#include <Rcpp.h>
// Copyright(c) 1997  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/Triangulation_2/include/CGAL/Regular_triangulation_2.h $
// $Id: Regular_triangulation_2.h d3fca65 2022-09-23T12:49:40+01:00 Andreas Fabri
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)     : Frederic Fichel, Mariette Yvinec, Julia Floetotto

#ifndef CGAL_REGULAR_TRIANGULATION_2_H
#define CGAL_REGULAR_TRIANGULATION_2_H

#include <CGAL/license/Triangulation_2.h>

#include <CGAL/Triangulation_2.h>
#include <CGAL/Regular_triangulation_face_base_2.h>
#include <CGAL/Regular_triangulation_vertex_base_2.h>

#include <CGAL/assertions.h>
#include <CGAL/utility.h>
#include <CGAL/Object.h>
#include <CGAL/STL_Extension/internal/Has_nested_type_Bare_point.h>

#include <boost/mpl/if.hpp>
#include <boost/mpl/identity.hpp>
#include <boost/utility/result_of.hpp>

#ifndef CGAL_TRIANGULATION_2_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
#include <CGAL/Spatial_sort_traits_adapter_2.h>
#include <CGAL/STL_Extension/internal/info_check.h>

#include <boost/iterator/zip_iterator.hpp>
#include <boost/mpl/and.hpp>
#include <boost/property_map/function_property_map.hpp>

#endif //CGAL_TRIANGULATION_2_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO

namespace CGAL {

template < class Gt,
           class Tds  = Triangulation_data_structure_2 <
                          Regular_triangulation_vertex_base_2<Gt>,
                          Regular_triangulation_face_base_2<Gt> > >
class Regular_triangulation_2
  : public Triangulation_2<Gt, Tds>
{
  typedef Regular_triangulation_2<Gt, Tds>     Self;
  typedef Triangulation_2<Gt, Tds>             Base;

public:
  typedef Self                                 Triangulation;
  typedef Triangulation_2<Gt, Tds>             Triangulation_base;
  typedef Tds                                  Triangulation_data_structure;
  typedef Gt                                   Geom_traits;

  // Traits are not supposed to define Bare_point, but leaving below
  // for backward compatibility
  typedef typename boost::mpl::eval_if_c<
      internal::Has_nested_type_Bare_point<Gt>::value,
      typename internal::Bare_point_type<Gt>,
      boost::mpl::identity<typename Gt::Point_2>
    >::type                                    Bare_point;
  typedef typename Gt::Weighted_point_2        Weighted_point;

  typedef typename Gt::FT                      Weight;
  typedef typename Gt::Construct_weighted_point_2 Construct_weighted_point_2;
  typedef typename Base::size_type             size_type;
  typedef typename Base::Face_handle           Face_handle;
  typedef typename Base::Vertex_handle         Vertex_handle;
  typedef typename Base::Vertex                Vertex;
  typedef typename Base::Edge                  Edge;
  typedef typename Base::Locate_type           Locate_type;
  typedef typename Base::Face_circulator       Face_circulator;
  typedef typename Base::Edge_circulator       Edge_circulator;
  typedef typename Base::Vertex_circulator     Vertex_circulator;
  typedef typename Base::Finite_edges_iterator Finite_edges_iterator;
  typedef typename Base::All_edges_iterator    All_edges_iterator;
  typedef typename Base::Finite_faces_iterator Finite_faces_iterator;
  typedef typename Base::All_faces_iterator    All_faces_iterator;
  typedef typename Base::Face::Vertex_list     Vertex_list;
  typedef typename Vertex_list::iterator       Vertex_list_iterator;

#ifndef CGAL_CFG_USING_BASE_MEMBER_BUG_2
  using Base::cw;
  using Base::ccw;
  using Base::dimension;
  using Base::geom_traits;
  using Base::infinite_vertex;
  using Base::finite_vertex;
  using Base::create_face;
  using Base::number_of_faces;
  using Base::all_faces_begin;
  using Base::all_faces_end;
  using Base::all_edges_begin;
  using Base::all_edges_end;
  using Base::finite_faces_begin;
  using Base::finite_faces_end;
  using Base::finite_edges_begin;
  using Base::finite_edges_end;
  using Base::OUTSIDE_AFFINE_HULL;
  using Base::VERTEX;
  using Base::FACE;
  using Base::EDGE;
  using Base::OUTSIDE_CONVEX_HULL;
  using Base::orientation;
  using Base::locate;
#ifndef CGAL_NO_STRUCTURAL_FILTERING
  using Base::inexact_locate;
#endif
  using Base::incident_faces;
  using Base::is_infinite;
  using Base::degree;
  using Base::delete_vertex;
  using Base::delete_face;
  using Base::incident_vertices;
  using Base::make_hole;
  using Base::mirror_index;
  using Base::show_vertex;
  using Base::test_dim_down;
  using Base::oriented_side;
  using Base::compare_x;
  using Base::compare_y;
#endif

private:
  typedef std::list<Face_handle>      Faces_around_stack;

  class Hidden_tester
  {
  public:
    bool operator()(const typename Base::All_vertices_iterator&  it){
      return it->is_hidden();
     }
    bool operator()(const typename Base::Finite_vertices_iterator&  it){
      return it->is_hidden();
    }
  };

  class Unhidden_tester
  {
  public:
    bool operator()(const typename Base::Finite_vertices_iterator&  it){
      return ! it->is_hidden();
    }
  };

  typedef typename Base::All_vertices_iterator     All_vib;
  typedef typename Base::Finite_vertices_iterator  Finite_vib;

public:
  // We derive in order to add a conversion to handle.
  class All_vertices_iterator :
    public Filter_iterator<All_vib, Hidden_tester>
  {
    typedef Filter_iterator<All_vib, Hidden_tester> Base;
    typedef All_vertices_iterator                     Self;
  public:
    All_vertices_iterator() : Base() {}
    All_vertices_iterator(const Base &b) : Base(b) {}
    Self & operator++() { Base::operator++(); return *this; }
    Self & operator--() { Base::operator--(); return *this; }
    Self operator++(int) { Self tmp(*this); ++(*this); return tmp; }
    Self operator--(int) { Self tmp(*this); --(*this); return tmp; }
    operator Vertex_handle() const { return Base::base(); }
  };

  typedef Iterator_range<Prevent_deref<All_vertices_iterator> > All_vertex_handles;

  class Finite_vertices_iterator :
    public Filter_iterator<Finite_vib, Hidden_tester>
  {
    typedef Filter_iterator<Finite_vib, Hidden_tester> Base;
    typedef Finite_vertices_iterator                          Self;
  public:
    Finite_vertices_iterator() : Base() {}
    Finite_vertices_iterator(const Base &b) : Base(b) {}
    Self & operator++() { Base::operator++(); return *this; }
    Self & operator--() { Base::operator--(); return *this; }
    Self operator++(int) { Self tmp(*this); ++(*this); return tmp; }
    Self operator--(int) { Self tmp(*this); --(*this); return tmp; }
    operator Vertex_handle() const { return Base::base(); }
 };

  typedef Iterator_range<Prevent_deref<Finite_vertices_iterator> > Finite_vertex_handles;

  class Hidden_vertices_iterator :
    public Filter_iterator<Finite_vib, Unhidden_tester>
  {
    typedef Filter_iterator<Finite_vib, Unhidden_tester> Base;
    typedef Hidden_vertices_iterator                     Self;
  public:
    Hidden_vertices_iterator() : Base() {}
    Hidden_vertices_iterator(const Base &b) : Base(b) {}
    Self & operator++() { Base::operator++(); return *this; }
    Self & operator--() { Base::operator--(); return *this; }
    Self operator++(int) { Self tmp(*this); ++(*this); return tmp; }
    Self operator--(int) { Self tmp(*this); --(*this); return tmp; }
    operator Vertex_handle() const { return Base::base(); }
 };

  typedef Iterator_range<Prevent_deref<Hidden_vertices_iterator> > Hidden_vertex_handles;

 //for backward compatibility
  typedef Finite_faces_iterator                Face_iterator;
  typedef Finite_edges_iterator                Edge_iterator;
  typedef Finite_vertices_iterator             Vertex_iterator;

  //Tag to distinguish Delaunay from regular triangulations
  typedef Tag_true  Weighted_tag;

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

private:
  size_type _hidden_vertices;

public:
  Regular_triangulation_2()
    : Base(), _hidden_vertices(0) {}

  Regular_triangulation_2(const Gt& gt)
    : Base(gt), _hidden_vertices(0) {}

  Regular_triangulation_2(const Regular_triangulation_2 &rt);

  template < class InputIterator >
  Regular_triangulation_2(InputIterator first, InputIterator last,
                          const Gt& gt)
    : Base(gt), _hidden_vertices(0)
  {
    insert(first, last);
  }

  template < class InputIterator >
  Regular_triangulation_2(InputIterator first, InputIterator last)
    : Base(), _hidden_vertices(0)
  {
    insert(first, last);
  }

  Regular_triangulation_2 & operator=(const Regular_triangulation_2 &tr);

  size_type number_of_vertices() const {
    return Base::number_of_vertices() - _hidden_vertices;
  }

  size_type number_of_hidden_vertices() const {
    return _hidden_vertices;
  }

  // CHECK - QUERY

  Oriented_side power_test(const Weighted_point &p,
                           const Weighted_point &q,
                           const Weighted_point &r,
                           const Weighted_point &s, bool perturb) const;
  Oriented_side power_test(const Weighted_point &p,
                           const Weighted_point &q,
                           const Weighted_point &r) const;
  Oriented_side power_test(const Weighted_point &p,
                           const Weighted_point &r) const;
  Oriented_side power_test(const Face_handle &f,
                           const Weighted_point &p, bool perturb=false) const;
  Oriented_side power_test(const Face_handle& f, int i,
                           const Weighted_point &p) const;

  bool is_valid(bool verbose = false, int level = 0) const;
  bool test_conflict(const Weighted_point  &p, Face_handle fh) const;
  void show_face(Face_handle fh) const;
  void show_all() const;

//  // template member functions, declared and defined at the end
//  template <class OutputItFaces, class OutputItBoundaryEdges,
//            class OutputItHiddenVertices>
//  Triple<OutputItFaces,OutputItBoundaryEdges, OutputItHiddenVertices>
//  get_conflicts_and_boundary_and_hidden_vertices(const
//                                                 Weighted_point  &p,
//                                                 OutputItFaces fit,
//                                                 OutputItBoundaryEdges eit,
//                                                 OutputItHiddenVertices vit,
//                                                 Face_handle start = Face_handle()) const;

//  template <class OutputItFaces, class OutputItBoundaryEdges>
//  std::pair<OutputItFaces,OutputItBoundaryEdges>
//  get_conflicts_and_boundary(const Weighted_point  &p,
//                             OutputItFaces fit,
//                             OutputItBoundaryEdges eit,
//                             Face_handle start) const;

//  template <class OutputItFaces>
//  OutputItFaces
//  get_conflicts(const Weighted_point  &p,
//                OutputItFaces fit,
//                Face_handle start) const;

//  template <class OutputItBoundaryEdges>
//  OutputItBoundaryEdges
//  get_boundary_of_conflicts(const Weighted_point  &p,
//                            OutputItBoundaryEdges eit,
//                            Face_handle start) const;
//  template <class OutputItBoundaryEdges, class OutputItHiddenVertices>
//  std::pair<OutputItBoundaryEdges, OutputItHiddenVertices>
//  get_boundary_of_conflicts_and_hidden_vertices(const Weighted_point  &p,
//                                                OutputItBoundaryEdges eit,
//                                                OutputItHiddenVertices vit,
//                                                Face_handle start = Face_handle()) const;
//  template <class OutputItHiddenVertices>
//  OutputItHiddenVertices
//  get_hidden_vertices(const Weighted_point  &p,
//                      OutputItHiddenVertices vit,
//                      Face_handle start =  Face_handle()) const;

  // DUAL
  Bare_point dual(Face_handle f) const;
  Object dual(const Edge &e) const ;
  Object dual(const Edge_circulator& ec) const;
  Object dual(const Finite_edges_iterator& ei) const;
  Bare_point weighted_circumcenter(Face_handle f) const;
  Bare_point weighted_circumcenter(const Weighted_point& p0,
                                   const Weighted_point& p1,
                                   const Weighted_point& p2) const;

  // Insertion, Deletion and Flip
  Vertex_handle push_back(const Weighted_point &p);
  Vertex_handle insert(const Weighted_point &p,
                       Face_handle f = Face_handle());
  Vertex_handle insert(const Weighted_point &p,
                       Locate_type  lt,
                       Face_handle loc, int li);
  Vertex_handle insert_in_face(const Weighted_point &p, Face_handle f);
  Vertex_handle insert_in_edge(const Weighted_point &p, Face_handle f, int i);
  void flip(Face_handle f, int i);
  void remove_degree_3(Vertex_handle v,
                       Face_handle f = Face_handle());
  void remove(Vertex_handle v);

  All_vertices_iterator all_vertices_begin() const;
  All_vertices_iterator all_vertices_end() const;
  All_vertex_handles all_vertex_handles() const;

  Finite_vertices_iterator finite_vertices_begin() const;
  Finite_vertices_iterator finite_vertices_end() const;
  Finite_vertex_handles finite_vertex_handles() const;

  Vertex_handle finite_vertex() const;

  Hidden_vertices_iterator hidden_vertices_begin() const;
  Hidden_vertices_iterator hidden_vertices_end() const;
  Hidden_vertex_handles hidden_vertex_handles() const;

//  Vertex_handle file_input(std::istream& is);
//  void file_output(std::ostream& os) const;

public:
  void clear();
  void copy_triangulation(const Self& tr);

private:
  void copy_triangulation_();
  Vertex_handle reinsert(Vertex_handle v, Face_handle start);
  void regularize(Vertex_handle v);
  void remove_hidden(Vertex_handle v);
  void remove_2D(Vertex_handle v);
  void fill_hole_regular(std::list<Edge> & hole);
  void set_face(Vertex_list& vl, const Face_handle& fh);
  void update_hidden_points_3_1(const Face_handle& f1, const Face_handle& f2,
                                const Face_handle& f3);
  void update_hidden_points_2_2(const Face_handle& f1, const Face_handle& f2);
  void update_hidden_points_1_3(const Face_handle& f1, const Face_handle& f2,
                                const Face_handle& f3);

  Vertex_handle hide_new_vertex(Face_handle f, const Weighted_point& p);
  void hide_remove_degree_3(Face_handle fh, Vertex_handle vh);
  void hide_vertex(Face_handle f, Vertex_handle v);
  void exchange_incidences(Vertex_handle va, Vertex_handle vb);
  void exchange_hidden(Vertex_handle va, Vertex_handle vb);

  void stack_flip(Vertex_handle v, Faces_around_stack &faces_around);
  void stack_flip_4_2(Face_handle f, int i, int j,
                      Faces_around_stack &faces_around);
  void stack_flip_3_1(Face_handle f, int i, int j,
                      Faces_around_stack &faces_around);
  void stack_flip_2_2(Face_handle f, int i,
                      Faces_around_stack &faces_around);
  void stack_flip_dim1(Face_handle f, int i,
                       Faces_around_stack &faces_around);
  bool is_valid_face(Face_handle fh) const;
  bool is_valid_vertex(Vertex_handle fh) const;

public:
#ifndef CGAL_TRIANGULATION_2_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
  template < class InputIterator >
  std::ptrdiff_t
  insert(InputIterator first, InputIterator last,
          std::enable_if_t<
              boost::is_convertible<
                  typename std::iterator_traits<InputIterator>::value_type,
                  Weighted_point
              >::value
          >* = nullptr
)
#else
  template < class InputIterator >
  std::ptrdiff_t
  insert(InputIterator first, InputIterator last)
#endif //CGAL_TRIANGULATION_2_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
  {
    size_type n = number_of_vertices();

    std::vector<Weighted_point> points(first, last);

    // spatial sorting must use bare points, so we need an adapter
    typedef typename Geom_traits::Construct_point_2 Construct_point_2;
    typedef typename boost::result_of<const Construct_point_2(const Weighted_point&)>::type Ret;
    typedef boost::function_property_map<Construct_point_2, Weighted_point, Ret> fpmap;
    typedef CGAL::Spatial_sort_traits_adapter_2<Geom_traits, fpmap> Search_traits_2;

    spatial_sort(points.begin(), points.end(),
                 Search_traits_2(
                   boost::make_function_property_map<Weighted_point, Ret, Construct_point_2>(
                     geom_traits().construct_point_2_object()), geom_traits()));

    Face_handle hint;
    for(typename std::vector<Weighted_point>::const_iterator p = points.begin(),
         end = points.end();
         p != end; ++p)
      hint = insert(*p, hint)->face();

    return number_of_vertices() - n;
  }

#ifndef CGAL_TRIANGULATION_2_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
private:
  //top stands for tuple-or-pair
  template <class Info>
  const Weighted_point& top_get_first(const std::pair<Weighted_point,Info>& pair) const { return pair.first; }
  template <class Info>
  const Info& top_get_second(const std::pair<Weighted_point,Info>& pair) const { return pair.second; }
  template <class Info>
  const Weighted_point& top_get_first(const boost::tuple<Weighted_point,Info>& tuple) const { return boost::get<0>(tuple); }
  template <class Info>
  const Info& top_get_second(const boost::tuple<Weighted_point,Info>& tuple) const { return boost::get<1>(tuple); }

  // Functor to go from an index of a container of Weighted_point to
  // the corresponding Bare_point
  template<class Construct_bare_point, class Container>
  struct Index_to_Bare_point
  {
    typename boost::result_of<const Construct_bare_point(const Weighted_point&)>::type
    operator()(const std::size_t& i) const
    {
      return cp(c[i]);
    }

    Index_to_Bare_point(const Container& c, const Construct_bare_point& cp)
      : c(c), cp(cp) { }

    const Container& c;
    const Construct_bare_point cp;
  };

  template <class Tuple_or_pair,class InputIterator>
  std::ptrdiff_t insert_with_info(InputIterator first,InputIterator last)
  {
    size_type n = number_of_vertices();
    std::vector<std::size_t> indices;
    std::vector<Weighted_point> points;
    std::vector<typename Triangulation_data_structure::Vertex::Info> infos;
    std::size_t index=0;
    for(InputIterator it=first;it!=last;++it){
      Tuple_or_pair pair = *it;
      points.push_back(top_get_first(pair));
      infos.push_back(top_get_second(pair));
      indices.push_back(index++);
    }

    // We need to sort the points and their info at the same time through
    // the `indices` vector AND spatial sort can only handle Gt::Point_2.
    typedef typename Geom_traits::Construct_point_2 Construct_point_2;
    typedef Index_to_Bare_point<Construct_point_2,
                                std::vector<Weighted_point> > Access_bare_point;
    typedef typename boost::result_of<const Construct_point_2(const Weighted_point&)>::type Ret;
    typedef boost::function_property_map<Access_bare_point, std::size_t, Ret> fpmap;
    typedef CGAL::Spatial_sort_traits_adapter_2<Gt, fpmap> Search_traits_2;

    Access_bare_point accessor(points, geom_traits().construct_point_2_object());
    spatial_sort(indices.begin(), indices.end(),
                 Search_traits_2(
                   boost::make_function_property_map<
                     std::size_t, Ret, Access_bare_point>(accessor),
                   geom_traits()));

    Face_handle hint;
    Vertex_handle v_hint;
    for(typename std::vector<std::size_t>::const_iterator
      it = indices.begin(), end = indices.end();
      it != end; ++it)
    {
      v_hint = insert(points[*it], hint);

      if(v_hint!=Vertex_handle()){
        v_hint->info()=infos[*it];
        hint=v_hint->face();
      }
    }

    return number_of_vertices() - n;
  }

public:

  template < class InputIterator >
  std::ptrdiff_t
  insert(InputIterator first,
          InputIterator last,
          std::enable_if_t<
              boost::is_convertible<
                typename std::iterator_traits<InputIterator>::value_type,
                std::pair<Weighted_point,typename internal::Info_check<typename Triangulation_data_structure::Vertex>::type>
              >::value
          >* = nullptr
)
  {return insert_with_info< std::pair<Weighted_point,typename internal::Info_check<typename Triangulation_data_structure::Vertex>::type> >(first,last);}

  template <class  InputIterator_1,class InputIterator_2>
  std::ptrdiff_t
  insert(boost::zip_iterator< boost::tuple<InputIterator_1,InputIterator_2> > first,
          boost::zip_iterator< boost::tuple<InputIterator_1,InputIterator_2> > last,
          std::enable_if_t<
            boost::mpl::and_<
              typename boost::is_convertible< typename std::iterator_traits<InputIterator_1>::value_type, Weighted_point >,
              typename boost::is_convertible< typename std::iterator_traits<InputIterator_2>::value_type, typename internal::Info_check<typename Triangulation_data_structure::Vertex>::type >
            >::value
          >* =nullptr
)
  {return insert_with_info< boost::tuple<Weighted_point,typename internal::Info_check<typename Triangulation_data_structure::Vertex>::type> >(first,last);}
#endif //CGAL_TRIANGULATION_2_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO

  template < class Stream>
  Stream& draw_dual(Stream & ps) const
  {
    Finite_edges_iterator eit = finite_edges_begin();
    for(; eit != finite_edges_end(); ++eit) {
      Object o = dual(eit);
      typename Geom_traits::Line_2  l;
      typename Geom_traits::Ray_2   r;
      typename Geom_traits::Segment_2 s;
      if(CGAL::assign(s,o)) ps << s;
      if(CGAL::assign(r,o)) ps << r;
      if(CGAL::assign(l,o)) ps << l;
    }
    return ps;
  }

  template <class OutputItFaces, class OutputItBoundaryEdges,
            class OutputItHiddenVertices>
  Triple<OutputItFaces,OutputItBoundaryEdges, OutputItHiddenVertices>
  get_conflicts_and_boundary_and_hidden_vertices(const Weighted_point  &p,
                                                 OutputItFaces fit,
                                                 OutputItBoundaryEdges eit,
                                                 OutputItHiddenVertices vit,
                                                 Face_handle start =
      Face_handle()) const
  {
    CGAL_precondition(dimension() == 2);
    int li;
    Locate_type lt;
    Face_handle fh = locate(p,lt,li, start);
    switch(lt) {
      case OUTSIDE_AFFINE_HULL:
        return make_triple(fit, eit, vit);
      case VERTEX:
      case FACE:
      case EDGE:
      case OUTSIDE_CONVEX_HULL:
        //test whether p is not in conflict
        // with the first face:
        // this includes the cases that p is located
        // on a vertex and either equal or no conflict
        if(!test_conflict(p,fh))
          return make_triple(fit, eit, vit);

        // region includes all faces in conflict so far detected
        // stack includes the faces in the region whose neighbors
        // have not yet been looked at
        std::set<Face_handle> region;
        std::stack<Edge> st;

        //collection of all boundary_vertices:
        std::set< Vertex_handle> boundary_vertices;
        //collection of potential_intern_vertices = vertices incident
        // to an edge incident to two faces in conflict and met
        // twice during the "walk":
        std::set< Vertex_handle> potential_intern_vertices;

        *fit++ = fh; //put fh in OutputItFaces
        region.insert(fh);
        st.push(Edge(fh,2));
        st.push(Edge(fh,1));
        st.push(Edge(fh,0));

        while(! st.empty()){
          Edge e = st.top();
          st.pop();
          Face_handle fh = e.first;
          Face_handle fn = fh->neighbor(e.second);
          int i = fn->index(fh);
          if(region.find(fn) == region.end()){
            if(test_conflict(p,fn))
            {
              region.insert(fn);
              st.push(Edge(fn, cw(i)));
              st.push(Edge(fn,ccw(i)));
              *fit++ = fn;
            }
            else{
              e = Edge(fn,i);
              *eit++ = e;
              if(!is_infinite(fn->vertex(cw(i))))
                boundary_vertices.insert(fn->vertex(cw(i)));
              if(!is_infinite(fn->vertex(ccw(i))))
                boundary_vertices.insert(fn->vertex(ccw(i)));
            }
          }
          else {
            //insert the vertices of the last edge into the set of
            // potential intern vertices:
            potential_intern_vertices.insert(fn->vertex(ccw(i)));
            potential_intern_vertices.insert(fn->vertex(cw(i)));
          }
        }
        if(!potential_intern_vertices.empty()){
          //determine the hidden vertices:
          std::set_difference(potential_intern_vertices.begin(),
                              potential_intern_vertices.end(),
                              boundary_vertices.begin(),
                              boundary_vertices.end(),
                              vit);
        }
        return  make_triple(fit, eit, vit);
    }
    CGAL_assertion(false);
    return make_triple(fit, eit, vit);
  }

  template <class OutputItFaces, class OutputItBoundaryEdges>
  std::pair<OutputItFaces,OutputItBoundaryEdges>
  get_conflicts_and_boundary(const Weighted_point &p,
                              OutputItFaces fit,
                              OutputItBoundaryEdges eit,
                              Face_handle start = Face_handle()) const
  {
    Triple<OutputItFaces, OutputItBoundaryEdges, Emptyset_iterator>
        pp = get_conflicts_and_boundary_and_hidden_vertices(p, fit, eit,
                                                            Emptyset_iterator(),
                                                            start);
    return std::make_pair(pp.first, pp.second);
  }
  template <class OutputItFaces, class OutputItHiddenVertices>
  std::pair<OutputItFaces, OutputItHiddenVertices>
  get_conflicts_and_hidden_vertices(const Weighted_point &p,
                                    OutputItFaces fit,
                                    OutputItHiddenVertices vit,
                                    Face_handle start = Face_handle()) const
  {
    Triple<OutputItFaces, Emptyset_iterator, OutputItHiddenVertices>
        pp = get_conflicts_and_boundary_and_hidden_vertices(p,fit,
                                                            Emptyset_iterator(),
                                                            vit,
                                                            start);
    return std::make_pair(pp.first,pp.third);
  }

  template <class OutputItBoundaryEdges, class OutputItHiddenVertices>
  std::pair<OutputItBoundaryEdges, OutputItHiddenVertices>
  get_boundary_of_conflicts_and_hidden_vertices(const Weighted_point &p,
                                                OutputItBoundaryEdges eit,
                                                OutputItHiddenVertices vit,
                                                Face_handle start =
      Face_handle()) const
  {
    Triple<Emptyset_iterator, OutputItBoundaryEdges, OutputItHiddenVertices>
        pp = get_conflicts_and_boundary_and_hidden_vertices(p,
                                                            Emptyset_iterator(),
                                                            eit,vit,
                                                            start);
    return std::make_pair(pp.second,pp.third);
  }

  template <class OutputItFaces>
  OutputItFaces
  get_conflicts(const Weighted_point  &p,
                OutputItFaces fit,
                Face_handle start= Face_handle()) const
  {
    Triple<OutputItFaces, Emptyset_iterator, Emptyset_iterator>
        pp =
        get_conflicts_and_boundary_and_hidden_vertices(p, fit,
                                                       Emptyset_iterator(),
                                                       Emptyset_iterator(),
                                                       start);
    return pp.first;
  }

  template <class OutputItBoundaryEdges>
  OutputItBoundaryEdges
  get_boundary_of_conflicts(const Weighted_point  &p,
                            OutputItBoundaryEdges eit,
                            Face_handle start= Face_handle()) const
  {
    Triple<Emptyset_iterator, OutputItBoundaryEdges, Emptyset_iterator>
        pp =
        get_conflicts_and_boundary_and_hidden_vertices(p,
                                                       Emptyset_iterator(),
                                                       eit,
                                                       Emptyset_iterator(),
                                                       start);
    return pp.second;
  }

  template <class OutputItHiddenVertices>
  OutputItHiddenVertices
  get_hidden_vertices(const Weighted_point  &p, OutputItHiddenVertices vit,
                      Face_handle start= Face_handle()) const
  {
    Triple<Emptyset_iterator, Emptyset_iterator, OutputItHiddenVertices>
        pp = get_conflicts_and_boundary_and_hidden_vertices(p, Emptyset_iterator(),
                                                            Emptyset_iterator(),
                                                            vit,
                                                            start);
    return pp.third;
  }

  // nearest power vertex query
  Vertex_handle nearest_power_vertex(const Bare_point& p) const;
};

template < class Gt, class Tds >
inline bool
Regular_triangulation_2<Gt,Tds>::
test_conflict(const Weighted_point  &p, Face_handle fh) const
{
  return(power_test(fh,p) == ON_POSITIVE_SIDE);
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
clear()
{
  Base::clear();
  _hidden_vertices = 0;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
copy_triangulation_()
{
  // the list of vertices have been copied member for member and are
  // not good
  // clear them and next
  // scan the hidden vertices to retablish the list in faces
  typename Tds::Face_iterator baseit = this->_tds.face_iterator_base_begin();
  for(; baseit !=  this->_tds.face_iterator_base_end(); baseit++){
    baseit->vertex_list().clear();
  }

  Hidden_vertices_iterator hvit = hidden_vertices_begin();
  for(; hvit !=  hidden_vertices_end() ; ++hvit){
    hvit->face()->vertex_list().push_back(hvit);
  }
  CGAL_postcondition(is_valid());
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
copy_triangulation(const Self &tr)
{
  Base::copy_triangulation(tr);
  _hidden_vertices = tr._hidden_vertices;
  copy_triangulation_();
}

template < class Gt, class Tds >
Regular_triangulation_2<Gt,Tds>::
Regular_triangulation_2(const Self &tr)
  : Base(tr), _hidden_vertices(tr._hidden_vertices)
{
  copy_triangulation_();
}

template <class Gt, class Tds >
Regular_triangulation_2<Gt,Tds> &
Regular_triangulation_2<Gt, Tds>::
operator=(const Self &tr)
{
  copy_triangulation(tr);
  return *this;
}

template < class Gt, class Tds >
Oriented_side
Regular_triangulation_2<Gt,Tds>::
power_test(const Face_handle &f, const Weighted_point &p, bool perturb) const
{
  // p is supposed to be a finite point
  // if f is a finite face,
  // return  ON_NEGATIVE_SIDE if p is above f
  //(p has to be hidden)
  if(dimension() == 1)
    return power_test(f->vertex(0)->point(), f->vertex(1)->point(), p);

  int i;
  if(! f->has_vertex(infinite_vertex(), i))
    return power_test(f->vertex(0)->point(),
                      f->vertex(1)->point(),
                      f->vertex(2)->point(), p, perturb);

  Orientation o = orientation(f->vertex(ccw(i))->point(),
                              f->vertex(cw(i))->point(),
                              p);
  if(o==COLLINEAR)
    return power_test(f->vertex(ccw(i))->point(),
                      f->vertex(cw(i))->point(),p);

  return o;
}

template < class Gt, class Tds >
Oriented_side
Regular_triangulation_2<Gt,Tds>::
power_test(const Face_handle& f, int i, const Weighted_point &p) const
{
  // f,i is supposed to be a finite edge
  // p is supposed to be on  edge(f,i)
  // return ON_NEGATIVE_SIDE if p is above(f,i)
  // (p has to be hidden)
  CGAL_precondition(!is_infinite(f,i) &&
                                   orientation(f->vertex(ccw(i))->point(),
                                               f->vertex(cw(i))->point(),
                                               p) == COLLINEAR);
  return  power_test(f->vertex(ccw(i))->point(),
                     f->vertex(cw(i))->point(),
                     p);
}

template < class Gt, class Tds >
inline
Oriented_side
Regular_triangulation_2<Gt,Tds>::
power_test(const Weighted_point &p0,
           const Weighted_point &p1,
           const Weighted_point &p2,
           const Weighted_point &p,
           bool perturb) const
{
  CGAL_precondition(orientation(p0, p1, p2) == POSITIVE);

  using namespace boost;

  Oriented_side os = geom_traits().power_side_of_oriented_power_circle_2_object()(p0, p1, p2, p);

  if((os != ON_ORIENTED_BOUNDARY) ||(! perturb))
    return os;

  // We are now in a degenerate case => we do a symbolic perturbation.

  // We sort the points lexicographically.
  const Weighted_point * points[4] = {&p0, &p1, &p2, &p};
  std::sort(points, points + 4, typename Base::Perturbation_order(this));
  // We successively look whether the leading monomial, then 2nd monomial
  // of the determinant has non null coefficient.
  // 2 iterations are enough(cf paper)
  for(int i=3; i>1; --i) {
    if(points[i] == &p)
      return ON_NEGATIVE_SIDE; // since p0 p1 p2 are non collinear
                               // and positively oriented
    Orientation o;
    if(points[i] == &p2 &&(o = orientation(p0,p1,p)) != COLLINEAR)
      return o;
    if(points[i] == &p1 &&(o = orientation(p0,p,p2)) != COLLINEAR)
      return o;
    if(points[i] == &p0 &&(o = orientation(p,p1,p2)) != COLLINEAR)
      return o;
  }

  CGAL_assertion(false);
  return ON_NEGATIVE_SIDE;
}

template < class Gt, class Tds >
inline
Oriented_side
Regular_triangulation_2<Gt,Tds>::
power_test(const Weighted_point &p,
           const Weighted_point &q,
           const Weighted_point &r) const
{
  return geom_traits().power_side_of_oriented_power_circle_2_object()(p,q,r);
}

template < class Gt, class Tds >
inline
Oriented_side
Regular_triangulation_2<Gt,Tds>::
power_test(const Weighted_point &p,
           const Weighted_point &r) const
{
  return geom_traits().power_side_of_oriented_power_circle_2_object()(p,r);
}

template < class Gt, class Tds >
bool
Regular_triangulation_2<Gt,Tds>::
is_valid_face(Face_handle fh) const
{
  bool result = true;
  if(is_infinite(fh)) result = result && fh->vertex_list().empty();
  if(!result) { show_face(fh);}
  CGAL_assertion(result);

  typename Vertex_list::iterator vlit = fh->vertex_list().begin(),
                                 vldone = fh->vertex_list().end();
  for(; vlit != vldone; vlit++) {
    result = result && power_test(fh,(*vlit)->point()) == ON_NEGATIVE_SIDE;
    result = result &&((*vlit)->face() == fh);
    if(!result)     show_face(fh);
    CGAL_assertion(result);
  }
  return result;
}

template < class Gt, class Tds >
bool
Regular_triangulation_2<Gt,Tds>::
is_valid_vertex(Vertex_handle vh) const
{
  bool result = true;
  if(vh->is_hidden()) {
    Locate_type lt;
    int li;
    Face_handle loc = locate(vh->point(), lt, li, vh->face());
    if(dimension() == 0) {
      result = result && lt == Base::VERTEX;
      result = result && power_test(vh->face()->vertex(0)->point(), vh->point()) <= 0;
    } else {
      result = result && (!is_infinite(vh->face()));
      result = result && (loc == vh->face() ||
                         (lt == Base::VERTEX &&
                           vh->face()->has_vertex(loc->vertex(li))) ||
                         (lt == Base::EDGE && vh->face() ==
                           loc->neighbor(li)));

      result = result && power_test(vh->face(),vh->point()) == ON_NEGATIVE_SIDE;

//      if(!result) {
//        Rcpp::Rcerr << " from is_valid_vertex " << std::endl;
//        Rcpp::Rcerr << "sommet cache " << &*(vh)
//                  << "vh_point " <<vh->point() << " " << std::endl;
//        Rcpp::Rcerr << "vh_>face " << &*(vh->face())  << " " << std::endl;
//        Rcpp::Rcerr <<  "loc      " <<  &*(loc)
//                   << " lt " << lt  << " li " << li << std::endl;
//        show_face(vh->face());
//        show_face(loc);
//      }
    }
  }
  else { // normal vertex
    result = result && vh->face()->has_vertex(vh);
//    if(!result) {
//      Rcpp::Rcerr << " from is_valid_vertex " << std::endl;
//      Rcpp::Rcerr << "normal vertex " << &(*vh) << std::endl;
//      Rcpp::Rcerr << vh->point() << " " << std::endl;
//      Rcpp::Rcerr << "vh_>face " << &*(vh->face())  << " " << std::endl;
//      show_face(vh->face());
//    }
  }
  CGAL_assertion(result);
  return result;
}

template < class Gt, class Tds >
bool
Regular_triangulation_2<Gt,Tds>::
is_valid(bool verbose, int /* level */) const
{
  // cannot call for is_valid() of Base Triangulation class
  // because 1) number of vertices of base class does not match
  // tds.is_valid calls is_valid for each vertex
  // and the test is not fulfilled by  hidden vertices ...
  // result = result && Triangulation_2<Gt,Tds>::is_valid(verbose, level);
  bool result = true;
  for(All_faces_iterator fit = all_faces_begin();
      fit != all_faces_end(); ++fit) {
    result = result && is_valid_face(fit);
  }

  for(All_vertices_iterator vit = all_vertices_begin();
                            vit != all_vertices_end(); ++vit) {
    result = result && is_valid_vertex(vit);
  }

  for(Hidden_vertices_iterator hvit = hidden_vertices_begin();
      hvit != hidden_vertices_end(); ++hvit) {
    result = result && is_valid_vertex(hvit);
  }

  switch(dimension()) {
    case 0 :
      break;
    case 1:
      if(number_of_vertices() > 2) {
        Finite_vertices_iterator it1 = finite_vertices_begin(),
            it2(it1), it3(it1);
        ++it2;
        ++it3; ++it3;
        while(it3 != finite_vertices_end()) {
          Orientation s = orientation(it1->point(),
                                      it2->point(),
                                      it3->point());
          result = result && s == COLLINEAR ;
          CGAL_assertion(result);
          ++it1 ; ++it2; ++it3;
        }
      }
      break;
    case 2 :
      for(Finite_faces_iterator it=finite_faces_begin();
          it!=finite_faces_end(); it++) {
        CGAL_assertion(! is_infinite(it));
        Orientation s = orientation(it->vertex(0)->point(),
                                    it->vertex(1)->point(),
                                    it->vertex(2)->point());
        CGAL_assertion(s == LEFT_TURN);
        result = result && (s == LEFT_TURN);

        for(int i = 0 ; i < 3 ; i++) {
          if(!is_infinite(it->vertex(i)))
            result = result && ON_POSITIVE_SIDE !=
                power_test(it->neighbor(i), it->vertex(i)->point());
          CGAL_assertion(result);
        }
      }

      Vertex_circulator start = incident_vertices(infinite_vertex());
      Vertex_circulator pc(start);
      Vertex_circulator qc(start); ++qc;
      Vertex_circulator rc(start); ++rc; ++rc;
      do{
        Orientation s = orientation(pc->point(),
                                    qc->point(),
                                    rc->point());
        CGAL_assertion(s != LEFT_TURN);
        result = result && (s != LEFT_TURN);
        ++pc ; ++qc ; ++rc;
      } while(pc != start);

      // check number of faces. This cannot be done by the Tds
      // which does not know the number of components nor the genus
      result = result && (number_of_faces() == 2*(number_of_vertices()+1)
                          - 4
                          - degree(infinite_vertex()));
      CGAL_assertion(result);
      break;
  }

  // in any dimension
  if(verbose) {
    Rcpp::Rcerr << " nombres de sommets " << number_of_vertices() << "\t"
              << "nombres de sommets  caches " << number_of_hidden_vertices()
              << std::endl;
  }
  result = result && (Base::number_of_vertices() ==
                       number_of_vertices() + number_of_hidden_vertices());
  CGAL_assertion(result);
  return result;
}

template <class Gt, class Tds >
void
 Regular_triangulation_2<Gt, Tds>::
show_face(Face_handle fh) const
{
  Base::show_face(fh);

  typename Vertex_list::iterator current;
  Rcpp::Rcerr << "  +++++>>>    ";
  for(current= fh->vertex_list().begin();
       current!= fh->vertex_list().end() ; current++) {
        Rcpp::Rcerr <<"[ "<< ((*current)->point()) << " ] ,  ";
  }
  Rcpp::Rcerr <<std::endl;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
show_all() const
{
  Rcpp::Rcerr << "PRINT THE FULL TRIANGULATION :" << std::endl;
  Rcpp::Rcerr << std::endl << "====> "<< this ;
  Rcpp::Rcerr << " dimension " <<  dimension() << std::endl;
  Rcpp::Rcerr << " nb of vertices " << number_of_vertices()
            << " nb of hidden vertices " << number_of_hidden_vertices()
            <<   std::endl;

  if(dimension() < 1) return;
  if(dimension() == 1) {
    Rcpp::Rcerr << " all edges " <<std::endl;
    All_edges_iterator aeit;
    for(aeit = all_edges_begin(); aeit != all_edges_end(); aeit++){
      show_face(aeit->first);
    }
  }

  else{ //dimension ==2
    Rcpp::Rcerr << " finite faces " << std::endl;
    Finite_faces_iterator fi;
    for(fi = finite_faces_begin(); fi != finite_faces_end(); fi++) {
      show_face(fi);
    }

    Rcpp::Rcerr << " infinite faces " << std::endl;
    All_faces_iterator afi;
    for(afi = all_faces_begin(); afi != all_faces_end(); afi++) {
      if(is_infinite(afi)) show_face(afi);
    }
  }

  if(number_of_vertices()>1) {
    Rcpp::Rcerr << "printing vertices of the regular triangulation" << std::endl;
    All_vertices_iterator vi;
    for(vi = all_vertices_begin(); vi != all_vertices_end(); vi++){
      show_vertex(vi);
      Rcpp::Rcerr << "  / associated face : "
                << &*(vi->face()) << std::endl;
    }
    Rcpp::Rcerr << std::endl;
  }

  Rcpp::Rcerr << "hidden vertices " << std::endl;
  Hidden_vertices_iterator hvi = hidden_vertices_begin();
  for(; hvi != hidden_vertices_end(); hvi++) {
    show_vertex(hvi);
    Rcpp::Rcerr << "  / associated face : "
              << &*(hvi->face()) << std::endl;
  }
  return;
}

//DUALITY
template < class Gt, class Tds >
inline
typename Regular_triangulation_2<Gt,Tds>::Bare_point
Regular_triangulation_2<Gt,Tds>::
dual(Face_handle f) const
{
  return weighted_circumcenter(f);
}

template < class Gt, class Tds >
inline
typename Regular_triangulation_2<Gt,Tds>::Bare_point
Regular_triangulation_2<Gt,Tds>::
weighted_circumcenter(Face_handle f) const
{
  CGAL_precondition(dimension() == 2 || !is_infinite(f));
  return weighted_circumcenter(f->vertex(0)->point(),
                               f->vertex(1)->point(),
                               f->vertex(2)->point());
}

template<class Gt, class Tds>
inline
typename Regular_triangulation_2<Gt,Tds>::Bare_point
Regular_triangulation_2<Gt,Tds>::
weighted_circumcenter(const Weighted_point& p0,
                      const Weighted_point& p1,
                      const Weighted_point& p2) const
{
  return geom_traits().construct_weighted_circumcenter_2_object()(p0,p1,p2);
}

template < class Gt, class Tds >
inline
Object
Regular_triangulation_2<Gt,Tds>::
dual(const Edge &e) const
{
  typedef typename Geom_traits::Line_2        Line;
  typedef typename Geom_traits::Ray_2         Ray;
  typedef typename Geom_traits::Segment_2     Segment;

  CGAL_precondition(! is_infinite(e));
  if(dimension() == 1){
    const Weighted_point& p = (e.first)->vertex(cw(e.second))->point();
    const Weighted_point& q = (e.first)->vertex(ccw(e.second))->point();
    Line l = geom_traits().construct_radical_axis_2_object()(p,q);
    return make_object(l);
  }

  // dimension==2
  if((! is_infinite(e.first)) &&
     (! is_infinite(e.first->neighbor(e.second)))) {
    Segment s = geom_traits().construct_segment_2_object()(
                  dual(e.first),dual(e.first->neighbor(e.second)));
    return make_object(s);
  }

  // one of the adjacent faces is infinite
  Face_handle f; int i;
  if(is_infinite(e.first)) {
    f = e.first->neighbor(e.second);
    i = f->index(e.first);
  }
  else {
    f = e.first;
    i = e.second;
  }
  const Weighted_point& p = f->vertex(cw(i))->point();
  const Weighted_point& q = f->vertex(ccw(i))->point();
  Line l  = geom_traits().construct_radical_axis_2_object()(p,q);
  Ray r = geom_traits().construct_ray_2_object()(dual(f), l);
  return make_object(r);
}

template < class Gt, class Tds >
inline
Object
Regular_triangulation_2<Gt,Tds>::
dual(const Edge_circulator& ec) const
{
  return dual(*ec);
}

template < class Gt, class Tds >
inline
Object
Regular_triangulation_2<Gt,Tds>::
dual(const Finite_edges_iterator& ei) const
{
  return dual(*ei);
}

//INSERTION-REMOVAL
template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
push_back(const Weighted_point &p)
{
  return insert(p);
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
insert(const Weighted_point &p, Face_handle start)
{
  Locate_type lt;
  int li;
  Face_handle loc = locate(p, lt, li, start);
  return insert(p, lt, loc, li);
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
insert(const Weighted_point &p, Locate_type lt, Face_handle loc, int li)
{
  Vertex_handle v;
  switch(lt) {
    case Base::VERTEX:
    {
      CGAL_precondition(dimension() >= 0);
      if(dimension() == 0) {
        // in this case locate() oddly returns loc = nullptr and li = 4,
        // so we work around it.
        loc = finite_vertex()->face();
        li = 0;
      }

      Vertex_handle vv = loc->vertex(li);
      CGAL::Oriented_side side = power_test(vv->point(), p);

      switch(side) {
        case ON_NEGATIVE_SIDE:
          return hide_new_vertex(loc, p);

        case ON_POSITIVE_SIDE:
          v = this->_tds.create_vertex();
          v->set_point(p);
          exchange_incidences(v,vv);
          hide_vertex(loc, vv);
          regularize(v);
          return v;

        default: // that is ON_ORIENTED_BOUNDARY:
          return vv;
      }
    }
    case Base::EDGE:
    {
      CGAL_precondition(dimension() >= 1);
      Oriented_side os = dimension() == 1 ? power_test(loc, li, p) :
                                            power_test(loc, p, true);

      if(os < 0) {
        if(is_infinite(loc)) loc = loc->neighbor(li);
        return hide_new_vertex(loc, p);
      }
      v = insert_in_edge(p, loc, li);
      break;
    }

    case Base::FACE:
    {
      CGAL_precondition(dimension() >= 2);
      if(power_test(loc, p, true) < 0) {
        return hide_new_vertex(loc,p);
      }
      v = insert_in_face(p, loc);
      break;
    }
    default:
      v = Base::insert(p, lt, loc, li);
  }

  if(lt == OUTSIDE_AFFINE_HULL) {
    //clear vertex list of infinite faces which have been copied
    for(All_faces_iterator afi = all_faces_begin();
          afi != all_faces_end(); afi++) {
      if(is_infinite(afi))
        afi->vertex_list().clear();
    }
  }

  regularize(v);
  return v;
}

/*
The reinsert function  insert a weighted point which was in a hidden
vertex.
The new and old vertices are then exchanged ; this is required
if the regular triangulation is used with a hierarchy because
the old vertex has its up and down pointers set and other vertices
pointing on him
*/
template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
reinsert(Vertex_handle v, Face_handle start)
{
  CGAL_assertion(v->is_hidden());
  v->set_hidden(false);
  _hidden_vertices--;

//  // to debug
//  Rcpp::Rcerr << "from reinsert " << std::endl;
//  show_vertex(v);
//  Locate_type lt;
//  int li;
//  Face_handle loc = locate(v->point(), lt, li, start);
//  Rcpp::Rcerr << "locate " << &(*loc) << "\t" << lt << "\t" << li << std::endl;
//  show_face(loc);
//  Rcpp::Rcerr << std::endl;

  Vertex_handle vh = insert(v->point(), start);
  if(vh->is_hidden())
    exchange_hidden(v,vh);
  else
    exchange_incidences(v,vh);

  this->_tds.delete_vertex(vh);
  return v;
}

// push va instead of vb in the list of the face fb hiding vb
// vb must be the last inserted vertex in the list of fb
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
exchange_hidden(Vertex_handle va, Vertex_handle vb)
{
  CGAL_assertion(vb->is_hidden());
  CGAL_assertion(vb == vb->face()->vertex_list().back());

//  //to debug
//  Rcpp::Rcerr << "from exchange hidden 1" << std::endl;
//  show_vertex(vb);
//  Rcpp::Rcerr << "  / associated face : "
//            << &*(vb->face()) << std::endl;

  vb->face()->vertex_list().pop_back();
  _hidden_vertices--;
  hide_vertex(vb->face(), va);

//  //to debug
//  Rcpp::Rcerr << "from exchange hidden 1" << std::endl;
//  show_vertex(va);
//  Rcpp::Rcerr << "  / associated face : "
//            << &*(va->face()) << std::endl << std::endl;
}

// set to va the incidences of vb
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
exchange_incidences(Vertex_handle va, Vertex_handle vb)
{
  CGAL_assertion(!vb->is_hidden());
  std::list<Face_handle> faces;
  if(dimension() == 0) {
    faces.push_back(vb->face());
  } else if(dimension() == 1) {
    faces.push_back(vb->face());
    int i = vb->face()->index(vb);
    faces.push_back(vb->face()->neighbor(1-i));
  }
  else {
    CGAL_assertion(dimension() == 2);
    Face_circulator fc = incident_faces(vb), done(fc);
    do {
      faces.push_back(fc);
      fc++;
    } while(fc != done);
  }

  va->set_face(*(faces.begin()));
  for(typename std::list<Face_handle>::iterator it = faces.begin();
      it != faces.end(); it++){
    Face_handle fh = *it;
    fh->set_vertex(fh->index(vb), va);
  }
  return;
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
insert_in_face(const Weighted_point &p, Face_handle f)
{
  Vertex_handle v = Base::insert_in_face(p,f);
  update_hidden_points_1_3(f,
         f->neighbor(ccw(f->index(v))),
         f->neighbor(cw(f->index(v))));
  return v;
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
insert_in_edge(const Weighted_point &p, Face_handle f, int i)
{
  Vertex_handle v;
  if(dimension() == 1) {
    v = Base::insert_in_edge(p,f,i);
    Face_handle g = f->neighbor(1 - f->index(v));
    update_hidden_points_2_2(f,g);
  }
  else { //dimension()==2
    // don't use update_hidden_points_2_2 any more to split
    // hidden vertices list because new affectation of f and n
    // around new vertex is unknown
    Face_handle n = f->neighbor(i);
    Vertex_list p_list;
    p_list.splice(p_list.begin(),f->vertex_list());
    p_list.splice(p_list.begin(),n->vertex_list());
    v = Base::insert_in_edge(p,f,i);
    Face_handle loc;
    while(! p_list.empty()){
      loc = locate(p_list.front()->point(), n);
      if(is_infinite(loc)) loc = loc->neighbor(loc->index(infinite_vertex()));
      hide_vertex(loc, p_list.front());
      p_list.pop_front();
    }
  }
  return v;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
regularize(Vertex_handle v)
{
  CGAL_precondition(v != infinite_vertex());
  Faces_around_stack faces_around;

  if(dimension() < 1) return;

  //initialise faces_around
  if(dimension() == 1) {
    faces_around.push_back(v->face());
    faces_around.push_back(v->face()->neighbor(1- v->face()->index(v)));
  }
  else{ //dimension==2
    Face_circulator fit = incident_faces(v), done(fit);
    do {
      faces_around.push_back(fit++);
    } while(fit != done);
  }

  while(! faces_around.empty())
    stack_flip(v, faces_around);
  return;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
flip(Face_handle f, int i)
{
  Face_handle n = f->neighbor(i);
  Base::flip(f,i);
  update_hidden_points_2_2(f,n);
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
remove_degree_3(Vertex_handle v, Face_handle f)
{
  if(f == Face_handle())
    f=v->face();

  update_hidden_points_3_1(f,
                           f->neighbor(cw(f->index(v))),
                           f->neighbor(ccw(f->index(v))));
  Base::remove_degree_3(v,f);
  if(is_infinite(f)) { //the list of f is given to its finite neighbor
    Face_handle fn = f->neighbor(f->index(infinite_vertex()));
    set_face(f->vertex_list(),fn);
    fn->vertex_list().splice(fn->vertex_list().begin(), f->vertex_list());
  }
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
remove_hidden(Vertex_handle v)
{
  _hidden_vertices--;
  v->face()->vertex_list().remove(v);
  delete_vertex(v);
  return;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
remove(Vertex_handle v)
{
  CGAL_precondition(v != Vertex_handle());
  CGAL_precondition(!is_infinite(v));

  if(v->is_hidden())
    return remove_hidden(v);

  Face_handle hint;
  int ihint = 0;

  Vertex_list to_reinsert;
  switch(dimension()) {
    case 0:
      to_reinsert.splice(to_reinsert.begin(), v->face()->vertex_list());
      break;
    case 1:
    {
      Face_handle f1 = v->face();
      ihint = f1->index(v);
      hint = f1->neighbor(ihint);
      Face_handle f2 = f1->neighbor(1 - ihint);
      ihint = mirror_index(f1, ihint);

      to_reinsert.splice(to_reinsert.begin(), f1->vertex_list());
      to_reinsert.splice(to_reinsert.begin(), f2->vertex_list());
      break;
    }
    case 2:
    {
      Face_circulator f = incident_faces(v), end = f;
      ihint = f->index(v);
      hint = f->neighbor(ihint);
      ihint = mirror_index(f, ihint);
      do to_reinsert.splice(to_reinsert.begin(), f->vertex_list());
      while(++f != end);
      break;
    }
  }

  if(number_of_vertices() <= 2) {
    this->_tds.remove_dim_down(v);
  } else if(dimension() < 2) {
    Base::remove(v);
  } else {
    remove_2D(v);
  }

  if(hint != Face_handle()) hint = hint->neighbor(ihint);

  for(typename Vertex_list::iterator i = to_reinsert.begin();
      i != to_reinsert.end(); ++i)
  {
    hint = reinsert(*i, hint)->face();
  }
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
remove_2D(Vertex_handle v)
{
  if(test_dim_down(v)) { this->_tds.remove_dim_down(v); }
  else {
    std::list<Edge> hole;
    make_hole(v, hole);
    fill_hole_regular(hole);
    delete_vertex(v);
  }
  return;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
fill_hole_regular(std::list<Edge> & first_hole)
{
  typedef std::list<Edge> Hole;
  typedef std::list<Hole> Hole_list;

  Hole hole;
  Hole_list hole_list;
  Face_handle ff, fn;
  int i, ii, in;

  hole_list.push_front(first_hole);

  while(! hole_list.empty())
  {
    hole = hole_list.front();
    hole_list.pop_front();
    typename Hole::iterator hit = hole.begin();

    // if the hole has only three edges, create the triangle
    if(hole.size() == 3)
    {
      Face_handle  newf = create_face();
      hit = hole.begin();
      for(int j=0; j<3; j++)
      {
        ff = (*hit).first;
        ii = (*hit).second;
        hit++;
        ff->set_neighbor(ii,newf);
        newf->set_neighbor(j,ff);
        newf->set_vertex(newf->ccw(j),ff->vertex(ff->cw(ii)));
      }
      continue;
    }

    // else find an edge with two finite vertices
    // on the hole boundary
    // and the new triangle adjacent to that edge
    //  cut the hole and push it back

    // first, ensure that a neighboring face
    // whose vertices on the hole boundary are finite
    // is the first of the hole
    bool finite = false;
    while(!finite)
    {
      ff = hole.front().first;
      ii = hole.front().second;
      if(is_infinite(ff->vertex(cw(ii))) ||
           is_infinite(ff->vertex(ccw(ii))))
      {
        hole.push_back(hole.front());
        hole.pop_front();
      }
      else
        finite = true;
    }

    // take the first neighboring face and pop it;
    ff = hole.front().first;
    ii = hole.front().second;
    hole.pop_front();

    Vertex_handle  v0 = ff->vertex(ff->cw(ii));
    const Weighted_point& p0 = v0->point();
    Vertex_handle  v1 = ff->vertex(ff->ccw(ii));
    const Weighted_point& p1 = v1->point();
    Vertex_handle  v2 = infinite_vertex();
    Weighted_point p2;
    Vertex_handle  vv;
    Weighted_point p;

    typename Hole::iterator hdone = hole.end();
    hit = hole.begin();
    typename Hole::iterator cut_after(hit);

    // if tested vertex is c with respect to the vertex opposite
    // to nullptr neighbor,
    // stop at the before last face;
    hdone--;
    while(hit != hdone)
    {
      fn = (*hit).first;
      in = (*hit).second;
      vv = fn->vertex(ccw(in));
      if(is_infinite(vv))
      {
        if(is_infinite(v2))
          cut_after = hit;
      }
      else
      { // vv is a finite vertex
        p = vv->point();
        if(orientation(p0,p1,p) == COUNTERCLOCKWISE)
        {
          if(is_infinite(v2))
          {
            v2=vv;
            p2=p;
            cut_after=hit;
          }
          else if(power_test(p0,p1,p2,p,true) == ON_POSITIVE_SIDE)
          {
            v2=vv;
            p2=p;
            cut_after=hit;
          }
        }
      }
      ++hit;
    }

    // create new triangle and update adjacency relations
    Face_handle newf = create_face(v0,v1,v2);
    newf->set_neighbor(2,ff);
    ff->set_neighbor(ii, newf);

    //update the hole and push back in the Hole_List stack
    // if v2 belongs to the neighbor following or preceding *f
    // the hole remain a single hole
    // otherwise it is split in two holes

    fn = hole.front().first;
    in = hole.front().second;
    if(fn->has_vertex(v2, i) && i == (int)fn->ccw(in))
    {
      newf->set_neighbor(0,fn);
      fn->set_neighbor(in,newf);
      hole.pop_front();
      hole.push_front(Edge(newf,1));
      hole_list.push_front(hole);
    }
    else
    {
      fn = hole.back().first;
      in = hole.back().second;
      if(fn->has_vertex(v2, i) && i == (int)fn->cw(in))
      {
        newf->set_neighbor(1,fn);
        fn->set_neighbor(in,newf);
        hole.pop_back();
        hole.push_back(Edge(newf,0));
        hole_list.push_front(hole);
      }
      else
      { // split the hole in two holes
        Hole new_hole;
        ++cut_after;
        while(hole.begin() != cut_after)
        {
          new_hole.push_back(hole.front());
          hole.pop_front();
        }

        hole.push_front(Edge(newf,1));
        new_hole.push_front(Edge(newf,0));
        hole_list.push_front(hole);
        hole_list.push_front(new_hole);
      }
    }
  }
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
set_face(Vertex_list& vl, const Face_handle& fh)
{
  for(typename Vertex_list::iterator it = vl.begin(); it != vl.end(); it++)
   (*it)->set_face(fh);
}

// add the vertex_list of f2 and f3 to the point list of f1
// for the 3-1 flip
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
update_hidden_points_3_1(const Face_handle& f1, const Face_handle& f2,
       const Face_handle& f3)
{
  set_face(f2->vertex_list(), f1);
  set_face(f3->vertex_list(), f1);
 (f1->vertex_list()).splice(f1->vertex_list().begin(),f2->vertex_list());
 (f1->vertex_list()).splice(f1->vertex_list().begin(),f3->vertex_list());
  return;
}


// the points of the lists of 2 faces are sorted
// because of a 2-2 flip
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
update_hidden_points_2_2(const Face_handle& f1, const Face_handle& f2)
{
  CGAL_assertion(f1->has_neighbor(f2));

  Vertex_list p_list;
  p_list.splice(p_list.begin(), f1->vertex_list());
  p_list.splice(p_list.begin(), f2->vertex_list());

  // if one of the face is infinite,
  // the other face hide all the points
  if(is_infinite(f1)) {
    set_face(p_list, f2);
   (f2->vertex_list()).splice(f2->vertex_list().begin(), p_list);
    return;
  }
  if(is_infinite(f2)) {
    set_face(p_list, f1);
   (f1->vertex_list()).splice(f1->vertex_list().begin(), p_list);
    return;
  }

  if(dimension() == 1) {
    const Weighted_point& a1 = f1->vertex(f1->index(f2))->point();
    const Weighted_point& a  = f1->vertex(1-f1->index(f2))->point();
    while(! p_list.empty()) {
      if(compare_x(a, p_list.front()->point()) ==
           compare_x(a, a1)  &&
           compare_y(a, p_list.front()->point()) ==
           compare_y(a, a1))
      {
        hide_vertex(f1, p_list.front());
      } else {
        hide_vertex(f2, p_list.front());
      }
      p_list.pop_front();
    }
    return;
  }

  // from here f1 and f2 are finite 2-dimensional faces
  int idx2 = f1->index(f2);
  Vertex_handle v0=f1->vertex(ccw(idx2));
  Vertex_handle v1=f1->vertex(cw(idx2));
  CGAL_assertion(!is_infinite(v0) && !is_infinite(v1));

  while(! p_list.empty())
  {
    if(orientation(v0->point(), v1->point(), p_list.front()->point()) ==
        COUNTERCLOCKWISE)
      hide_vertex(f1, p_list.front());
    else
      hide_vertex(f2, p_list.front());
    p_list.pop_front();
  }
}

// The point list of f1 is separated into 3 lists
// for a 1-3 flip
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
update_hidden_points_1_3(const Face_handle& f1, const Face_handle& f2,
                         const Face_handle& f3)
{
  CGAL_assertion(f1->has_neighbor(f2) &&
                               f2->has_neighbor(f3) &&
                               f3->has_neighbor(f1));


  Vertex_list p_list;
  p_list.splice(p_list.begin(),f1->vertex_list());
  if(p_list.empty())
    return;

  // the following does not work if
  // two of f1,f2 and f3 are twice neighbors
  // but this cannot appear taking the assertion into account;
  int idx2 = f1->index(f2),
      idx3 = f1->index(f3);
  Vertex_handle v2 = f1->vertex(idx2),
      v3 = f1->vertex(idx3),
      v0 = f1->vertex(3-(idx2+idx3)),
      v1 = f2->vertex(f2->index(f1));

  CGAL_assertion(f2->has_vertex(v0) && f1->has_vertex(v0));
  CGAL_assertion(f3->has_vertex(v1));
  CGAL_assertion(! is_infinite(v0));

  // if two of f1, f2,and f3 are infinite
  // the list goes entirely to the third finite face
  // no orientation test necessary
  // because the point list of an infinite face
  // is only made of point projecting on its finite edge
  if(is_infinite(f1) && is_infinite(f2)) {
    set_face(p_list, f3);
    f3->vertex_list().splice(f3->vertex_list().begin(), p_list);
    return;
  }
  if(is_infinite(f1) && is_infinite(f3)) {
    set_face(p_list, f2);
    f2->vertex_list().splice(f2->vertex_list().begin(), p_list);
    return;
  }
  if(is_infinite(f2) && is_infinite(f3)){
    set_face(p_list, f1);
    f1->vertex_list().splice(f1->vertex_list().begin(), p_list);
    return;
  }

  // if here, v1,v2,v3 and v0 are finite vertices
  while(! p_list.empty())
  {
    Vertex_handle v(p_list.front());
//    if(orientation(v2->point(),v0->point(), v->point()) !=
//       orientation(v2->point(),v0->point(),v3->point()))
//    { // not in f1
//      if(orientation(v1->point(), v0->point(), v->point()) !=
//          orientation(v1->point(), v0->point(), v3->point()))
//        // not in f2
//        hide_vertex(f3, v);
//      else
//        hide_vertex(f2, v);
//    }
//    else
//      hide_vertex(f1, v);
    if(orientation(v2->point(),v0->point(), v->point()) ==
       orientation(v2->point(),v0->point(),v3->point())  &&
       orientation(v3->point(),v0->point(), v->point()) ==
       orientation(v3->point(),v0->point(), v2->point()))
      hide_vertex(f1, v);
    else if(orientation(v1->point(), v0->point(), v->point()) ==
             orientation(v1->point(), v0->point(), v3->point()))
      hide_vertex(f2,v);
    else
      hide_vertex(f3,v);
    p_list.pop_front();
  }
}

// the vertex is a degree three vertex which has to removed
// and hidden
// create first  a new hidden vertex and exchange with the vertex
// to be removed by the tds :
// this is required to keep up and down pointers right when using a hierarchy
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
hide_remove_degree_3(Face_handle fh, Vertex_handle vh)
{
 Vertex_handle vnew= this->_tds.create_vertex();
 exchange_incidences(vnew,vh);
 remove_degree_3(vnew, fh);
 hide_vertex(fh,vh);
}

// create a vertex and hide it
template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
hide_new_vertex(Face_handle f, const Weighted_point& p)
{
  Vertex_handle v = this->_tds.create_vertex();
  v->set_point(p);
  hide_vertex(f, v);
  return v;
}

// insert the vertex to the hidden vertex list
template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
hide_vertex(Face_handle f, Vertex_handle vh)
{
  // no hidden vertex in infinite face
  if(is_infinite(f) && dimension() > 0)
    f = f->neighbor(f->index(infinite_vertex()));

  if(! vh->is_hidden()) {
    vh->set_hidden(true);
    _hidden_vertices++;
  }
  vh->set_face(f);
  f->vertex_list().push_back(vh);
}

// template < class Gt, class Tds >
// void
// Regular_triangulation_2<Gt,Tds>::
// hide_vertex(Face_handle f, void* ptr)
// {
//   Vertex_handle v(static_cast<Vertex*>(ptr));
//   hide_vertex(f, v);
// }

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
stack_flip(Vertex_handle v, Faces_around_stack &faces_around)
{
  Face_handle f=faces_around.front();
  faces_around.pop_front();
  int i = f->index(v);
  Face_handle n = f->neighbor(i);

  if(dimension() == 1) {
    if(is_infinite(f)  || is_infinite(n))
      return;

    if(power_test(v->point(),
                     n->vertex(n->index(f))->point(),
                     f->vertex(1-i)->point()) ==  ON_NEGATIVE_SIDE)
      stack_flip_dim1(f,i,faces_around);
    return;
  }

  // now dimension() == 2
  //test the regularity of edge(f,i)
  //if(power_test(n, v->point()) == ON_NEGATIVE_SIDE)
  if(power_test(n, v->point(), true) != ON_POSITIVE_SIDE)
    return;

  if(is_infinite(f,i))
  {
    int j = 3 - (i + f->index(infinite_vertex()));
    if(degree(f->vertex(j)) == 4)
      stack_flip_4_2(f,i,j,faces_around);
    return;
  }

  // now f and n are both finite faces
  int ni = n->index(f);
  Orientation occw = orientation(f->vertex(i)->point(),
                                 f->vertex(ccw(i))->point(),
                                 n->vertex(ni)->point());
  Orientation ocw  = orientation(f->vertex(i)->point(),
                                 f->vertex(cw(i))->point(),
                                 n->vertex(ni)->point());
  if(occw == LEFT_TURN && ocw == RIGHT_TURN) {
    // quadrilater(f,n) is convex
    stack_flip_2_2(f,i, faces_around);
    return;
  }
  if(occw == RIGHT_TURN && degree(f->vertex(ccw(i))) == 3) {
    stack_flip_3_1(f,i,ccw(i),faces_around);
    return;
  }
  if(ocw == LEFT_TURN && degree(f->vertex(cw(i))) == 3) {
    stack_flip_3_1(f,i,cw(i),faces_around);
    return;
  }
  if(occw == COLLINEAR && degree(f->vertex(ccw(i))) == 4) {
    stack_flip_4_2(f,i,ccw(i),faces_around);
    return;
  }
  if(ocw == COLLINEAR && degree(f->vertex(cw(i))) == 4)
    stack_flip_4_2(f,i,cw(i),faces_around);

  return;
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
stack_flip_4_2(Face_handle f, int i, int j, Faces_around_stack & faces_around)
{
  int k = 3-(i+j);
  Face_handle g=f->neighbor(k);
  if(!faces_around.empty())
  {
    if(faces_around.front() == g)
      faces_around.pop_front();
    else if(faces_around.back() == g)
      faces_around.pop_back();
  }

  //union f with  g and f->neihgbor(i) with g->f->neihgbor(i)
  Face_handle fn = f->neighbor(i);
  //Face_handle gn = g->neighbor(g->index(f->vertex(i)));
  Vertex_handle vq = f->vertex(j);

  this->_tds.flip(f, i); //not using flip because the vertex j is flat.
  update_hidden_points_2_2(f,fn);
  Face_handle h1 = (j == ccw(i) ? fn : f);
  //hide_vertex(h1, vq);
  hide_remove_degree_3(g,vq);
  if(j == ccw(i)) {
    faces_around.push_front(h1);
    faces_around.push_front(g);
  }
  else {
    faces_around.push_front(g);
    faces_around.push_front(h1);
  }
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
stack_flip_3_1(Face_handle f, int i, int j, Faces_around_stack & faces_around)
{
  int k = 3-(i+j);
  Face_handle g=f->neighbor(k);
  if(!faces_around.empty())
  {
    if(faces_around.front()== g)
      faces_around.pop_front();
    else if(faces_around.back() == g)
      faces_around.pop_back();
  }

  Vertex_handle vq= f->vertex(j);
  //hide_vertex(f,vq);
  hide_remove_degree_3(f,vq);
  faces_around.push_front(f);
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
stack_flip_2_2(Face_handle f, int i, Faces_around_stack & faces_around)
{
  Vertex_handle vq = f->vertex(ccw(i));
  flip(f,i);
  if(f->has_vertex(vq)) {
    faces_around.push_front(f->neighbor(ccw(i)));
    faces_around.push_front(f);
  }
  else {
    faces_around.push_front(f);
    faces_around.push_front(f->neighbor(cw(i)));
  }
}

template < class Gt, class Tds >
void
Regular_triangulation_2<Gt,Tds>::
stack_flip_dim1(Face_handle f, int i, Faces_around_stack &faces_around)
{
  Vertex_handle va = f->vertex(1-i);
  Face_handle n= f->neighbor(i);
  int in = n->index(f);
  Vertex_handle vb = n->vertex(in);
  f->set_vertex(1-i, n->vertex(in));
  vb->set_face(f);
  f->set_neighbor(i, n->neighbor(1-in));
  n->neighbor(1-in)->set_neighbor(n->neighbor(1-in)->index(n), f);
 (f->vertex_list()).splice(f->vertex_list().begin(),n->vertex_list());
  set_face(f->vertex_list(),f);
  delete_face(n);
  hide_vertex(f,va);
  faces_around.push_front(f);
  return;
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::All_vertices_iterator
Regular_triangulation_2<Gt,Tds>::
all_vertices_begin() const
{
  return CGAL::filter_iterator(Base::all_vertices_end(),
                               Hidden_tester(),
                               Base::all_vertices_begin());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::All_vertices_iterator
Regular_triangulation_2<Gt,Tds>::
all_vertices_end() const
{
  return CGAL::filter_iterator(Base::all_vertices_end(),
                               Hidden_tester());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::All_vertex_handles
Regular_triangulation_2<Gt,Tds>::
all_vertex_handles() const
{
  return make_prevent_deref_range(all_vertices_begin(),all_vertices_end());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Finite_vertices_iterator
Regular_triangulation_2<Gt,Tds>::
finite_vertices_begin() const
{
  return CGAL::filter_iterator(Base::finite_vertices_end(),
                               Hidden_tester(),
                               Base::finite_vertices_begin());
}


template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
finite_vertex() const
{
  CGAL_precondition(number_of_vertices() >= 1);
  return(finite_vertices_begin());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Finite_vertices_iterator
Regular_triangulation_2<Gt,Tds>::
finite_vertices_end() const
{

  return CGAL::filter_iterator(Base::finite_vertices_end(),
                               Hidden_tester());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Finite_vertex_handles
Regular_triangulation_2<Gt,Tds>::
finite_vertex_handles() const
{
  return make_prevent_deref_range(finite_vertices_begin(),finite_vertices_end());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Hidden_vertices_iterator
Regular_triangulation_2<Gt,Tds>::
hidden_vertices_begin() const
{
  return CGAL::filter_iterator(Base::finite_vertices_end(),
                               Unhidden_tester(),
                               Base::finite_vertices_begin());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Hidden_vertices_iterator
Regular_triangulation_2<Gt,Tds>::
hidden_vertices_end() const
{
  return CGAL::filter_iterator(Base::finite_vertices_end(),
                               Unhidden_tester());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Hidden_vertex_handles
Regular_triangulation_2<Gt,Tds>::
hidden_vertex_handles() const
{
  return make_prevent_deref_range(hidden_vertices_begin(),hidden_vertices_end());
}

template < class Gt, class Tds >
typename Regular_triangulation_2<Gt,Tds>::Vertex_handle
Regular_triangulation_2<Gt,Tds>::
nearest_power_vertex(const Bare_point& p) const
{
  if(dimension() == -1) { return Vertex_handle(); }

  if(dimension() == 0) { return finite_vertex(); }

  typename Geom_traits::Compare_power_distance_2 cmp_power_distance =
      geom_traits().compare_power_distance_2_object();

  Vertex_handle vclosest;
  Vertex_handle v = finite_vertex();

  //  if(dimension() == 1) {
  //  }

  do {
    vclosest = v;
    Weighted_point wp = v->point();
    Vertex_circulator vc_start = incident_vertices(v);
    Vertex_circulator vc = vc_start;
    do {
      if(!is_infinite(vc)) {
        if(cmp_power_distance(p, vc->point(), wp) == SMALLER) {
          v = vc;
          break;
        }
      }
      ++vc;
    } while(vc != vc_start);
  } while(vclosest != v);

  return vclosest;
}

} //namespace CGAL

#endif // CGAL_REGULAR_TRIANGULATION_2_H

