    Check_type_equality_for_info()
    {
      ERROR__INFO_TYPES_OF_insert_AND_Storage_traits_with_info_2_MUST_MATCH
        (T(), U());
    }
  };

  template<typename T>
  struct Check_type_equality_for_info<T,T>
  {
  };

} // namespace Internal

} //namespace SegmentDelaunayGraph_2


template<class Gt, class ST, class STag, class D_S, class LTag, class SDGLx >
class Segment_Delaunay_graph_hierarchy_2;

template<class Gt, class ST, class D_S, class LTag >
class Segment_Delaunay_graph_Linf_2;


template<class Gt,
         class ST = Segment_Delaunay_graph_storage_traits_2<Gt>,
         class D_S = Triangulation_data_structure_2 <
                Segment_Delaunay_graph_vertex_base_2<ST>,
                Segment_Delaunay_graph_face_base_2<Gt> >,
         class LTag = Tag_false >
class Segment_Delaunay_graph_2
  : private Triangulation_2<
          Segment_Delaunay_graph_traits_wrapper_2<Gt>, D_S >
{
  friend class Segment_Delaunay_graph_Linf_2<Gt,ST,D_S,LTag>;
  friend class Segment_Delaunay_graph_hierarchy_2<Gt,ST,Tag_true,D_S,LTag,
   Segment_Delaunay_graph_2<Gt,ST,D_S,LTag> >;
  friend class Segment_Delaunay_graph_hierarchy_2<Gt,ST,Tag_false,D_S,LTag,
   Segment_Delaunay_graph_2<Gt,ST,D_S,LTag> >;
  friend class Segment_Delaunay_graph_hierarchy_2<Gt,ST,Tag_true,D_S,LTag,
   Segment_Delaunay_graph_Linf_2<Gt,ST,D_S,LTag> >;
  friend class Segment_Delaunay_graph_hierarchy_2<Gt,ST,Tag_false,D_S,LTag,
   Segment_Delaunay_graph_Linf_2<Gt,ST,D_S,LTag> >;
protected:
  // LOCAL TYPES
  //------------
  typedef Segment_Delaunay_graph_2<Gt,ST,D_S,LTag>       Self;

  typedef Segment_Delaunay_graph_traits_wrapper_2<Gt>   Modified_traits;
  typedef Triangulation_2<Modified_traits,D_S>           DG;
  typedef DG                                            Delaunay_graph;

  typedef LTag                                          List_tag;

public:
  // PUBLIC TYPES
  //-------------
  typedef D_S                                     Data_structure;
  typedef D_S                                     Triangulation_data_structure;
  typedef Gt                                     Geom_traits;
  typedef ST                                     Storage_traits;
  typedef typename Gt::Site_2                    Site_2;
  typedef typename Gt::Point_2                   Point_2;

  typedef typename D_S::Edge                      Edge;
  typedef typename D_S::Vertex_handle             Vertex_handle;
  typedef typename D_S::Face_handle               Face_handle;
  typedef typename D_S::Vertex                    Vertex;
  typedef typename D_S::Face                      Face;

  typedef typename D_S::size_type                 size_type;

  typedef typename D_S::Vertex_circulator         Vertex_circulator;
  typedef typename D_S::Edge_circulator           Edge_circulator;
  typedef typename D_S::Face_circulator           Face_circulator;

  typedef typename D_S::Face_iterator             All_faces_iterator;
  typedef typename D_S::Vertex_iterator           All_vertices_iterator;
  typedef typename D_S::Edge_iterator             All_edges_iterator;

  typedef typename DG::Finite_faces_iterator     Finite_faces_iterator;
  typedef typename DG::Finite_vertices_iterator  Finite_vertices_iterator;
  typedef typename DG::Finite_edges_iterator     Finite_edges_iterator;

  typedef typename Storage_traits::Point_container     Point_container;
  typedef typename Storage_traits::Point_handle        Point_handle;
  typedef typename Storage_traits::const_Point_handle  const_Point_handle;

protected:
  typedef typename Geom_traits::Arrangement_type_2  AT2;
  typedef typename AT2::Arrangement_type            Arrangement_type;

  // these containers should have point handles and should replace the
  // point container...
  typedef boost::tuples::tuple<Point_handle,Point_handle,bool>  Site_rep_2;

  struct Site_rep_less_than {
    // less than for site reps
    bool operator()(const Site_rep_2& x, const Site_rep_2& y) const {
      Point_handle x1 = boost::tuples::get<0>(x);
      Point_handle y1 = boost::tuples::get<0>(y);

      if ( &(*x1) < &(*y1) ) { return true; }
      if ( &(*y1) < &(*x1) ) { return false; }

      Point_handle x2 = boost::tuples::get<1>(x);
      Point_handle y2 = boost::tuples::get<1>(y);

      return &(*x2) < &(*y2);
    }
  };

  typedef std::set<Site_rep_2,Site_rep_less_than>  Input_sites_container;
  typedef typename Input_sites_container::const_iterator
  All_inputs_iterator;

  typedef
  CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::Internal::
  Project_input_to_site_2<Site_rep_2, Site_2>
  Proj_input_to_site;

  typedef CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::Internal::Project_site_2<Vertex>
  Proj_site;

  struct Point_handle_less_than {
    // less than
    bool operator()(const const_Point_handle& x,
                    const const_Point_handle& y) const {
      return &(*x) < &(*y);
    }
  };

  typedef std::pair<Point_handle,Point_handle>   Point_handle_pair;

  typedef std::map<Point_handle,Point_handle,Point_handle_less_than>
  Handle_map;

public:
  typedef boost::transform_iterator<Proj_input_to_site, All_inputs_iterator>
  Input_sites_iterator;

  typedef boost::transform_iterator<Proj_site, Finite_vertices_iterator>
  Output_sites_iterator;
protected:
  // LOCAL VARIABLE(S)
  //------------------
  // the container of points
  Point_container pc_;
  Input_sites_container isc_;
  Storage_traits st_;
#ifdef CGAL_SDG_NO_FACE_MAP
  std::vector<Face_handle> fhc_;
#endif

protected:
  // MORE LOCAL TYPES
  //-----------------
  typedef typename Gt::Intersections_tag        Intersections_tag;

  typedef std::map<Face_handle,bool>            Face_map;
  typedef std::vector<Edge>                     Edge_vector;

  typedef std::list<Vertex_handle>              Vertex_list;
  typedef typename Vertex_list::iterator        Vertex_list_iterator;

  typedef Triple<Vertex_handle,Vertex_handle,Vertex_handle>
  Vertex_triple;

  typedef Vertex_handle                         Vh_triple[3];
  typedef std::map<Face_handle,Sign>            Sign_map;

  typedef std::pair<Face_handle,Face_handle>    Face_pair;

  typedef typename Storage_traits::Storage_site_2   Storage_site_2;

  // the edge list
  typedef typename
  CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::Internal::Which_list<Edge,List_tag>::List
  List;


protected:
  // types for insert on segment functions
  typedef Vertex_triple (Self::*Insert_on_Type)(
      const Storage_site_2& ss, const Site_2& t,
      Vertex_handle v, const Tag_true&);
  Insert_on_Type insert_point_on_segment_ptr;

  typedef Vertex_triple (Self::*Insert_Exact_on_Type)(
      const Storage_site_2& ss, const Site_2& t,
      Vertex_handle v);
  Insert_Exact_on_Type insert_exact_point_on_segment_ptr;

  Vertex_triple
  insert_point_on_segment(const Storage_site_2& ss, const Site_2& t,
                          Vertex_handle v, const Tag_true&);

  Vertex_triple
  insert_exact_point_on_segment(const Storage_site_2& ss, const Site_2& t,
                                Vertex_handle v);

private:
  // CREATION helper
  template<class ITag>
  inline
  void setup_if_intersecting_pointer(ITag tag) {
    setup_if_intersecting_pointer_with_tag(tag);
  }

  void setup_if_intersecting_pointer_with_tag(Tag_false) {
    insert_point_on_segment_ptr = nullptr;
  }

  void setup_if_intersecting_pointer_with_tag(Tag_true) {
    insert_point_on_segment_ptr = &Self::insert_point_on_segment;
  }

  void setup_insert_on_pointers_l2(void) {
    Intersections_tag itag;
    setup_if_intersecting_pointer(itag);
    insert_exact_point_on_segment_ptr = &Self::insert_exact_point_on_segment;
  }

public:
  // CREATION
  //---------
  Segment_Delaunay_graph_2(const Geom_traits& gt = Geom_traits(),
                           const Storage_traits& st = Storage_traits())
    : DG(gt), st_(st)
  {
    setup_insert_on_pointers_l2();
  }

  template< class Input_iterator >
  Segment_Delaunay_graph_2(Input_iterator first, Input_iterator beyond,
                           const Geom_traits& gt = Geom_traits(),
                           const Storage_traits& st = Storage_traits())
    : DG(gt), st_(st)
  {
    setup_insert_on_pointers_l2();
    insert(first, beyond);
  }

  Segment_Delaunay_graph_2(const Self& other);
  Self& operator=(const Self& other);

public:
  // ACCESS METHODS
  // --------------
  const Geom_traits&  geom_traits() const { return DG::geom_traits(); }

  const Storage_traits&  storage_traits() const { return st_; }

  const Data_structure&   data_structure() const { return this->_tds; }
  const Triangulation_data_structure& tds() const { return this->_tds; }
  const Point_container&  point_container() const { return pc_; }

  inline size_type number_of_input_sites() const {
    return isc_.size();
  }

  inline size_type number_of_output_sites() const {
    return number_of_vertices();
  }

  inline size_type number_of_vertices() const {
    return DG::number_of_vertices();
  }

  inline size_type number_of_faces() const {
    return DG::number_of_faces();
  }

  inline Vertex_handle infinite_vertex() const {
    return DG::infinite_vertex();
  }

  inline Face_handle infinite_face() const {
    return DG::infinite_face();
  }

  inline Vertex_handle finite_vertex() const {
    return DG::finite_vertex();
  }

  inline int dimension() const {
    return DG::dimension();
  }

  using Delaunay_graph::cw;
  using Delaunay_graph::ccw;
  using Delaunay_graph::delete_vertex;
  using Delaunay_graph::delete_face;
public:
  // TRAVERSAL OF THE DUAL GRAPH
  //----------------------------
  inline Finite_faces_iterator finite_faces_begin() const {
    return DG::finite_faces_begin();
  }

  inline Finite_faces_iterator finite_faces_end() const {
    return DG::finite_faces_end();
  }

  inline Finite_vertices_iterator finite_vertices_begin() const {
    return DG::finite_vertices_begin();
  }

  inline Finite_vertices_iterator finite_vertices_end() const {
    return DG::finite_vertices_end();
  }

  inline Finite_edges_iterator finite_edges_begin() const {
    return DG::finite_edges_begin();
  }

  inline Finite_edges_iterator finite_edges_end() const {
    return DG::finite_edges_end();
  }

  inline All_faces_iterator all_faces_begin() const {
    return DG::all_faces_begin();
  }

  inline All_faces_iterator all_faces_end() const {
    return DG::all_faces_end();
  }

  inline All_vertices_iterator all_vertices_begin() const {
    return DG::all_vertices_begin();
  }

  inline All_vertices_iterator all_vertices_end() const {
    return DG::all_vertices_end();
  }

  inline All_edges_iterator all_edges_begin() const {
    return DG::all_edges_begin();
  }

  inline All_edges_iterator all_edges_end() const {
    return DG::all_edges_end();
  }

  inline Input_sites_iterator input_sites_begin() const {
    return Input_sites_iterator(isc_.begin());
  }

  inline Input_sites_iterator input_sites_end() const {
    return Input_sites_iterator(isc_.end());
  }

  inline Output_sites_iterator output_sites_begin() const {
    return Output_sites_iterator(finite_vertices_begin());
  }

  inline Output_sites_iterator output_sites_end() const {
    return Output_sites_iterator(finite_vertices_end());
  }

public:
  // CIRCULATORS
  //------------
  inline Face_circulator
  incident_faces(Vertex_handle v,
                 Face_handle f = Face_handle()) const {
    return DG::incident_faces(v, f);
  }

  inline Vertex_circulator
  incident_vertices(Vertex_handle v,
                    Face_handle f = Face_handle()) const {
    return DG::incident_vertices(v, f);
  }

  inline Edge_circulator
  incident_edges(Vertex_handle v,
                 Face_handle f = Face_handle()) const {
    return DG::incident_edges(v, f);
  }

public:
  // PREDICATES
  //-----------
  inline bool is_infinite(const Vertex_handle& v) const {
    return DG::is_infinite(v);
  }

  inline bool is_infinite(const Face_handle& f) const {
    return DG::is_infinite(f);
  }

  inline bool is_infinite(const Face_handle& f, int i) const {
    return DG::is_infinite(f, i);
  }

  inline bool is_infinite(const Edge& e) const {
    return is_infinite(e.first, e.second);
  }

  inline bool is_infinite(const Edge_circulator& ec) const {
    return DG::is_infinite(ec);
  }

public:
  // INSERTION METHODS
  //------------------
  template<class Input_iterator>
  inline size_type insert(Input_iterator first, Input_iterator beyond) {
    return insert(first, beyond, Tag_false());
  }

  template<class Input_iterator>
  size_type insert(Input_iterator first, Input_iterator beyond,        Tag_false)
  {
    // do it the obvious way: insert them as they come;
    // one might think though that it might be better to first insert
    // all end points and then all segments, or a variation of that.
    size_type n_before = number_of_vertices();
    for (Input_iterator it = first; it != beyond; ++it) {
      insert(*it);
    }
    size_type n_after = number_of_vertices();
    return n_after - n_before;
  }

  //insert a range of points using spatial sorting
  std::size_t insert_points(std::vector<Point_2>& points)
  {
    size_type n = this->number_of_vertices();
    spatial_sort (points.begin(), points.end(), geom_traits());
    Vertex_handle hint;
    for (typename std::vector<Point_2>::const_iterator
          p = points.begin(), end = points.end(); p != end; ++p)
    {
      hint = insert(*p, hint);
    }

    return this->number_of_vertices() - n;
  }

  template <class PointIterator>
  std::size_t insert_points(PointIterator first, PointIterator beyond)
  {
    std::vector<Point_2> points (first, beyond);
    return insert_points(points);
  }

  template <class IndicesIterator>
  std::size_t insert_segments( const std::vector<Point_2>& points,
                               IndicesIterator indices_first,
                               IndicesIterator indices_beyond )
  {
    typedef std::vector<std::size_t> Vertex_indices;
    typedef std::vector<Vertex_handle> Vertices;

    Vertex_indices vertex_indices(boost::counting_iterator<std::size_t>(0),
                                  boost::counting_iterator<std::size_t>(points.size()));

    size_type n = this->number_of_vertices();
    Spatial_sort_traits_adapter_2<Gt,
                                  typename Pointer_property_map<Point_2>::const_type >
      sort_traits(make_property_map(points));

    spatial_sort(vertex_indices.begin(), vertex_indices.end(), sort_traits);

    Vertices vertices;
    vertices.resize(points.size());

    Vertex_handle hint;
    for(typename Vertex_indices::const_iterator
          it_pti = vertex_indices.begin(), end = vertex_indices.end();
          it_pti != end; ++it_pti)
    {
      hint = insert(points[*it_pti], hint);
      vertices[*it_pti] = hint;
    }

    for(IndicesIterator it_cst=indices_first, end=indices_beyond;
        it_cst!=end; ++it_cst)
    {
      Vertex_handle v1 = vertices[it_cst->first];
      Vertex_handle v2 = vertices[it_cst->second];
      if(v1 != v2) insert(v1, v2);
    }

    return this->number_of_vertices() - n;
  }

  template <class PointIterator, class IndicesIterator>
  std::size_t insert_segments(PointIterator points_first,
                              PointIterator points_beyond,
                              IndicesIterator indices_first,
                              IndicesIterator indices_beyond)
  {
    std::vector<Point_2> points (points_first, points_beyond);
    return insert_segments(points, indices_first, indices_beyond);
  }

  static const Point_2& get_source(const std::pair<Point_2, Point_2>& segment){
    return segment.first;
  }
  static const Point_2& get_target(const std::pair<Point_2, Point_2>& segment){
    return segment.second;
  }

  template <class Segment_2>
  static Point_2 get_source(const Segment_2& segment){
    return segment.source();
  }
  template <class Segment_2>
  static Point_2 get_target(const Segment_2& segment){
    return segment.target();
  }

  static const Point_2& get_source(const Site_2& segment){
    return segment.source_of_supporting_site();
  }

  static const Point_2& get_target(const Site_2& segment){
    return segment.target_of_supporting_site();
  }

  template <class SegmentIterator>
  std::size_t insert_segments(SegmentIterator first, SegmentIterator beyond)
  {
    std::vector<Point_2> points;
    for (SegmentIterator s_it=first; s_it!=beyond; ++s_it)
    {
      points.push_back( get_source(*s_it) );
      points.push_back( get_target(*s_it) );
    }

    std::vector< std::pair<std::size_t, std::size_t> > segment_indices;
    std::size_t nb_segments = points.size() / 2;
    segment_indices.reserve( nb_segments );
    for (std::size_t k=0; k < nb_segments; ++k)
      segment_indices.push_back( std::make_pair(2*k,2*k+1) );

    return insert_segments( points,
                            segment_indices.begin(),
                            segment_indices.end() );
  }

  template <class Input_iterator>
  inline size_type
  insert_range(Input_iterator first, Input_iterator beyond, Site_2){
    std::vector<Point_2> points;
    std::vector<Point_2> segment_points;
    std::vector< std::pair<std::size_t, std::size_t> > segment_indices;
    std::vector<Site_2> non_input_segments;

    for (Input_iterator it=first; it!=beyond; ++it)
    {
      if ( it->is_input() )
      {
        if (it->is_point() ) points.push_back( it->point() );
        else{
          segment_points.push_back( it->source_of_supporting_site() );
          segment_points.push_back( it->target_of_supporting_site() );
          segment_indices.push_back( std::make_pair(segment_points.size()-2,
                                                    segment_points.size()-1 ));
        }
      }
      else
        non_input_segments.push_back(*it);
    }
    //insert the points
    size_type n = insert_points(points);
    //insert the segments
    n += insert_segments( segment_points,
                          segment_indices.begin(),
                          segment_indices.end() );
    //insert non-input sites
    CGAL::cpp98::random_shuffle( non_input_segments.begin(), non_input_segments.end() );
    n += insert(non_input_segments.begin(),
                non_input_segments.end(), Tag_false() );
    return n;
  }

  template <class Input_iterator>
  inline size_type
  insert_range(Input_iterator first, Input_iterator beyond, Point_2){
    return insert_points(first, beyond);
  }

  template <class Input_iterator, class Segment_2>
  inline size_type
  insert_range(Input_iterator first, Input_iterator beyond, Segment_2){
    return insert_segments(first, beyond);
  }

  template<class Input_iterator>
  inline size_type
  insert(Input_iterator first, Input_iterator beyond, Tag_true)
  {
    return
      insert_range(first, beyond,
                   typename std::iterator_traits<Input_iterator>::value_type()
      );
  }

  // insert a point
  inline Vertex_handle insert(const Point_2& p) {
    // update input site container
    Point_handle ph = register_input_site(p);
    Storage_site_2 ss = st_.construct_storage_site_2_object()(ph);
    return insert_point(ss, p, Vertex_handle());
  }

  inline Vertex_handle insert(const Point_2& p, Vertex_handle vnear) {
    // update input site container
    Point_handle ph = register_input_site(p);
    Storage_site_2 ss = st_.construct_storage_site_2_object()(ph);
    return insert_point(ss, p, vnear);
  }

protected:
  // insert a point without registering it in the input sites
  // container: useful for the hierarchy
  inline Vertex_handle insert_no_register(const Storage_site_2& ss,
                                          const Point_2& p,
                                          Vertex_handle vnear) {
    return insert_point(ss, p, vnear);
  }

public:
  // insert a segment
  inline Vertex_handle insert(const Point_2& p0, const Point_2& p1) {
    // update input site container
    Point_handle_pair php = register_input_site(p0, p1);
    Storage_site_2 ss =
      st_.construct_storage_site_2_object()(php.first, php.second);
    Vertex_handle v = insert_segment(ss, Site_2::construct_site_2(p0, p1),
                                     Vertex_handle());
    if ( v == Vertex_handle() ) {
      unregister_input_site(php.first, php.second);
    }
    return v;
  }

  // inserting a segment whose endpoints have already been inserted
  // update input site container
  inline Vertex_handle insert(const Vertex_handle& v0,
                              const Vertex_handle& v1) {
    CGAL_precondition( v0->storage_site().is_point() &&
                       v1->storage_site().is_point() );

    Point_handle h0 = v0->storage_site().point();
    Point_handle h1 = v1->storage_site().point();
    Storage_site_2 ss = st_.construct_storage_site_2_object()(h0, h1);

    // update input site container
    Point_handle_pair php = register_input_site(h0, h1);

    if ( number_of_vertices() == 2 ) {
      return insert_third(ss, v0, v1);
    }

    Vertex_handle v = insert_segment_interior(ss.site(), ss, v0);
    if ( v == Vertex_handle() ) {
      unregister_input_site(php.first, php.second);
    }
    return v;
  }

  inline Vertex_handle insert(const Point_2& p0, const Point_2& p1,
                              Vertex_handle vnear) {
    // update input site container
    Point_handle_pair php = register_input_site(p0, p1);
    Storage_site_2 ss =
      st_.construct_storage_site_2_object()(php.first, php.second);
    Vertex_handle v =
      insert_segment(ss, Site_2::construct_site_2(p0, p1), vnear);
    if ( v == Vertex_handle() ) {
      unregister_input_site(php.first, php.second);
    }
    return v;
  }

  inline Vertex_handle insert(const Site_2& t) {
    return insert(t, Vertex_handle());
  }

  Vertex_handle insert(const Site_2& t, Vertex_handle vnear)
  {
    // the intended use is to unify the calls to insert(...);
    // thus the site must be an exact one;
    CGAL_precondition( t.is_input() );

    // update input site container

    if ( t.is_segment() ) {
      Point_handle_pair php =
        register_input_site( t.source_of_supporting_site(),
                             t.target_of_supporting_site() );
      Storage_site_2 ss =
        st_.construct_storage_site_2_object()(php.first, php.second);
      Vertex_handle v = insert_segment(ss, t, vnear);
      if ( v == Vertex_handle() ) {
        unregister_input_site( php.first, php.second );
      }
      return v;
    } else if ( t.is_point() ) {
      Point_handle ph = register_input_site( t.point() );
      Storage_site_2 ss = st_.construct_storage_site_2_object()(ph);
      return insert_point(ss, t.point(), vnear);
    } else {
      CGAL_precondition ( t.is_defined() );
      return Vertex_handle(); // to avoid compiler error
    }
  }

protected:
  template<class SSite>
  inline void convert_info1(SSite& ss_trg, const SSite& ss_src,
                            bool is_src, int,
                            typename SSite::Has_info_tag const* = 0) const
  {
    //    Rcpp::Rcerr << "converting info..." << std::flush;
    typename Storage_traits::Convert_info convert = st_.convert_info_object();

    ss_trg.set_info( convert(ss_src.info(), is_src) );
    //    Rcpp::Rcerr << " done!" << std::endl;
  }

  template<class SSite>
  inline void convert_info1(SSite& /*  ss_trg */,
                            const SSite& /* ss_src */, bool, char) const
  {
  }

  void convert_info(Storage_site_2& ss_trg,
                    const Storage_site_2& ss_src, bool is_src) const {
    CGAL_precondition( ss_src.is_segment() && ss_trg.is_point() );
    CGAL_precondition( ss_src.is_input() && ss_trg.is_input() );
    CGAL_assertion( (is_src && same_points(ss_src.source_site(), ss_trg)) ||
                    (!is_src && same_points(ss_src.target_site(), ss_trg))
                    );
    convert_info1(ss_trg, ss_src, is_src, 0);
  }

  template<class SSite>
  inline void merge_info1(Vertex_handle v, const SSite& ss, int,
                          typename SSite::Has_info_tag const* = 0)
  {
    //    Rcpp::Rcerr << "merging info..." << std::flush;
    Storage_site_2 ss_v = v->storage_site();

    typename Storage_traits::Merge_info merge = st_.merge_info_object();

    ss_v.set_info( merge(ss_v.info(), ss.info()) );
    v->set_site(ss_v);
    //    Rcpp::Rcerr << " done!" << std::endl;
  }

  template<class SSite>
  inline void merge_info1(Vertex_handle, const SSite&, char) const
  {
  }

  // merges the info of the storage site of the vertex handle with the
  // info of the given site; the vertex_handle contains the storage
  // site with the new info
  inline void merge_info(Vertex_handle v, const Storage_site_2& ss)  {
    CGAL_precondition( (v->storage_site().is_segment() &&
                        ss.is_segment() &&
                        same_segments(v->site(), ss.site())) ||
                       (v->storage_site().is_point() &&
                        ss.is_point() &&
                        same_points(v->site(), ss.site())) ||
                           (v->storage_site().is_point() &&        ss.is_segment())
                           );
    merge_info1(v, ss, 0);
  }

public:
  template<typename Info_t>
  inline Vertex_handle insert(const Site_2& t,
                              const Info_t& info) {
    return insert(t, info, Vertex_handle());
  }

  template<typename Info_t>
  Vertex_handle insert(const Site_2& t,
                       const Info_t& info,
                       Vertex_handle vnear)
  {
    typedef typename Storage_traits::Info Info;
    CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::Internal::
      Check_type_equality_for_info<Info_t, Info>();
    // the intended use is to unify the calls to insert(...);
    // thus the site must be an exact one;
    CGAL_precondition( t.is_input() );

    // update input site container

    if ( t.is_segment() ) {
      Point_handle_pair php =
        register_input_site( t.source_of_supporting_site(),
                             t.target_of_supporting_site() );
      Storage_site_2 ss =
        st_.construct_storage_site_2_object()(php.first, php.second);
      ss.set_info(info);
      Vertex_handle v = insert_segment(ss, t, vnear);
      if ( v == Vertex_handle() ) {
        unregister_input_site( php.first, php.second );
      }
      return v;
    } else if ( t.is_point() ) {
      Point_handle ph = register_input_site( t.point() );
      Storage_site_2 ss = st_.construct_storage_site_2_object()(ph);
      ss.set_info(info);
      return insert_point(ss, t.point(), vnear);
    } else {
      CGAL_precondition ( t.is_defined() );
      return Vertex_handle(); // to avoid compiler error
    }
  }

  // REMOVAL METHODS
  //----------------
protected:
  bool is_star(const Vertex_handle& v) const;
  bool is_linear_chain(const Vertex_handle& v0, const Vertex_handle& v1,
                       const Vertex_handle& v2) const;
  bool is_flippable(const Face_handle& f, int i) const;

  void minimize_degree(const Vertex_handle& v);

  // this method does not really do the job as intended, i.e., for removal
  void equalize_degrees(const Vertex_handle& v, Self& small_d,
                        std::map<Vertex_handle,Vertex_handle>& vmap,
                        List& l) const;

  void expand_conflict_region_remove(const Face_handle& in_f,
                                     const Site_2& t,
                                     List& l,
                                     Face_map& fm,
                                     Sign_map& sign_map);

  void find_conflict_region_remove(const Vertex_handle& v,
                                   const Vertex_handle& vnearest,
                                   List& l, Face_map& fm, Sign_map& vm);

  template<class OutputItFaces>
  OutputItFaces get_faces(const List& l, OutputItFaces fit) const
  {
    // map that determines if a face has been visited
    std::map<Face_handle,bool> fmap;

    // compute the initial face
    Edge e_front = l.front();
    Face_handle fstart = e_front.first->neighbor(e_front.second);

    // do the recursion
    return get_faces(l, fstart, fmap, fit);
  }

  template<class OutputItFaces>
  OutputItFaces get_faces(const List& l, Face_handle f,
                          std::map<Face_handle,bool>& fmap,
                          OutputItFaces fit) const
  {
    // if the face has been visited return
    if ( fmap.find(f) != fmap.end() ) { return fit; }

    // mark the face as visited
    fmap[f] = true;

    // output the face
    *fit++ = f;

    // recursively go to neighbors
    for (int i = 0; i < 3; i++) {
      Face_handle n = f->neighbor(i);
      Edge ee(n, n->index( this->_tds.mirror_vertex(f,i) ));
      if ( !l.is_in_list(ee) ) {
        fit = get_faces(l, n, fmap, fit);
      }
    }
    return fit;
  }

  size_type count_faces(const List& l) const;

  void fill_hole(const Self& small_d, const Vertex_handle& v, const List& l,
                 std::map<Vertex_handle,Vertex_handle>& vmap);

  bool remove_first(const Vertex_handle& v);
  bool remove_second(const Vertex_handle& v);
  bool remove_third(const Vertex_handle& v);

  void compute_small_diagram(const Vertex_handle& v, Self& small_d) const;
  void compute_vertex_map(const Vertex_handle& v, const Self& small_d,
                          std::map<Vertex_handle,Vertex_handle>& vmap) const;
  void remove_degree_d_vertex(const Vertex_handle& v);

  bool remove_base(const Vertex_handle& v);

public:
  bool remove(const Vertex_handle& v);

protected:
  inline void unregister_input_site(const Point_handle& h)
  {
    Site_rep_2 rep(h, h, true);
    typename Input_sites_container::iterator it = isc_.find(rep);
    CGAL_assertion( it != isc_.end() );

    pc_.erase(h);
    isc_.erase(it);
  }

  inline void unregister_input_site(const Point_handle& h1,
                                    const Point_handle& h2)
  {
    Site_rep_2 rep(h1, h2, false);
    typename Input_sites_container::iterator it = isc_.find(rep);

    Site_rep_2 sym_rep(h2, h1, false);
    typename Input_sites_container::iterator sym_it = isc_.find(sym_rep);

    CGAL_assertion( it != isc_.end() || sym_it != isc_.end() );

    if ( it != isc_.end() ) { isc_.erase(it); }
    if ( sym_it != isc_.end() ) { isc_.erase(sym_it); }

    Site_rep_2 r1(h1, h1, true);
    if ( isc_.find(r1) == isc_.end() ) { isc_.insert(r1); }

    Site_rep_2 r2(h2, h2, true);
    if ( isc_.find(r2) == isc_.end() ) { isc_.insert(r2); }
  }

  inline Point_handle register_input_site(const Point_2& p)
  {
    std::pair<Point_handle,bool> it = pc_.insert(p);
    Site_rep_2 rep(it.first, it.first, true);
    isc_.insert( rep );
    return it.first;
  }

  inline
  Point_handle_pair register_input_site(const Point_2& p0, const Point_2& p1)
  {
    std::pair<Point_handle,bool> it1 = pc_.insert(p0);
    std::pair<Point_handle,bool> it2 = pc_.insert(p1);
    Site_rep_2 rep(it1.first, it2.first, false);
    isc_.insert( rep );
    return Point_handle_pair(it1.first, it2.first);
  }

  inline
  Point_handle_pair register_input_site(const Point_handle& h0,
                                        const Point_handle& h1)
  {
    CGAL_precondition( h0 != h1 );
    Site_rep_2 rep(h0, h1, false);
    isc_.insert( rep );
    return Point_handle_pair(h0, h1);
  }

  Vertex_handle  insert_first(const Storage_site_2& ss, const Point_2& p);
  Vertex_handle  insert_second(const Storage_site_2& ss, const Point_2& p);
  Vertex_handle  insert_third(const Storage_site_2& ss, const Point_2& p);
  Vertex_handle  insert_third(const Site_2& t, const Storage_site_2& ss);
  Vertex_handle  insert_third(const Storage_site_2& ss, Vertex_handle v0,
                              Vertex_handle v1);

  Vertex_handle insert_point(const Storage_site_2& ss, const Point_2& p,
                             Vertex_handle vnear);
  Vertex_handle insert_point(const Storage_site_2& ss,
                             const Site_2& t, Vertex_handle vnear);
  Vertex_handle insert_point2(const Storage_site_2& ss,
                              const Site_2& t, Vertex_handle vnear);

  Vertex_handle insert_segment(const Storage_site_2& ss, const Site_2& t,
                               Vertex_handle vnear);

  Vertex_handle insert_segment_interior(const Site_2& t,
                                        const Storage_site_2& ss,
                                        Vertex_handle vnear);

  template<class ITag>
  inline
  Vertex_handle insert_intersecting_segment(const Storage_site_2& ss,
                                            const Site_2& t,
                                            Vertex_handle v,
                                            ITag tag) {
    return insert_intersecting_segment_with_tag(ss, t, v, tag);
  }

  Vertex_handle
  insert_intersecting_segment_with_tag(const Storage_site_2& ss,
                                       const Site_2& t,
                                       Vertex_handle v, Tag_false);

  Vertex_handle
  insert_intersecting_segment_with_tag(const Storage_site_2& ss,
                                       const Site_2& t,
                                       Vertex_handle v, Tag_true);



public:
  // NEAREST NEIGHBOR LOCATION
  //--------------------------
  inline Vertex_handle nearest_neighbor(const Point_2& p) const {
    return nearest_neighbor(Site_2::construct_site_2(p), Vertex_handle());
  }

  inline Vertex_handle nearest_neighbor(const Point_2& p,
                                        Vertex_handle vnear) const {
    return nearest_neighbor(Site_2::construct_site_2(p), vnear);
  }

protected:
  Vertex_handle nearest_neighbor(const Site_2& p,
                                 Vertex_handle vnear) const;


protected:
  // I/O METHODS
  //------------
  typedef std::map<const_Point_handle,size_type,Point_handle_less_than>
  Point_handle_mapper;

  typedef std::vector<Point_handle> Point_handle_vector;

  void file_output(std::ostream&, const Storage_site_2&,
                   Point_handle_mapper&) const;

  void file_output(std::ostream&, Point_handle_mapper&,
                   bool print_point_container) const;

  void file_input(std::istream&, Storage_site_2&,
                  const Point_handle_vector&, const Tag_true&) const;
  void file_input(std::istream&, Storage_site_2&,
                  const Point_handle_vector&, const Tag_false&) const;
  void file_input(std::istream&, bool read_handle_vector,
                  Point_handle_vector&);

public:
  void file_input(std::istream& is) {
    Point_handle_vector P;
    file_input(is, true, P);
  }

  void file_output(std::ostream& os) const {
    Point_handle_mapper P;
    size_type inum = 0;
    for (const_Point_handle ph = pc_.begin(); ph != pc_.end(); ++ph) {
      P[ph] = inum++;
    }
    file_output(os, P, true);
  }

  template< class Stream >
  Stream& draw_dual(Stream& str) const
  {
    Finite_edges_iterator eit = finite_edges_begin();
    for (; eit != finite_edges_end(); ++eit) {
      draw_dual_edge(*eit, str);
    }
    return str;
  }

  template < class Stream >
  Stream& draw_skeleton(Stream& str) const
  {
    Finite_edges_iterator eit = finite_edges_begin();
    for (; eit != finite_edges_end(); ++eit) {
      Site_2 p = eit->first->vertex(  cw(eit->second) )->site();
      Site_2 q = eit->first->vertex( ccw(eit->second) )->site();

      bool is_endpoint_of_seg =
        ( p.is_segment() && q.is_point() &&
          is_endpoint_of_segment(q, p) ) ||
        ( p.is_point() && q.is_segment() &&
          is_endpoint_of_segment(p, q) );

      if ( !is_endpoint_of_seg ) {
        draw_dual_edge(*eit, str);
      }
    }
    return str;
  }

  // MK: this has to be rewritten. all the checking must be done in
  // the geometric traits class.
  template< class Stream >
  Stream& draw_dual_edge(Edge e, Stream& str) const
  {
    CGAL_precondition( !is_infinite(e) );

    typename Geom_traits::Line_2              l;
    typename Geom_traits::Segment_2           s;
    typename Geom_traits::Ray_2               r;
    CGAL::Parabola_segment_2<Gt>              ps;

    Object o = primal(e);

    if (CGAL::assign(l, o))   str << l;
    if (CGAL::assign(s, o))   str << s;
    if (CGAL::assign(r, o))   str << r;
    if (CGAL::assign(ps, o))  ps.draw(str);

    return str;
  }

  template< class Stream >
  inline
  Stream& draw_dual_edge(Edge_circulator ec, Stream& str) const {
    return draw_dual_edge(*ec, str);
  }

  template< class Stream >
  inline
  Stream& draw_dual_edge(Finite_edges_iterator eit, Stream& str) const {
    return draw_dual_edge(*eit, str);
  }

public:
  // VALIDITY CHECK
  //---------------
  bool is_valid(bool verbose = false, int level = 1) const;

public:
  // MISCELLANEOUS
  //--------------
  void clear() {
    DG::clear();
    pc_.clear();
    isc_.clear();
  }

  void swap(Segment_Delaunay_graph_2& sdg) {
    DG::swap(sdg);
    pc_.swap(sdg.pc_);
    isc_.swap(sdg.isc_);
  }

  //////////////////////////////////////////////////////////////////////
  // THE METHODS BELOW ARE LOCAL
  //////////////////////////////////////////////////////////////////////

protected:
  // THE COPY METHOD
  //------------------------------------------------------------------
  // used in the copy constructor and assignment operator

  Storage_site_2
  copy_storage_site(const Storage_site_2& ss_other,
                    Handle_map& hm, const Tag_false&);

  Storage_site_2
  copy_storage_site(const Storage_site_2& ss_other,
                    Handle_map& hm, const Tag_true&);

  void copy(Segment_Delaunay_graph_2& other);
  void copy(Segment_Delaunay_graph_2& other, Handle_map& hm);

protected:
  // HELPER METHODS FOR COMBINATORIAL OPERATIONS ON THE DATA STRUCTURE
  //------------------------------------------------------------------

  // getting the degree of a vertex
  inline
  typename Data_structure::size_type degree(const Vertex_handle& v) const {
    return this->_tds.degree(v);
  }

  // getting the symmetric edge
  inline Edge sym_edge(const Edge e) const {
    return sym_edge(e.first, e.second);
  }

  inline Edge sym_edge(const Face_handle& f, int i) const {
    Face_handle f_sym = f->neighbor(i);
#ifdef CGAL_SDG_ALTERNATE_SYMEDGE_IMPLEMENTATION_BY_AF
    int count = ( f_sym->neighbor(0) == f );
    int i_sym = 0;
    if ( f_sym->neighbor(1) == f ) {
      ++count;
      i_sym = 1;
    }
    if ( f_sym->neighbor(2) == f ) {
      ++count;
      i_sym = 2;
    }
    if ( count == 1 ) {
      return Edge(f_sym, i_sym);
    }
    return Edge(f_sym, f_sym->index( f->vertex(i) ));
#else
    return Edge(  f_sym, f_sym->index( this->_tds.mirror_vertex(f, i) )  );
#endif
  }

  Edge flip(Face_handle& f, int i) {
    CGAL_precondition ( f != Face_handle() );
    CGAL_precondition (i == 0 || i == 1 || i == 2);
    CGAL_precondition( this->dimension()==2 );

    CGAL_precondition( f->vertex(i) != this->_tds.mirror_vertex(f, i) );

    this->_tds.flip(f, i);

    return Edge(f, ccw(i));
  }

  inline Edge flip(Edge e) {
    return flip(e.first, e.second);
  }

  inline bool is_degree_2(const Vertex_handle& v) const {
    Face_circulator fc = incident_faces(v);
    Face_circulator fc1 = fc;
    ++(++fc1);
    return ( fc == fc1 );
  }

  inline Vertex_handle insert_degree_2(Edge e) {
    return this->_tds.insert_degree_2(e.first,e.second);
  }

  inline Vertex_handle insert_degree_2(Edge e, const Storage_site_2& ss) {
    Vertex_handle v = insert_degree_2(e);
    v->set_site(ss);
    return v;
  }

  inline void remove_degree_2(Vertex_handle v) {
    CGAL_precondition( is_degree_2(v) );
    this->_tds.remove_degree_2(v);
  }

  inline void remove_degree_3(Vertex_handle v) {
    CGAL_precondition( degree(v) == 3 );
    this->_tds.remove_degree_3(v, Face_handle());
  }

  inline Vertex_handle create_vertex(const Storage_site_2& ss) {
    Vertex_handle v = this->_tds.create_vertex();
    v->set_site(ss);
    return v;
  }

  inline Vertex_handle create_vertex_dim_up(const Storage_site_2& ss) {
    Vertex_handle v = this->_tds.insert_dim_up(infinite_vertex());
    v->set_site(ss);
    return v;
  }

protected:
  // HELPER METHODS FOR CREATING STORAGE SITES
  //------------------------------------------
  inline
  Storage_site_2 split_storage_site(const Storage_site_2& ss0,
                                    const Storage_site_2& ss1,
                                    bool first)
  {
    // Split the first storage site which is a segment using the
    // second storage site which is an exact point
    // i denotes whether the first or second half is to be created
    CGAL_precondition( ss0.is_segment() && ss1.is_point() );

    return st_.construct_storage_site_2_object()(ss0, ss1, first);
  }

public:
  // METHODS FOR ACCESSING THE PRIMAL GRAPH
  //---------------------------------------
  // used primarily for visualization
  inline Point_2 primal(const Face_handle& f) const {
    return circumcenter(f);
  }

  Object primal(const Edge e) const;

  inline Object primal(const Edge_circulator& ec) const {
    return primal(*ec);
  }

  inline Object primal(const Finite_edges_iterator& ei) const {
    return primal(*ei);
  }

protected:
  void print_error_message() const;

public:
  // wrappers for constructions
  inline Point_2 circumcenter(const Face_handle& f) const {
    CGAL_precondition( this->dimension()==2 || !is_infinite(f) );
    return circumcenter(f->vertex(0)->site(),
                        f->vertex(1)->site(),
                        f->vertex(2)->site());
  }

  inline Point_2 circumcenter(const Site_2& t0, const Site_2& t1,
                              const Site_2& t2) const {
    return
    geom_traits().construct_svd_vertex_2_object()(t0, t1, t2);
  }

protected:
  // HELPER METHODS FOR INSERTION
  //-----------------------------
  void initialize_conflict_region(const Face_handle& f, List& l);

  std::pair<Face_handle,Face_handle>
  find_faces_to_split(const Vertex_handle& v, const Site_2& t) const;

  bool check_unregistered_face(const Face_handle& n,
                               const Site_2& t,
                               List& l,
#ifndef CGAL_SDG_NO_FACE_MAP
                               Face_map& fm,
#endif
                               Triple<bool, Vertex_handle, Arrangement_type>& vcross);

  void expand_conflict_region(const Face_handle& in_f,
                              const Site_2& t,
                              List& l,
#ifndef CGAL_SDG_NO_FACE_MAP
                              Face_map& fm,
                              std::map<Face_handle, Sign>& sign_map,
#endif
                              Triple<bool, Vertex_handle, Arrangement_type>& vcross);

  Vertex_handle add_bogus_vertex(Edge e, List& l);
  Vertex_list   add_bogus_vertices(List& l);
  void          remove_bogus_vertices(Vertex_list& vl);

#ifdef CGAL_SDG_NO_FACE_MAP
  void retriangulate_conflict_region(Vertex_handle v, List& l);
#else
  void retriangulate_conflict_region(Vertex_handle v, List& l,
                                     Face_map& fm);
#endif


// choosing the correct bisector constructors
private:
  template <typename T, typename Tag_has_bisector_constructions>
  struct ConstructionHelper {};

  // take constructors from L2
  template <typename T>
  struct ConstructionHelper<T, Tag_false>
  {
    typedef CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::
              Construct_sdg_bisector_2<Gt,
                Integral_domain_without_division_tag>
            tagbis;
    typedef CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::
              Construct_sdg_bisector_ray_2<Gt,
                Integral_domain_without_division_tag>
            tagbisray;
    typedef CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::
              Construct_sdg_bisector_segment_2<Gt,
                Integral_domain_without_division_tag>
            tagbisseg;
  };

  // constructors from traits
  template <typename T>
  struct ConstructionHelper<T, Tag_true>
  {
    typedef
            typename T:: template Construct_sdg_bisector_2
              <Gt, Integral_domain_without_division_tag>
            tagbis;
    typedef
            typename T:: template Construct_sdg_bisector_ray_2
              <Gt, Integral_domain_without_division_tag>
            tagbisray;
    typedef
            typename T:: template Construct_sdg_bisector_segment_2
              <Gt, Integral_domain_without_division_tag>
            tagbisseg;
  };

  template <typename T>
  struct ConstructionChooser
  {
    typedef typename ConstructionHelper<T, typename T::Tag_has_bisector_constructions>::tagbis tagbis;
    typedef typename ConstructionHelper<T, typename T::Tag_has_bisector_constructions>::tagbisray tagbisray;
    typedef typename ConstructionHelper<T, typename T::Tag_has_bisector_constructions>::tagbisseg tagbisseg;
  };


protected:
  // TYPES AND ACCESS METHODS FOR VISUALIZATION
  //-------------------------------------------

  // types
  typedef
  CGAL_SEGMENT_DELAUNAY_GRAPH_2_NS::Construct_sdg_circle_2<Gt,Integral_domain_without_division_tag>
  Construct_sdg_circle_2;

  typedef typename
          ConstructionChooser<Geom_traits>::tagbis
          Construct_sdg_bisector_2;
  typedef typename
          ConstructionChooser<Geom_traits>::tagbisray
          Construct_sdg_bisector_ray_2;
  typedef typename
          ConstructionChooser<Geom_traits>::tagbisseg
          Construct_sdg_bisector_segment_2;

  // access
  inline Construct_sdg_circle_2
  construct_sdg_circle_2_object() const{
    return Construct_sdg_circle_2();
  }

  inline Construct_sdg_bisector_2
  construct_sdg_bisector_2_object() const {
    return Construct_sdg_bisector_2();
  }

  inline Construct_sdg_bisector_ray_2
  construct_sdg_bisector_ray_2_object() const {
    return Construct_sdg_bisector_ray_2();
  }

  inline Construct_sdg_bisector_segment_2
  construct_sdg_bisector_segment_2_object() const {
    return Construct_sdg_bisector_segment_2();
  }

protected:
  // WRAPPERS FOR GEOMETRIC PREDICATES
  //----------------------------------
  inline
  bool same_points(const Storage_site_2& p, const Storage_site_2& q) const {
    return geom_traits().equal_2_object()(p.site(), q.site());
  }

  inline
  bool same_segments(const Storage_site_2& t, Vertex_handle v) const {
    if ( is_infinite(v) ) { return false; }
    if ( t.is_point() || v->storage_site().is_point() ) { return false; }
    return same_segments(t.site(), v->site());
  }

  inline
  bool is_endpoint_of_segment(const Storage_site_2& p,
                              const Storage_site_2& s) const
  {
    CGAL_precondition( p.is_point() && s.is_segment() );
    return ( same_points(p, s.source_site()) ||
             same_points(p, s.target_site()) );
  }

  inline
  bool is_degenerate_segment(const Storage_site_2& s) const {
    CGAL_precondition( s.is_segment() );
    return same_points(s.source_site(), s.target_site());
  }

  // returns:
  //   ON_POSITIVE_SIDE if q is closer to t1
  //   ON_NEGATIVE_SIDE if q is closer to t2
  //   ON_ORIENTED_BOUNDARY if q is on the bisector of t1 and t2
  inline
  Oriented_side side_of_bisector(const Storage_site_2 &t1,
                                 const Storage_site_2 &t2,
                                 const Storage_site_2 &q) const {
    return
      geom_traits().oriented_side_of_bisector_2_object()(t1.site(),
                                                         t2.site(),
                                                         q.site());
  }

  inline
  Sign incircle(const Storage_site_2 &t1, const Storage_site_2 &t2,
                const Storage_site_2 &t3, const Storage_site_2 &q) const {
#ifdef CGAL_PROFILE_SDG_DUMP_INCIRCLE
    typedef typename Geom_traits::FT  FT;
    if ( !Algebraic_structure_traits<FT>::Is_exact::value ) {
      std::ofstream ofs("incircle.cin", std::ios_base::app);
      ofs.precision(16);
      ofs << t1.site() << " ";
      ofs << t2.site() << " ";
      ofs << t3.site() << " ";
      ofs <<  q.site() << std::endl;
      ofs.close();
    }
#endif
    return geom_traits().vertex_conflict_2_object()(t1.site(),
                                                    t2.site(),
                                                    t3.site(),
                                                    q.site());
  }

  inline
  Sign incircle(const Storage_site_2 &t1, const Storage_site_2 &t2,
                const Storage_site_2 &q) const {
    return geom_traits().vertex_conflict_2_object()(t1.site(),
                                                    t2.site(),
                                                    q.site());
  }

  inline
  Sign incircle(const Face_handle& f, const Storage_site_2& q) const {
    return incircle(f, q.site());
  }

  inline
  bool finite_edge_interior(const Storage_site_2& t1,
                            const Storage_site_2& t2,
                            const Storage_site_2& t3,
                            const Storage_site_2& t4,
                            const Storage_site_2& q, Sign sgn) const {
    return
      geom_traits().finite_edge_interior_conflict_2_object()
      (t1.site(), t2.site(), t3.site(), t4.site(), q.site(), sgn);
  }

  inline
  bool finite_edge_interior(const Face_handle& f, int i,
                            const Storage_site_2& q, Sign sgn) const {
    CGAL_precondition( !is_infinite(f) &&
                       !is_infinite(f->neighbor(i)) );
    return finite_edge_interior( f->vertex( ccw(i) )->site(),
                                 f->vertex(  cw(i) )->site(),
                                 f->vertex(     i  )->site(),
                                 this->_tds.mirror_vertex(f, i)->site(),
                                 q.site(), sgn);
  }

  inline
  bool finite_edge_interior(const Storage_site_2& t1,
                            const Storage_site_2& t2,
                            const Storage_site_2& t3,
                            const Storage_site_2& q,
                            Sign sgn) const {
    return geom_traits().finite_edge_interior_conflict_2_object()(t1.site(),
                                                                  t2.site(),
                                                                  t3.site(),
                                                                  q.site(),
                                                                  sgn);
  }

  inline
  bool finite_edge_interior(const Storage_site_2& t1,
                            const Storage_site_2& t2,
                            const Storage_site_2& q,
                            Sign sgn) const {
    return
      geom_traits().finite_edge_interior_conflict_2_object()(t1.site(),
                                                             t2.site(),
                                                             q.site(),
                                                             sgn);
  }

  bool finite_edge_interior(const Face_handle& f, int i,
                            const Storage_site_2& p, Sign sgn,
                            int j) const {
    return finite_edge_interior(f, i, p.site(), sgn, j);
  }

  inline
  bool infinite_edge_interior(const Storage_site_2& t2,
                              const Storage_site_2& t3,
                              const Storage_site_2& t4,
                              const Storage_site_2& q, Sign sgn) const {
    return
      geom_traits().infinite_edge_interior_conflict_2_object()
      (t2.site(), t3.site(), t4.site(), q.site(), sgn);
  }

  inline
  bool infinite_edge_interior(const Face_handle& f, int i,
                              const Storage_site_2& q, Sign sgn) const
  {
    return infinite_edge_interior(f, i, q, sgn);
  }

  inline
  bool edge_interior(const Face_handle& f, int i,
                     const Storage_site_2& t, Sign sgn) const {
    return edge_interior(f, i, t.site(), sgn);
  }

  inline
  bool edge_interior(const Edge& e,
                     const Storage_site_2& t, Sign sgn) const {
    return edge_interior(e.first, e.second, t, sgn);
  }

  inline Arrangement_type
  arrangement_type(const Storage_site_2& t, const Vertex_handle& v) const {
    if ( is_infinite(v) ) { return AT2::DISJOINT; }
    return arrangement_type(t, v->storage_site());
  }

  inline
  Arrangement_type arrangement_type(const Storage_site_2& p,
                                    const Storage_site_2& q) const {
    return arrangement_type(p.site(), q.site());
  }

  inline
  bool are_parallel(const Storage_site_2& p, const Storage_site_2& q) const {
    return geom_traits().are_parallel_2_object()(p.site(), q.site());
  }

  inline Oriented_side
  oriented_side(const Storage_site_2& s1, const Storage_site_2& s2,
                const Storage_site_2& supp,
                const Storage_site_2& p) const
  {
    CGAL_precondition( supp.is_segment() && p.is_point() );
    return
      geom_traits().oriented_side_2_object()(
          s1.site(), s2.site(), supp.site(), p.site());
  }

  inline Oriented_side
  oriented_side(const Storage_site_2& s1, const Storage_site_2& s2,
                const Storage_site_2& s3, const Storage_site_2& supp,
                const Storage_site_2& p) const {
    CGAL_precondition( supp.is_segment() && p.is_point() );
    return geom_traits().oriented_side_2_object()(s1.site(),
                                                  s2.site(),
                                                  s3.site(),
                                                  supp.site(), p.site());
  }


  //-------

  inline
  bool same_points(const Site_2& p, const Site_2& q) const {
    return geom_traits().equal_2_object()(p, q);
  }

  inline
  bool same_segments(const Site_2& t, Vertex_handle v) const {
    if ( is_infinite(v) ) { return false; }
    if ( t.is_point() || v->site().is_point() ) { return false; }
    return same_segments(t, v->site());
  }

  inline
  bool same_segments(const Site_2& p, const Site_2& q) const {
    CGAL_precondition( p.is_segment() && q.is_segment() );

    return
      (same_points(p.source_site(), q.source_site()) &&
       same_points(p.target_site(), q.target_site())) ||
      (same_points(p.source_site(), q.target_site()) &&
       same_points(p.target_site(), q.source_site()));
  }

  inline
  bool is_endpoint_of_segment(const Site_2& p, const Site_2& s) const
  {
    CGAL_precondition( p.is_point() && s.is_segment() );
    return ( same_points(p, s.source_site()) ||
             same_points(p, s.target_site()) );
  }

  inline
  bool is_degenerate_segment(const Site_2& s) const {
    CGAL_precondition( s.is_segment() );
    return same_points(s.source_site(), s.target_site());
  }

  // returns:
  //   ON_POSITIVE_SIDE if q is closer to t1
  //   ON_NEGATIVE_SIDE if q is closer to t2
  //   ON_ORIENTED_BOUNDARY if q is on the bisector of t1 and t2
  inline
  Oriented_side side_of_bisector(const Site_2 &t1, const Site_2 &t2,
                                 const Site_2 &q) const {
    return geom_traits().oriented_side_of_bisector_2_object()(t1, t2, q);
  }

  inline
  Sign incircle(const Site_2 &t1, const Site_2 &t2,
                const Site_2 &t3, const Site_2 &q) const {
#ifdef CGAL_PROFILE_SDG_DUMP_INCIRCLE
    typedef typename Geom_traits::FT  FT;
    if ( !Algebraic_structure_traits<FT>::Is_exact::value ) {
      std::ofstream ofs("incircle.cin", std::ios_base::app);
      ofs.precision(16);
      ofs << t1 << " ";
      ofs << t2 << " ";
      ofs << t3 << " ";
      ofs <<  q << std::endl;
      ofs.close();
    }
#endif
    return geom_traits().vertex_conflict_2_object()(t1, t2, t3, q);
  }

  inline
  Sign incircle(const Site_2 &t1, const Site_2 &t2,
                const Site_2 &q) const {
    return geom_traits().vertex_conflict_2_object()(t1, t2, q);
  }

  inline
  Sign incircle(const Face_handle& f, const Site_2& q) const;

  inline
  Sign incircle(const Vertex_handle& v0, const Vertex_handle& v1,
                const Vertex_handle& v) const {
    CGAL_precondition( !is_infinite(v0) && !is_infinite(v1)
                       && !is_infinite(v) );

    return incircle( v0->site(), v1->site(), v->site());
  }

  Sign incircle(const Vertex_handle& v0, const Vertex_handle& v1,
                const Vertex_handle& v2, const Vertex_handle& v) const;

  inline
  bool finite_edge_interior(const Site_2& t1, const Site_2& t2,
                            const Site_2& t3, const Site_2& t4,
                            const Site_2& q,  Sign sgn) const {
    return
      geom_traits().finite_edge_interior_conflict_2_object()
      (t1,t2,t3,t4,q,sgn);
  }

  inline
  bool finite_edge_interior(const Face_handle& f, int i,
                            const Site_2& q, Sign sgn) const {
    CGAL_precondition( !is_infinite(f) &&
                       !is_infinite(f->neighbor(i)) );
    return finite_edge_interior( f->vertex( ccw(i) )->site(),
                                 f->vertex(  cw(i) )->site(),
                                 f->vertex(     i  )->site(),
                                 this->_tds.mirror_vertex(f, i)->site(),
                                 q, sgn);
  }

  inline
  bool finite_edge_interior(const Vertex_handle& v1, const Vertex_handle& v2,
                            const Vertex_handle& v3, const Vertex_handle& v4,
                            const Vertex_handle& v, Sign sgn) const {
    CGAL_precondition( !is_infinite(v1) && !is_infinite(v2) &&
                       !is_infinite(v3) && !is_infinite(v4) &&
                       !is_infinite(v) );
    return finite_edge_interior( v1->site(), v2->site(),
                                 v3->site(), v4->site(),
                                 v->site(), sgn);
  }

  inline
  bool finite_edge_interior(const Site_2& t1, const Site_2& t2,
                            const Site_2& t3, const Site_2& q,
                            Sign sgn) const {
    return
    geom_traits().finite_edge_interior_conflict_2_object()(t1,t2,t3,q,sgn);
  }

  inline
  bool finite_edge_interior(const Site_2& t1, const Site_2& t2,
                            const Site_2& q,  Sign sgn) const {
    return
    geom_traits().finite_edge_interior_conflict_2_object()(t1,t2,q,sgn);
  }

  bool finite_edge_interior(const Face_handle& f, int i,
                            const Site_2& p, Sign sgn, int) const;

  bool finite_edge_interior(const Vertex_handle& v1, const Vertex_handle& v2,
                            const Vertex_handle& v3, const Vertex_handle& v4,
                            const Vertex_handle& v, Sign, int) const;

  inline
  bool infinite_edge_interior(const Site_2& t2, const Site_2& t3,
                              const Site_2& t4, const Site_2& q,
                              Sign sgn) const {
    return
      geom_traits().infinite_edge_interior_conflict_2_object()
      (t2,t3,t4,q,sgn);
  }


  bool infinite_edge_interior(const Face_handle& f, int i,
                              const Site_2& q, Sign sgn) const;

  bool infinite_edge_interior(const Vertex_handle& v1,
                              const Vertex_handle& v2,
                              const Vertex_handle& v3,
                              const Vertex_handle& v4,
                              const Vertex_handle& v,
                              Sign sgn) const;

  bool edge_interior(const Face_handle& f, int i,
                     const Site_2& t, Sign sgn) const;

  bool edge_interior(const Edge& e,
                     const Site_2& t, Sign sgn) const {
    return edge_interior(e.first, e.second, t, sgn);
  }

  bool edge_interior(const Vertex_handle& v1,
                     const Vertex_handle& v2,
                     const Vertex_handle& v3,
                     const Vertex_handle& v4,
                     const Vertex_handle& v,
                     Sign sgn) const;

  inline Arrangement_type
  arrangement_type(const Site_2& t, const Vertex_handle& v) const {
    if ( is_infinite(v) ) { return AT2::DISJOINT; }
    return arrangement_type(t, v->site());
  }

  Arrangement_type arrangement_type(const Site_2& p, const Site_2& q) const;

  inline
  bool are_parallel(const Site_2& p, const Site_2& q) const {
    return geom_traits().are_parallel_2_object()(p, q);
  }

  inline Oriented_side
  oriented_side(const Site_2& s1, const Site_2&s2,
                const Site_2& supp, const Site_2& p) const
  {
    CGAL_precondition( supp.is_segment() && p.is_point() );
    return geom_traits().oriented_side_2_object()(s1, s2, supp, p);
  }

  inline Oriented_side
  oriented_side(const Site_2& s1, const Site_2& s2, const Site_2& s3,
                const Site_2& supp, const Site_2& p) const {
    CGAL_precondition( supp.is_segment() && p.is_point() );
    return geom_traits().oriented_side_2_object()(s1, s2, s3, supp, p);
  }

  bool is_degenerate_edge(const Site_2& p1,
                          const Site_2& p2,
                          const Site_2& p3,
                          const Site_2& p4) const {
    return geom_traits().is_degenerate_edge_2_object()
      (p1, p2, p3, p4);
  }

  bool is_degenerate_edge(const Vertex_handle& v1,
                          const Vertex_handle& v2,
                          const Vertex_handle& v3,
                          const Vertex_handle& v4) const {
    CGAL_precondition( !is_infinite(v1) && !is_infinite(v2) &&
                       !is_infinite(v3) && !is_infinite(v4) );

    return is_degenerate_edge(v1->site(), v2->site(),
                              v3->site(), v4->site());
  }

  bool is_degenerate_edge(const Face_handle& f, int i) const {
    Vertex_handle v1 = f->vertex( ccw(i) );
    Vertex_handle v2 = f->vertex(  cw(i) );
    Vertex_handle v3 = f->vertex(     i  );
    Vertex_handle v4 = this->_tds.mirror_vertex(f, i);

    return is_degenerate_edge(v1, v2, v3, v4);
  }

  bool is_degenerate_edge(const Edge& e) const {
    return is_degenerate_edge(e.first, e.second);
  }

  Vertex_handle first_endpoint_of_segment(const Vertex_handle& v) const;
  Vertex_handle second_endpoint_of_segment(const Vertex_handle& v) const;

}; // Segment_Delaunay_graph_2


template<class Gt, class D_S, class LTag>
std::istream& operator>>(std::istream& is,
                         Segment_Delaunay_graph_2<Gt,D_S,LTag>& sdg)
{
  sdg.file_input(is);
  return is;
}

template<class Gt, class D_S, class LTag>
std::ostream& operator<<(std::ostream& os,
                         const Segment_Delaunay_graph_2<Gt,D_S,LTag>& sdg)
{
  sdg.file_output(os);
  return os;
}

} //namespace CGAL


#include <CGAL/Segment_Delaunay_graph_2/Segment_Delaunay_graph_2_impl.h>

#include <CGAL/enable_warnings.h>

#endif // CGAL_SEGMENT_DELAUNAY_GRAPH_2_H

