// Copyright (c) 2016-2018  INRIA Sophia Antipolis, INRIA Nancy (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.5.2/Periodic_4_hyperbolic_triangulation_2/include/CGAL/Periodic_4_hyperbolic_Delaunay_triangulation_2.h $
// $Id: Periodic_4_hyperbolic_Delaunay_triangulation_2.h 98e4718 2021-08-26T11:33:39+02:00 Sébastien Loriot
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)     : Iordan Iordanov

#ifndef CGAL_PERIODIC_4_HYPERBOLIC_DELAUNAY_TRIANGULATION_2_H
#define CGAL_PERIODIC_4_HYPERBOLIC_DELAUNAY_TRIANGULATION_2_H

#include <CGAL/license/Periodic_4_hyperbolic_triangulation_2.h>

#include <CGAL/Periodic_4_hyperbolic_triangulation_2.h>
#include <CGAL/Periodic_4_hyperbolic_triangulation_vertex_base_2.h>
#include <CGAL/Periodic_4_hyperbolic_triangulation_face_base_2.h>
#include <CGAL/Periodic_4_hyperbolic_Delaunay_triangulation_traits_2.h>
#include <CGAL/Triangulation_vertex_base_2.h>
#include <CGAL/Triangulation_face_base_2.h>
#include <CGAL/Delaunay_triangulation_2.h>

#include <CGAL/intersections.h>
#include <CGAL/iterator.h>
#include <CGAL/Timer.h>

#include <iterator>
#include <map>
#include <set>
#include <vector>

#if defined PROFILING_MODE
#include <CGAL/Timer.h>
extern long calls_predicate_identity;
extern long calls_predicate_non_identity;
extern double time_predicate_identity;
extern double time_predicate_non_identity;
extern double time_remove_dp;
#endif

namespace CGAL {

template <class GT = Periodic_4_hyperbolic_Delaunay_triangulation_traits_2<>,
          class TDS = Triangulation_data_structure_2<
                        Periodic_4_hyperbolic_triangulation_vertex_base_2<GT>,
                        Periodic_4_hyperbolic_triangulation_face_base_2<GT> > >
class Periodic_4_hyperbolic_Delaunay_triangulation_2
  : public Periodic_4_hyperbolic_triangulation_2<GT, TDS>
{

  typedef Periodic_4_hyperbolic_Delaunay_triangulation_2<GT, TDS>   Self;
  typedef Periodic_4_hyperbolic_triangulation_2<GT, TDS>            Base;

public:
  typedef typename Base::Locate_type                                Locate_type;
  typedef typename Base::Geom_traits                                Geom_traits;
  typedef typename Base::Triangulation_data_structure               Triangulation_data_structure;
  typedef typename Base::Hyperbolic_translation                     Hyperbolic_translation;
  typedef typename Base::Point                                      Point;
  typedef typename Geom_traits::Hyperbolic_Voronoi_point_2          Hyperbolic_Voronoi_point;
  typedef typename Base::Hyperbolic_segment                         Hyperbolic_segment;
  typedef typename Base::Hyperbolic_triangle                        Hyperbolic_triangle;

  typedef typename Base::Periodic_point                             Periodic_point;
  typedef typename Base::Periodic_segment                           Periodic_segment;
  typedef typename Base::Periodic_triangle                          Periodic_triangle;

  typedef typename Base::Vertex                                     Vertex;
  typedef typename Base::Edge                                       Edge;
  typedef typename Base::Face                                       Face;

  typedef typename Base::Vertex_handle                              Vertex_handle;
  typedef typename Base::Face_handle                                Face_handle;

  typedef typename Base::size_type                                  size_type;
  typedef typename Base::difference_type                            difference_type;

  typedef typename Base::Face_iterator                              Face_iterator;
  typedef typename Base::Edge_iterator                              Edge_iterator;
  typedef typename Base::Vertex_iterator                            Vertex_iterator;
  typedef typename Base::Face_circulator                            Face_circulator;
  typedef typename Base::Edge_circulator                            Edge_circulator;
  typedef typename Base::Vertex_circulator                          Vertex_circulator;

public:
  typedef Point                                                     value_type;
  typedef const value_type&                                         const_reference;

  // Tag to distinguish regular triangulations from others
  typedef Tag_false                                                 Weighted_tag;

  // Tag to distinguish periodic triangulations from others
  typedef Tag_true                                                  Periodic_tag;

#ifndef CGAL_CFG_USING_BASE_MEMBER_BUG_2
  using Base::cw;
  using Base::ccw;
  using Base::geom_traits;
  using Base::tds;
  using Base::neighbor_translation;
#endif

private:
  typedef typename GT::FT                                           FT;

  class Dummy_point
  {
    Point _pt;
    bool  _is_inserted;
    Vertex_handle _vh;

  public:
    Dummy_point(FT x, FT y): _pt(x,y), _is_inserted(true) {}
    Dummy_point(const Point& p): _pt(p), _is_inserted(true) {}

    Point operator()() const { return _pt; }
    bool is_inserted() const {  return _is_inserted; }
    Vertex_handle vertex() const { return _vh; }
    void set_inserted(bool val) { _is_inserted = val; }
    void set(const Point& val) { _pt = val; }
    void set_vertex(Vertex_handle v) { _vh = v; }
  };

  std::vector<Dummy_point> dummy_points;

private:
  std::vector<Vertex_handle> insert_dummy_points(bool rational = true);

  bool is_removable(Vertex_handle v,
                    Delaunay_triangulation_2<GT, TDS>& dt,
                    std::map<Vertex_handle, Vertex_handle>& vmap);

public:
  Periodic_4_hyperbolic_Delaunay_triangulation_2(const Geom_traits &gt = Geom_traits())
    : Base(gt)
  {
    insert_dummy_points();
  }

  template < typename InputIterator >
  Periodic_4_hyperbolic_Delaunay_triangulation_2(InputIterator first, InputIterator last,
                                                 const Geom_traits &gt = Geom_traits())
    : Base(gt)
  {
    insert_dummy_points();
    insert(first, last);
  }

  Periodic_4_hyperbolic_Delaunay_triangulation_2(const Periodic_4_hyperbolic_Delaunay_triangulation_2& tr)
    : Base(tr)
  {
    insert_dummy_points();
  }

  Vertex_handle insert(const Point& p, Face_handle start = Face_handle());

  template <typename InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last,
                        const bool flag_clear_dummy_points = true)
  {
    size_type n_initial = this->number_of_vertices();
    std::vector<Point> points(first, last);
    spatial_sort(points.begin(), points.end(), geom_traits());

    Face_handle f;
    int cnt = 0;
    int cnt_good = 0;
    for(typename std::vector<Point>::const_iterator p=points.begin(), end = points.end(); p != end; ++p)
    {
      ++cnt;
      Vertex_handle v = insert(*p, f);
      if(v != Vertex_handle())
      {
        ++cnt_good;
        f = v->face();
      }
    }

    std::ptrdiff_t ret = this->number_of_vertices() - n_initial;

    if(flag_clear_dummy_points)
      try_to_remove_dummy_vertices();

    return ret;
  }

  /*!
    This function returns the faces in conflict with `p` in the output iterator `it`.
    Note that the optional parameter `store_translations` is _not_ documented. This
    flag indicates whether the translations in the faces in conflict with `p` should
    be stored in their incident vertices. The flag is `false` by default, and is set
    to `true` only when the point `p` is going to be inserted in the triangulation.
  */
  template<class OutputFaceIterator>
  void find_conflicts(const Point& p,
                      OutputFaceIterator it,
                      Face_handle start = Face_handle(),
                      Hyperbolic_translation ltr = Hyperbolic_translation(),
                      bool store_translations = false) const
  {
    // If no starting face is given, locate p here and obtain also the location translation
    if(start == Face_handle())
      start = this->euclidean_locate(p, ltr);

    // If start is Face_handle() here, it means p has not been located in the triangulation.
    // Nothing can be done at this point, so just return an empty set.
    if(start != Face_handle())
    {
      std::set<Face_handle > visited; // faces that have already been visited; avoid visiting faces again (also avoid infinite loops)
      std::set<Face_handle > to_visit; // faces to visit; should contain only new faces
      std::map<Face_handle, Hyperbolic_translation > trans; // Combination of location translation and neighbor translation for each face to examine
      typedef std::pair<Face_handle, Hyperbolic_translation> FTPair; // useful typedef for inserting elements into the map

      // insert first face in set of faces to visit...
      to_visit.insert(start);
      // ...and associate location translation with it
      trans.insert(FTPair(start, ltr));

      // repeat until there are still faces to check
      while(!to_visit.empty())
      {
        typename std::set<Face_handle>::iterator itf;
        itf = to_visit.begin(); // get an iterator to the face
        Face_handle face = *itf; // and the face itself (avoids dereferencing everywhere)
        Hyperbolic_translation tr = trans[face]; // get the translation associated with the face

        visited.insert(face); // the face has now been visited; it will NOT be visited again
        to_visit.erase(itf); // erase the face from the list of faces to visit (amortized constant time)

        if(_side_of_circle(face, p, tr) == ON_BOUNDED_SIDE)
        {
          if(store_translations)
          {
            for(int i=0; i<3; ++i)
              face->vertex(i)->set_translation(tr * face->translation(i));
          }

          it++ = face;
          for(int i=0; i<3; ++i)
          {
            if(visited.find(face->neighbor(i)) == visited.end())
            {
              to_visit.insert(face->neighbor(i));
              trans.insert(FTPair(face->neighbor(i), tr*neighbor_translation(face,i)));
            }
          }
        }
      } // end of while
    } // end if(start != Face_handle())
  } // end of function

  int try_to_remove_dummy_vertices()
  {
    // count of dummy points in the triangulation
    int cnt = 0;
    for(std::size_t i=0; i<dummy_points.size(); ++i)
    {
      if(dummy_points[i].is_inserted())
      {
        if(remove(dummy_points[i].vertex()))
          dummy_points[i].set_inserted(false);
        else
          ++cnt;
      }
    }

    return cnt;
  }

  Point get_dummy_point(int i) const
  {
    CGAL_triangulation_precondition(0 <= i && i <= static_cast<int>(dummy_points.size()));
    return dummy_points[i]();
  }

  bool remove(Vertex_handle v);

  template <class VertexRemoveIterator>
  void remove(VertexRemoveIterator first, VertexRemoveIterator last)
  {
    for (VertexRemoveIterator vit = first; vit != last; ++vit)
    {
      remove(*vit);
    }
  }

  bool is_dummy_vertex(Vertex_handle vh) const
  {
    for(std::size_t i=0; i<dummy_points.size(); ++i)
    {
      if(dummy_points[i]() == vh->point())
        return true;
    }

    return false;
  }

  int number_of_dummy_points() const
  {
    int cnt = 0;
    for(std::size_t i=0; i<dummy_points.size(); ++i)
    {
      if(dummy_points[i].is_inserted())
        ++cnt;
    }

    return cnt;
  }

protected:

  Bounded_side _side_of_circle(const Face_handle f, const Point& q,
                               const Hyperbolic_translation& translation) const
  {
    Point p[] = {f->vertex(0)->point(),
                 f->vertex(1)->point(),
                 f->vertex(2)->point()};

    Hyperbolic_translation o[]= {translation * f->translation(0),
                                 translation * f->translation(1),
                                 translation * f->translation(2)};

#if defined PROFILING_MODE
    if(o[0].is_identity() && o[1].is_identity() && o[2].is_identity())
      calls_predicate_identity++;
    else
      calls_predicate_non_identity++;
#endif

#if defined PROFILING_MODE
    CGAL::Timer tmr;
    tmr.start();
#endif

    Oriented_side os = this->side_of_oriented_circle(p[0], p[1], p[2], q,
                                                     o[0], o[1], o[2], Hyperbolic_translation());
#if defined PROFILING_MODE
    tmr.stop();
    if(o[0].is_identity() && o[1].is_identity() && o[2].is_identity())
      time_predicate_identity += tmr.time();
    else
      time_predicate_non_identity += tmr.time();
#endif

    if(os == ON_NEGATIVE_SIDE)
      return ON_UNBOUNDED_SIDE;
    else if(os == ON_POSITIVE_SIDE)
      return ON_BOUNDED_SIDE;
    else
      return ON_BOUNDARY;
  }

public:
  Point dual(const Face_handle f, const Hyperbolic_translation& nboff = Hyperbolic_translation()) const
  {
    Point res = geom_traits.construct_hyperbolic_circumcenter_2_object()(f->vertex(0)->point(),
                                                      f->vertex(1)->point(),
                                                      f->vertex(2)->point(),
                                                      nboff * f->translation(0),
                                                      nboff * f->translation(1),
                                                      nboff * f->translation(2));
    return res;
  }

  Hyperbolic_segment dual(const Edge &e) const
  {
    Hyperbolic_segment res = geom_traits().construct_hyperbolic_segment_2_object()(dual(e.first), dual(e.first->neighbor(e.second), neighbor_translation(e.first, e.second)));
    return res;
  }

  void clear()
  {
    Base::clear();
    insert_dummy_points();
  }

  bool is_valid(bool verbose = false) const { return Base::is_valid(verbose); }

};  // class Periodic_4_hyperbolic_Delaunay_triangulation_2

template <class Gt, class Tds>
bool
Periodic_4_hyperbolic_Delaunay_triangulation_2<Gt,Tds>::
is_removable(Vertex_handle v,
             Delaunay_triangulation_2<Gt,Tds>& dt,
             std::map<Vertex_handle,
             Vertex_handle>& vmap)
{
  typedef Delaunay_triangulation_2<Gt, Tds>           Delaunay;
  typedef typename Delaunay::Finite_faces_iterator    Finite_Delaunay_faces_iterator;

  // This is the exact value of the limit.
  // The systole is 2*acosh(1+sqrt(2)), and we want half of that.
  // The max _diameter_ of the hyperbolic circles must be less than this.
  double factor(1.0); // percentage of the limit to consider -- failsafe! iordanov 10/02/19: additional failsafe in remove(), this can be 1.0 now
  double lim(factor*acosh(1. + sqrt(2.)));

  std::vector<Vertex_handle> bdry_verts;
  Face_circulator nbf(tds().incident_faces(v)), done(nbf);
  do
  {
    int idx = nbf->index(v);
    Hyperbolic_translation off = nbf->translation(idx).inverse();
    off = off*nbf->translation(ccw(idx));

    Vertex_handle thisv = nbf->vertex(ccw(idx));
    bdry_verts.push_back(thisv);

    Point pt = this->construct_point(thisv->point(), off);
    Vertex_handle new_v = dt.insert(pt);
    vmap.insert(std::pair<Vertex_handle, Vertex_handle>(new_v, thisv));
  }
  while(++nbf != done);

  int n_verts = static_cast<int>(bdry_verts.size());
  double max_diam = 0.;
  for(Finite_Delaunay_faces_iterator fit  = dt.finite_faces_begin();
                                     fit != dt.finite_faces_end(); ++fit)
  {
    bool is_good = true;
    for(int i=0; i<3; ++i)
    {
      Vertex_handle this_v = vmap[fit->vertex(i)];
      Vertex_handle prev_v = bdry_verts[n_verts - 1];
      Vertex_handle curr_v = bdry_verts[0];
      for(std::size_t j=1; curr_v!=this_v; j=(j+1)%n_verts)
      {
        prev_v = curr_v;
        curr_v = bdry_verts[j];
      }

      if(vmap[fit->vertex(ccw(i))] == prev_v)
      {
        is_good = false;
        break;
      }
    }

    if(is_good)
    {
      typename Gt::Compute_approximate_hyperbolic_diameter cdiam;
      double diam = cdiam(fit->vertex(0)->point(), fit->vertex(1)->point(), fit->vertex(2)->point());
      if(max_diam < diam)
        max_diam = diam;
    }
  }

  return (max_diam < lim);
}

/*********************************************************************************/

template <typename Gt, class Tds >
inline
typename Periodic_4_hyperbolic_Delaunay_triangulation_2<Gt, Tds>::Vertex_handle
Periodic_4_hyperbolic_Delaunay_triangulation_2<Gt, Tds>::
insert(const Point& p, Face_handle hint)
{
  Hyperbolic_translation loff;
  Locate_type lt;
  int li;
  Face_handle start = this->euclidean_locate(p, lt, li, loff, hint);

  // If the point is not located in a face of the triangulation, we have nothing to do
  if(start == Face_handle())
    return Vertex_handle();

  if(lt == Periodic_4_hyperbolic_Delaunay_triangulation_2<Gt, Tds>::VERTEX)
  {
    for(std::size_t i=0; i<dummy_points.size(); ++i)
    {
      if(dummy_points[i].is_inserted())
      {
        if(dummy_points[i]() == p)
        {
          dummy_points[i].set_inserted(false);
          return dummy_points[i].vertex();
        }
      }
    }
    return Vertex_handle();
  }

  std::vector<Face_handle> faces;
  std::set<Face_handle> visited;
  find_conflicts(p, std::back_inserter(faces), start, loff, true);

  Vertex_handle v = this->_tds.insert_in_hole(faces.begin(), faces.end());
  v->set_point(p);

  Face_circulator ifc = tds().incident_faces(v), done(ifc);
  do
  {
    for(int i=0; i<3; ++i)
      ifc->set_translation(i, ifc->vertex(i)->translation());

    this->make_canonical(ifc);
  }
  while(++ifc != done);

  Vertex_circulator ivc = tds().incident_vertices(v), done_v(ivc);
  do
  {
    ivc->clear_translation();
  }
  while(++ivc != done_v);

  return v;
}

//------------------------------------------------------

template <typename Gt, class Tds >
bool
Periodic_4_hyperbolic_Delaunay_triangulation_2<Gt, Tds>::
remove(Vertex_handle v)
{
  typedef Delaunay_triangulation_2<Gt, Tds>           Delaunay;
  typedef typename Delaunay::Finite_faces_iterator    Finite_Delaunay_faces_iterator;
  typedef std::pair<Face_handle, int>                 Neighbor_pair;
  typedef std::pair<Edge, Neighbor_pair>              Edge_neighbor;

  Delaunay dt;
  std::map<Vertex_handle, Vertex_handle> vmap;

  if(is_removable(v, dt, vmap))
  {
    std::vector<Edge> bdry_edges;
    std::vector<Vertex_handle> bdry_verts;
    std::map<Edge, Neighbor_pair> bdry_nbrs;

    Face_circulator nb = tds().incident_faces(v), done(nb);
    std::vector<Face_handle> nbrs;
    do
    {
      int idx = nb->index(v);
      Edge e = Edge(nb, idx);
      bdry_edges.push_back(e);

      Face_handle nbf = nb->neighbor(idx);
      int nidx = 0;
      if(nbf->neighbor(1) == nb) nidx = 1;
      if(nbf->neighbor(2) == nb) nidx = 2;
      CGAL_triangulation_assertion(nbf->neighbor(nidx) == nb);

      bdry_nbrs.insert(Edge_neighbor(e, Neighbor_pair(nbf, nidx)));
      bdry_verts.push_back(nb->vertex(ccw(idx)));

      for(int i=0; i<3; ++i)
        nb->vertex(i)->set_translation(nb->translation(idx).inverse() * nb->translation(i));

      nbrs.push_back(nb);
      ++nb;
    }
    while(nb != done);

    std::size_t n_verts = bdry_verts.size();
    std::vector<Face_handle> new_f;
    for(Finite_Delaunay_faces_iterator fit = dt.finite_faces_begin();
                                       fit != dt.finite_faces_end(); ++fit)
    {
      bool is_good = true;
      for(int i=0; i<3; ++i)
      {
        Vertex_handle this_v = vmap[fit->vertex(i)];
        Vertex_handle prev_v = bdry_verts[n_verts - 1];
        Vertex_handle curr_v = bdry_verts[0];
        for(std::size_t j=1; curr_v!=this_v; j=(j+1)%n_verts)
        {
          prev_v = curr_v;
          curr_v = bdry_verts[j];
        }

        if(vmap[fit->vertex(ccw(i))] == prev_v)
        {
          is_good = false;
          break;
        }
      }

      if(is_good)
      {
        Face_handle f = tds().create_face();
        for(int j=0; j<3; ++j)
          f->set_vertex(j, vmap[fit->vertex(j)]);

        new_f.push_back(f);
      }
    }

    typedef std::pair<Face_handle, int>       Nbr_entry;
    typedef std::pair<Nbr_entry, Nbr_entry>   Nbr_pair;
    typedef std::vector<Nbr_pair>             Nbr_history;

    Nbr_history failsafe;

    int internb = 0;
    int bdrynb = 0;
    for(std::size_t i=0; i<new_f.size(); ++i)
    {
      for(int k=0; k< 3; k++)
      {
        bool found_bdry = false;
        for(int j=0; j<static_cast<int>(bdry_verts.size()); ++j)
        {
          if(new_f[i]->vertex(ccw(k)) == bdry_verts[j] &&
             new_f[i]->vertex(cw(k)) == bdry_verts[(j+1)%n_verts])
          {
            found_bdry = true;
            Neighbor_pair nb = bdry_nbrs[bdry_edges[j]];
            Face_handle nbf = nb.first;
            int nbidx = nb.second;

            Nbr_entry side1(nbf, nbidx);
            Nbr_entry side2(nbf->neighbor(nbidx), nbf->neighbor(nbidx)->index(nbf));

            CGAL_triangulation_assertion(side1.first->neighbor(side1.second) == side2.first);
            CGAL_triangulation_assertion(side2.first->neighbor(side2.second) == side1.first);

            Nbr_pair hist(side1, side2);
            failsafe.push_back(hist);

            tds().set_adjacency(nbf, nbidx, new_f[i], k);
            bdrynb++;
            break;
          }
        }

        if(!found_bdry)
        {
          for(unsigned int l=0; l<new_f.size(); ++l)
          {
            if(l == i)
              continue;

            for(int j=0; j<3; ++j)
            {
              if(new_f[i]->vertex(ccw(k)) == new_f[l]->vertex(cw(j)) &&
                 new_f[i]->vertex(cw(k))  == new_f[l]->vertex(ccw(j)))
              {
                tds().set_adjacency(new_f[i], k, new_f[l], j);
                internb++;
                break;
              }
            }
          }
        }
      }
    }

    /*
      This is a failsafe check: make sure that there are no cycles of length 2 before
      deleting the old faces. If everything is OK, then proceed with the actual removal
      and keep the new faces. Otherwise the new objects are deleted and the operation
      is canceled.
    */
    for (Face_iterator fit = this->faces_begin(); fit != this->faces_end(); ++fit) {
      if (std::find(nbrs.begin(), nbrs.end(), fit) == nbrs.end()) {
        for (int ii = 0; ii < 3; ii++) {
          for (int jj = ii+1; jj < 3; jj++) {
            if ( fit->neighbor(ii) == fit->neighbor(jj) ) {

              for (unsigned int safeit = 0; safeit < failsafe.size(); ++safeit) {
                Nbr_pair hist = failsafe[safeit];
                Nbr_entry side1 = hist.first;
                Nbr_entry side2 = hist.second;
                tds().set_adjacency(side1.first, side1.second, side2.first, side2.second);
              }

              for (unsigned int canit = 0; canit < nbrs.size(); ++canit) {
                for (int iit = 0; iit < 3; ++iit) {
                  nbrs[canit]->vertex(iit)->clear_translation();
                }
                this->make_canonical(nbrs[canit]);
              }

              for (unsigned int rit = 0; rit < new_f.size(); rit++) {
                tds().delete_face(new_f[rit]);
              }

              CGAL_triangulation_assertion(this->is_valid(true));

              return false;
            }
          }
        }
      }
    }

    for(int j=0; j<static_cast<int>(new_f.size()); ++j)
    {
      for(int i=0; i<3; ++i)
      {
        new_f[j]->vertex(i)->set_face(new_f[j]);
        new_f[j]->set_translation(i, new_f[j]->vertex(i)->translation());
      }

      this->make_canonical(new_f[j]);
    }

    for(std::size_t j=0; j<bdry_edges.size(); ++j)
    {
      Face_handle f = bdry_edges[j].first;
      int i = bdry_edges[j].second;
      f->vertex(ccw(i))->clear_translation();
    }

    for(int i=0; i<static_cast<int>(nbrs.size()); ++i)
      tds().delete_face(nbrs[i]);

    tds().delete_vertex(v);

    CGAL_triangulation_assertion(this->is_valid(true));

    return true;
  }
  else
  {
    // is not removable
    return false;
  }
}

} // namespace CGAL

#include <CGAL/Periodic_4_hyperbolic_triangulation_2/internal/Periodic_4_hyperbolic_triangulation_dummy_14.h>

#endif // CGAL_PERIODIC_4_HYPERBOLIC_DELAUNAY_TRIANGULATION_2_H
