// Copyright (c) 1999-2003  INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL$
// $Id$
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)     : Monique Teillaud <Monique.Teillaud@sophia.inria.fr>
//                 Sylvain Pion
//                 Clement Jamin

#ifndef CGAL_TRIANGULATION_3_H
#define CGAL_TRIANGULATION_3_H

#include <CGAL/license/Triangulation_3.h>

#include <CGAL/disable_warnings.h>
#include <CGAL/basic.h>

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
# define CGAL_PROFILE
# include <CGAL/Profile_counter.h>
#endif

#include <iostream>
#include <list>
#include <set>
#include <map>
#include <unordered_map>
#include <utility>
#include <stack>

#include <CGAL/Unique_hash_map.h>
#include <CGAL/triangulation_assertions.h>
#include <CGAL/Triangulation_utils_3.h>

#include <CGAL/Triangulation_data_structure_3.h>
#include <CGAL/Triangulation_cell_base_3.h>
#include <CGAL/Triangulation_vertex_base_3.h>

#include <CGAL/spatial_sort.h>
#include <CGAL/Spatial_sort_traits_adapter_3.h>

#include <CGAL/iterator.h>
#include <CGAL/function_objects.h>
#include <CGAL/Iterator_project.h>
#include <CGAL/Default.h>
#include <CGAL/internal/boost/function_property_map.hpp>

#include <CGAL/Bbox_3.h>
#include <CGAL/Spatial_lock_grid_3.h>

#include <boost/bind.hpp>
#include <boost/random/linear_congruential.hpp>
#include <boost/random/uniform_smallint.hpp>
#include <boost/random/variate_generator.hpp>
#include <boost/mpl/if.hpp>
#include <boost/unordered_map.hpp>
#include <boost/utility/result_of.hpp>
#include <boost/container/small_vector.hpp>

#ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
#include <CGAL/internal/info_check.h>
#include <boost/iterator/zip_iterator.hpp>
#endif

#ifndef CGAL_NO_STRUCTURAL_FILTERING
#include <CGAL/internal/Static_filters/tools.h>
#include <CGAL/Triangulation_structural_filtering_traits.h>
#include <CGAL/determinant.h>
#endif // no CGAL_NO_STRUCTURAL_FILTERING

#ifdef CGAL_LINKED_WITH_TBB
# include <tbb/scalable_allocator.h>
#endif

#define CGAL_TRIANGULATION_3_USE_THE_4_POINTS_CONSTRUCTOR

namespace CGAL {

template < class GT, class Tds = Default,
           class Lock_data_structure = Default >
class Triangulation_3;

template < class GT, class Tds, class Lds > std::istream& operator>>
(std::istream& is, Triangulation_3<GT,Tds,Lds>& tr);

#ifndef CGAL_NO_STRUCTURAL_FILTERING
namespace internal {

// structural filtering is performed only for EPIC
struct Structural_filtering_3_tag {};
struct No_structural_filtering_3_tag {};

template <bool filter>
struct Structural_filtering_selector_3
{
#ifdef FORCE_STRUCTURAL_FILTERING
  typedef Structural_filtering_3_tag     Tag;
#else
  typedef No_structural_filtering_3_tag  Tag;
#endif
};

template <>
struct Structural_filtering_selector_3<true>
{
  typedef Structural_filtering_3_tag     Tag;
};
} // namespace internal
#endif // no CGAL_NO_STRUCTURAL_FILTERING

/************************************************
// Class Triangulation_3_base
// Two versions: Sequential (no locking) / Parallel (with locking)
************************************************/

// Sequential (without locking)
template <typename Concurrency_tag, typename Lock_data_structure_>
class Triangulation_3_base
{
public:
  // If Lock_data_structure_ = Default => void
  typedef typename Default::Get<
  Lock_data_structure_, void>::type Lock_data_structure;

protected:
  Triangulation_3_base() {}
  Triangulation_3_base(Lock_data_structure *) {}

  void swap(Triangulation_3_base<Concurrency_tag, Lock_data_structure_>&) {}

  template <typename Vertex_triple, typename Facet>
  struct Vertex_triple_Facet_map_generator
  {
    typedef boost::unordered_map<Vertex_triple, Facet> type;
  };

  template <typename Vertex_handle>
  struct Vertex_handle_unique_hash_map_generator
  {
    typedef Unique_hash_map<Vertex_handle,
    Vertex_handle,
    Handle_hash_function> type;
  };

public:
  bool is_parallel() const
  {
    return false;
  }

  // LOCKS (no-op functions)
  template <typename Point_3>
  bool try_lock_point(const Point_3&, int = 0) const
  { return true; }

  template <typename Vertex_handle>
  bool try_lock_vertex(const Vertex_handle&, int = 0) const
  { return true; }

  template <typename Cell_handle>
  bool try_lock_cell(const Cell_handle&, int = 0) const
  { return true; }

  template <typename Facet>
  bool try_lock_facet(const Facet&, int = 0) const
  { return true; }

  template <typename P3>
  bool is_point_locked_by_this_thread(const P3&) const
  { return false; }

  template <typename Cell_handle>
  bool is_cell_locked_by_this_thread(const Cell_handle&) const
  { return false; }

  void *get_lock_data_structure() const
  {
    return 0;
  }

  void set_lock_data_structure(void *) const {}

  void unlock_all_elements() const {}
  template <typename P3> void unlock_all_elements_but_one_point(const P3&) const {}

  const Bbox_3 *get_bbox() const
  {
    return nullptr;
  }
};

#ifdef CGAL_LINKED_WITH_TBB
// Parallel (with locking)
template <typename Lock_data_structure_>
class Triangulation_3_base<Parallel_tag, Lock_data_structure_>
{
public:
  // If Lock_data_structure_ = Default => use Spatial_lock_grid_3
  typedef typename Default::Get<
    Lock_data_structure_,
    Spatial_lock_grid_3<Tag_priority_blocking> >::type Lock_data_structure;

protected:
  Triangulation_3_base()
    : m_lock_ds(0)
  {
  }

  Triangulation_3_base(Lock_data_structure *lock_ds)
    : m_lock_ds(lock_ds)
  {
  }

  void swap(Triangulation_3_base<Parallel_tag, Lock_data_structure_>& tr)
  {
    std::swap(tr.m_lock_ds, m_lock_ds);
  }

  template <typename Vertex_triple, typename Facet>
  struct Vertex_triple_Facet_map_generator
  {
    typedef boost::unordered_map
    <
    Vertex_triple,
    Facet,
    boost::hash<Vertex_triple>,
    std::equal_to<Vertex_triple>,
    tbb::scalable_allocator<std::pair<const Vertex_triple, Facet> >
    > type;
  };

  template <typename Vertex_handle>
  struct Vertex_handle_unique_hash_map_generator
  {
    typedef Unique_hash_map<Vertex_handle,
                            Vertex_handle,
                            Handle_hash_function,
                            tbb::scalable_allocator<Vertex_handle> > type;
  };

public:
  bool is_parallel() const
  {
    return m_lock_ds != 0;
  }

  // LOCKS
  template <typename Point_3>
  bool try_lock_point(const Point_3& p, int lock_radius = 0) const
  {
    bool locked = true;
    if(m_lock_ds)
    {
      locked = m_lock_ds->try_lock(p, lock_radius);
    }
    return locked;
  }

  template <typename Vertex_handle>
  bool try_lock_vertex(const Vertex_handle& vh, int lock_radius = 0) const
  {
    bool locked = true;
    if(m_lock_ds)
    {
      locked = m_lock_ds->try_lock(vh->point(), lock_radius);
    }
    return locked;
  }

  template <typename Cell_handle>
  bool try_lock_cell(const Cell_handle& cell_handle, int lock_radius = 0) const
  {
    bool success = true;
    // Lock the element area on the grid
    for(int iVertex = 0 ; success && iVertex < 4 ; ++iVertex)
    {
      success = try_lock_vertex(cell_handle->vertex(iVertex), lock_radius);
    }
    return success;
  }

  template <typename Facet>
  bool try_lock_facet(const Facet& facet, int lock_radius = 0) const
  {
    bool success = true;

    // Lock the element area on the grid
    for(int iVertex = (facet.second+1)&3 ;
         success && iVertex != facet.second ; iVertex = (iVertex+1)&3)
    {
      success = try_lock_vertex(facet.first->vertex(iVertex), lock_radius);
    }

    return success;
  }

  template <typename P3>
  bool is_point_locked_by_this_thread(const P3& p) const
  {
    bool locked = true;
    if(m_lock_ds)
    {
      locked = m_lock_ds->is_locked_by_this_thread(p);
    }
    return locked;
  }

  template <typename Cell_handle>
  bool is_cell_locked_by_this_thread(const Cell_handle& cell_handle) const
  {
    bool locked = true;
    if(m_lock_ds)
    {
      for(int iVertex = 0 ; locked && iVertex < 4 ; ++iVertex)
      {
        locked = m_lock_ds->is_locked_by_this_thread(
                   cell_handle->vertex(iVertex)->point());
      }
    }
    return locked;
  }

  Lock_data_structure *get_lock_data_structure() const
  {
    return m_lock_ds;
  }

  void set_lock_data_structure(Lock_data_structure *lock_ds) const
  {
    m_lock_ds = lock_ds;
  }

  void unlock_all_elements() const
  {
    if(m_lock_ds)
      m_lock_ds->unlock_all_points_locked_by_this_thread();
  }

  template <typename P3>
  void unlock_all_elements_but_one_point(const P3& point) const
  {
    if(m_lock_ds)
      m_lock_ds->unlock_all_tls_locked_locations_but_one_point(point);
  }

  const Bbox_3 *get_bbox() const
  {
    return &m_lock_ds->get_bbox();
  }

protected:
  mutable Lock_data_structure *m_lock_ds;
};
#endif // CGAL_LINKED_WITH_TBB

/************************************************
 *
 * Triangulation_3 class
 *
 ************************************************/

template < class GT, class Tds_, class Lock_data_structure_ >
class Triangulation_3
  : public Triangulation_3_base<
             // Get Concurrency_tag from TDS
             typename Default::Get<Tds_,
                                   Triangulation_data_structure_3<
                                     Triangulation_vertex_base_3<GT>,
                                     Triangulation_cell_base_3<GT> >
                                  >::type::Concurrency_tag,
             Lock_data_structure_>,
  public Triangulation_utils_3
{
  friend std::istream& operator>> <>
  (std::istream& is, Triangulation_3<GT,Tds_,Lock_data_structure_>& tr);

  typedef typename Default::Get<Tds_,
                     Triangulation_data_structure_3 <
                       Triangulation_vertex_base_3<GT>,
                       Triangulation_cell_base_3<GT> > >::type       Tds;

  typedef Triangulation_3<GT, Tds_, Lock_data_structure_>            Self;
  typedef Triangulation_3_base<typename Tds::Concurrency_tag,
                               Lock_data_structure_>                 Base;

public:
  typedef typename Base::Lock_data_structure   Lock_data_structure;
  typedef Tds                                  Triangulation_data_structure;
  typedef GT                                   Geom_traits;

  typedef typename GT::Segment_3               Segment;
  typedef typename GT::Triangle_3              Triangle;
  typedef typename GT::Tetrahedron_3           Tetrahedron;

  // point types
  typedef typename GT::Point_3                 Point_3;
  typedef typename Tds::Vertex::Point          Point;

  typedef typename Tds::Concurrency_tag        Concurrency_tag;

  typedef typename Tds::Vertex                 Vertex;
  typedef typename Tds::Cell                   Cell;
  typedef typename Tds::Facet                  Facet;
  typedef typename Tds::Edge                   Edge;

  typedef typename Tds::size_type              size_type;
  typedef typename Tds::difference_type        difference_type;

  typedef typename Tds::Vertex_handle          Vertex_handle;
  typedef typename Tds::Cell_handle            Cell_handle;

  typedef typename Tds::Cell_circulator        Cell_circulator;
  typedef typename Tds::Facet_circulator       Facet_circulator;

  // Not documented, see TDS.
  typedef typename Tds::Face_circulator        Face_circulator;

  typedef typename Tds::Cell_iterator          Cell_iterator;
  typedef typename Tds::Facet_iterator         Facet_iterator;
  typedef typename Tds::Edge_iterator          Edge_iterator;
  typedef typename Tds::Vertex_iterator        Vertex_iterator;

  typedef Cell_iterator                        All_cells_iterator;
  typedef Facet_iterator                       All_facets_iterator;
  typedef Edge_iterator                        All_edges_iterator;
  typedef Vertex_iterator                      All_vertices_iterator;

  typedef typename Tds::Cell_handles           All_cell_handles;
  typedef typename Tds::Vertex_handles         All_vertex_handles;
  typedef typename Tds::Facets                 All_facets;
  typedef typename Tds::Edges                  All_edges;
  
  typedef typename Tds::Simplex                Simplex;

private:
  // This class is used to generate the Finite_*_iterators.
  class Infinite_tester
  {
    const Self *t;

  public:
    Infinite_tester() {}

    Infinite_tester(const Self *tr)
      : t(tr) {}

    bool operator()(const Vertex_iterator& v) const
    {
      return t->is_infinite(v);
    }

    bool operator()(typename std::vector<Vertex_handle>::const_iterator v) const
    {
      return t->is_infinite(*v);
    }

    bool operator()(const Cell_iterator& c) const
    {
      return t->is_infinite(c);
    }

    bool operator()(const Edge_iterator& e) const
    {
      return t->is_infinite(*e);
    }

    bool operator()(const Facet_iterator& f) const
    {
      return t->is_infinite(*f);
    }
  };

public:
  // We derive in order to add a conversion to handle.
  class Finite_cells_iterator
    : public Filter_iterator<Cell_iterator, Infinite_tester>
  {
    typedef Filter_iterator<Cell_iterator, Infinite_tester> Base;
    typedef Finite_cells_iterator                           Self;

  public:
    Finite_cells_iterator() : Base() {}
    Finite_cells_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 Cell_handle() const { return Base::base(); }
  };

  // We derive in order to add a conversion to handle.
  class Finite_vertices_iterator
    : public Filter_iterator<Vertex_iterator, Infinite_tester>
  {
    typedef Filter_iterator<Vertex_iterator, Infinite_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_cells_iterator> >    Finite_cell_handles;
  typedef Iterator_range<Prevent_deref<Finite_vertices_iterator> > Finite_vertex_handles;

  typedef Filter_iterator<Edge_iterator, Infinite_tester>     Finite_edges_iterator;
  typedef Filter_iterator<Facet_iterator, Infinite_tester>    Finite_facets_iterator;

  typedef Iterator_range<Finite_edges_iterator> Finite_edges;
  typedef Iterator_range<Finite_facets_iterator> Finite_facets;
  
private:
  // Auxiliary iterators for convenience
  // do not use default template argument to please VC++
  typedef Project_point<Vertex>                               Proj_point;

public:
  typedef Iterator_project<Finite_vertices_iterator,
  Proj_point,
  const Point&,
  const Point*,
  std::ptrdiff_t,
  std::bidirectional_iterator_tag>   Point_iterator;


  typedef Iterator_range<Point_iterator> Points;
  
  // To have a back_inserter
  typedef Point                                               value_type;
  typedef const value_type&                                   const_reference;

  // Tag to distinguish triangulations with weighted_points
  typedef Tag_false                                           Weighted_tag;

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

  enum Locate_type
  {
    VERTEX=0,
    EDGE, //1
    FACET, //2
    CELL, //3
    OUTSIDE_CONVEX_HULL, //4
    OUTSIDE_AFFINE_HULL //5
  };

protected:
  Tds _tds;
  GT  _gt;
  Vertex_handle infinite; // infinite vertex

public:
  template<typename P> // Point or Point_3
  Point_3 construct_point(const P& p) const
  {
    return geom_traits().construct_point_3_object()(p);
  }

  template<typename P> // Point or Point_3
  Comparison_result compare_xyz(const P& p, const P& q) const
  {
    return geom_traits().compare_xyz_3_object()(construct_point(p),
                                                construct_point(q));
  }

  bool equal(const Point& p, const Point& q) const
  {
    return compare_xyz(p, q) == EQUAL;
  }

  template<typename P> // Point or Point_3
  Orientation orientation(const P& p, const P& q, const P& r, const P& s) const
  {
    return geom_traits().orientation_3_object()(construct_point(p),
                                                construct_point(q),
                                                construct_point(r),
                                                construct_point(s));
  }

  bool coplanar(const Point& p, const Point& q, const Point& r, const Point& s) const
  {
    return orientation(p, q, r, s) == COPLANAR;
  }

  template<typename P> // Point or Point_3
  Orientation coplanar_orientation(const P& p, const P& q, const P& r) const
  {
    return geom_traits().coplanar_orientation_3_object()(construct_point(p),
                                                         construct_point(q),
                                                         construct_point(r));
  }

  bool collinear(const Point& p, const Point& q, const Point& r) const
  {
    return coplanar_orientation(p, q, r) == COLLINEAR;
  }

  template<typename P> // Point or Point_3
  Segment construct_segment(const P& p, const P& q) const
  {
    return geom_traits().construct_segment_3_object()(construct_point(p),
                                                      construct_point(q));
  }

  template<typename P> // Point or Point_3
  Triangle construct_triangle(const P& p, const P& q, const P& r) const
  {
    return geom_traits().construct_triangle_3_object()(construct_point(p),
                                                       construct_point(q),
                                                       construct_point(r));
  }

  template<typename P> // Point or Point_3
  Tetrahedron construct_tetrahedron(const P& p, const P& q, const P& r, const P& s) const
  {
    return geom_traits().construct_tetrahedron_3_object()(construct_point(p),
                                                          construct_point(q),
                                                          construct_point(r),
                                                          construct_point(s));
  }

  enum COLLINEAR_POSITION
  {
    BEFORE,
    SOURCE,
    MIDDLE,
    TARGET,
    AFTER
  };

  COLLINEAR_POSITION collinear_position(const Point& s, const Point& p, const Point& t) const
  {
    // (s,t) defines a line, p is on that line.
    // Depending on the position of p wrt s and t, returns :
    // --------------- s ---------------- t --------------
    // BEFORE       SOURCE    MIDDLE    TARGET       AFTER

    CGAL_triangulation_precondition(!equal(s, t));
    CGAL_triangulation_precondition(collinear(s, p, t));

    Comparison_result ps = compare_xyz(p, s);
    if(ps == EQUAL)
      return SOURCE;
    Comparison_result st = compare_xyz(s, t);
    if(ps == st)
      return BEFORE;
    Comparison_result pt = compare_xyz(p, t);
    if(pt == EQUAL)
      return TARGET;
    if(pt == st)
      return MIDDLE;
    return AFTER;
  }

  // used as functor in std::sort in Delaunay and regular triangulations
  struct Perturbation_order
  {
    bool operator()(const Point* p, const Point* q) const {
      return t->compare_xyz(*p, *q) == SMALLER;
    }

    Perturbation_order(const Self *tr)
      : t(tr) {}

    const Self *t;
  };

  void init_tds()
  {
    infinite = _tds.insert_increase_dimension();
  }

  void init_tds(const Point& p0, const Point& p1, const Point& p2, const Point& p3)
  {
    Vertex_handle v0, v1, v2, v3;
    infinite = _tds.insert_first_finite_cell(v0, v1, v2, v3, infinite);
    v0->set_point(p0);
    v1->set_point(p1);
    v2->set_point(p2);
    v3->set_point(p3);
  }

  void init_tds(const Point& p0, const Point& p1,
                const Point& p2, const Point& p3,
                Vertex_handle& vh0, Vertex_handle& vh1,
                Vertex_handle& vh2, Vertex_handle& vh3)
  {
    infinite = _tds.insert_first_finite_cell(vh0, vh1, vh2, vh3, infinite);
    vh0->set_point(p0);
    vh1->set_point(p1);
    vh2->set_point(p2);
    vh3->set_point(p3);
  }

public:
  // CONSTRUCTORS
  Triangulation_3(const GT& gt = GT(), Lock_data_structure *lock_ds = nullptr)
    : Base(lock_ds), _tds(), _gt(gt)
  {
    init_tds();
  }

  Triangulation_3(Lock_data_structure *lock_ds, const GT& gt = GT())
    : Base(lock_ds), _tds(), _gt(gt)
  {
    init_tds();
  }

  // copy constructor duplicates vertices and cells
  Triangulation_3(const Triangulation_3& tr)
    : Base(tr.get_lock_data_structure()), _gt(tr._gt)
  {
    infinite = _tds.copy_tds(tr._tds, tr.infinite);
    CGAL_triangulation_expensive_postcondition(*this == tr);
  }

  template < typename InputIterator >
  Triangulation_3(InputIterator first, InputIterator last,
                  const GT& gt = GT(), Lock_data_structure *lock_ds = nullptr)
    : Base(lock_ds), _gt(gt)
  {
    init_tds();
    insert(first, last);
  }

  // Create the 3D triangulation of p0, p1, p3 and p4
  // Precondition: p0, p1, p3 and p4 MUST BE positively oriented
  Triangulation_3(const Point& p0, const Point& p1,
                  const Point& p3, const Point& p4,
                  const GT& gt = GT(), Lock_data_structure *lock_ds = nullptr)
    : Base(lock_ds), _gt(gt)
  {
    CGAL_triangulation_precondition(orientation(p0, p1, p3, p4) == POSITIVE);
    init_tds(p0, p1, p3, p4);
  }

  void clear()
  {
    _tds.clear();
    init_tds();
  }

  Triangulation_3& operator=(Triangulation_3 tr)
  {
    // Because the parameter tr is passed by value, the triangulation passed
    // as argument has been copied.
    // The following 'swap' consumes the *copy* and the original triangulation
    // is left untouched.
    swap(tr);
    return *this;
  }

  // HELPING FUNCTIONS
  void swap(Triangulation_3& tr)
  {
    std::swap(tr._gt, _gt);
    std::swap(tr.infinite, infinite);
    _tds.swap(tr._tds);
    Base::swap(tr);
  }

  //ACCESS FUNCTIONS
  const GT& geom_traits() const { return _gt; }
  const Tds& tds() const { return _tds; }
  Tds& tds() { return _tds; }
  int dimension() const { return _tds.dimension(); }

  size_type number_of_finite_cells() const;
  size_type number_of_cells() const;
  size_type number_of_finite_facets() const;
  size_type number_of_facets() const;
  size_type number_of_finite_edges() const;
  size_type number_of_edges() const;
  size_type number_of_vertices() const // number of finite vertices
  { return _tds.number_of_vertices()-1; }

  Vertex_handle infinite_vertex() const { return infinite; }
  void set_infinite_vertex(Vertex_handle v) { infinite = v; }

  Cell_handle infinite_cell() const
  {
    CGAL_triangulation_assertion(infinite_vertex()->cell()->has_vertex(infinite_vertex()));
    return infinite_vertex()->cell();
  }

  // GEOMETRIC ACCESS FUNCTIONS
  Tetrahedron tetrahedron(const Cell_handle c) const
  {
    CGAL_triangulation_precondition(dimension() == 3);
    CGAL_triangulation_precondition(! is_infinite(c));
    return construct_tetrahedron(c->vertex(0)->point(),
                                 c->vertex(1)->point(),
                                 c->vertex(2)->point(),
                                 c->vertex(3)->point());
  }

  template<typename P> // can be 'Point' or 'Point_3'
  Tetrahedron tetrahedron(const Facet& f, const P& p) const
  {
    const Cell_handle c = f.first;
    const int index = f.second;

    const P& fp1 = point(c, (index + 1)&3);
    const P& fp2 = point(c, (index + 2)&3);
    const P& fp3 = point(c, (index + 3)&3);

    return construct_tetrahedron(construct_point(p),
                                 construct_point(fp1),
                                 construct_point(fp2),
                                 construct_point(fp3));
  }

  Triangle triangle(const Cell_handle c, int i) const;
  Triangle triangle(const Facet& f) const { return triangle(f.first, f.second); }

  Segment segment(const Cell_handle c, int i, int j) const;
  Segment segment(const Edge& e) const {
    return segment(e.first, e.second, e.third);
  }

  void set_point(Cell_handle c, int i, const Point& p)
  {
    CGAL_triangulation_precondition(dimension() >= 0);
    CGAL_triangulation_precondition(i >= 0 && i <= dimension());
    CGAL_triangulation_precondition(! is_infinite(c->vertex(i)));
    c->vertex(i)->point() = p;
  }

  const Point& point(Cell_handle c, int i) const
  {
    CGAL_triangulation_precondition(dimension() >= 0);
    CGAL_triangulation_precondition(i >= 0 && i <= dimension());
    CGAL_triangulation_precondition(! is_infinite(c->vertex(i)));
    return c->vertex(i)->point();
  }

  void set_point(Vertex_handle v, const Point& p)
  {
    CGAL_triangulation_precondition(dimension() >= 0);
    CGAL_triangulation_precondition(! is_infinite(v));
    v->point() = p;
  }

  const Point& point(Vertex_handle v) const
  {
    CGAL_triangulation_precondition(number_of_vertices() > 0);
    CGAL_triangulation_precondition(! is_infinite(v));
    return v->point();
  }

  // TEST IF INFINITE FEATURES
  bool is_infinite(const Vertex_handle v) const { return v == infinite_vertex(); }
  bool is_infinite(const Cell_handle c) const
  {
    CGAL_triangulation_precondition(dimension() == 3);
    return c->has_vertex(infinite_vertex());
  }

  bool is_infinite(const Cell_handle c, int i) const;
  bool is_infinite(const Facet& f) const { return is_infinite(f.first,f.second); }
  bool is_infinite(const Cell_handle c, int i, int j) const;
  bool is_infinite(const Edge& e) const
  {
    return is_infinite(e.first, e.second, e.third);
  }

  // QUERIES
  bool is_vertex(const Point& p, Vertex_handle& v) const;
  bool is_vertex(Vertex_handle v) const;
  bool is_edge(Vertex_handle u, Vertex_handle v,
               Cell_handle& c, int& i, int& j) const;
  bool is_facet(Vertex_handle u, Vertex_handle v, Vertex_handle w,
                Cell_handle& c, int& i, int& j, int& k) const;
  bool is_cell(Cell_handle c) const;
  bool is_cell(Vertex_handle u, Vertex_handle v,
               Vertex_handle w, Vertex_handle t,
               Cell_handle& c, int& i, int& j, int& k, int& l) const;
  bool is_cell(Vertex_handle u, Vertex_handle v,
               Vertex_handle w, Vertex_handle t,
               Cell_handle& c) const;

  bool has_vertex(const Facet& f, Vertex_handle v, int& j) const;
  bool has_vertex(Cell_handle c, int i, Vertex_handle v, int& j) const;
  bool has_vertex(const Facet& f, Vertex_handle v) const;
  bool has_vertex(Cell_handle c, int i, Vertex_handle v) const;

  bool are_equal(Cell_handle c, int i, Cell_handle n, int j) const;
  bool are_equal(const Facet& f, const Facet& g) const;
  bool are_equal(const Facet& f, Cell_handle n, int j) const;

#ifdef CGAL_NO_STRUCTURAL_FILTERING
  Cell_handle
  locate(const Point& p,
         Locate_type& lt, int& li, int& lj,
         Cell_handle start = Cell_handle(),
         bool *could_lock_zone = nullptr) const;
#else // no CGAL_NO_STRUCTURAL_FILTERING
#  ifndef CGAL_T3_STRUCTURAL_FILTERING_MAX_VISITED_CELLS
#    define CGAL_T3_STRUCTURAL_FILTERING_MAX_VISITED_CELLS 2500
#  endif // no CGAL_T3_STRUCTURAL_FILTERING_MAX_VISITED_CELLS

public:
  Cell_handle inexact_locate(const Point& p,
                             Cell_handle start = Cell_handle(),
                             int max_num_cells = CGAL_T3_STRUCTURAL_FILTERING_MAX_VISITED_CELLS,
                             bool *could_lock_zone = nullptr) const;
protected:
  Cell_handle exact_locate(const Point& p,
                           Locate_type& lt,
                           int& li, int& lj,
                           Cell_handle start,
                           bool *could_lock_zone = nullptr) const;

  Cell_handle generic_locate(const Point& p,
                             Locate_type& lt,
                             int& li, int& lj,
                             Cell_handle start,
                             internal::Structural_filtering_3_tag,
                             bool *could_lock_zone = nullptr) const
  {
    Cell_handle ch = inexact_locate(
                       p, start, CGAL_T3_STRUCTURAL_FILTERING_MAX_VISITED_CELLS, could_lock_zone);
    if(could_lock_zone && *could_lock_zone == false)
      return ch; // = Cell_handle() here
    else
      return exact_locate(p, lt, li, lj, ch, could_lock_zone);
  }

  Cell_handle generic_locate(const Point& p,
                             Locate_type& lt,
                             int& li, int& lj,
                             Cell_handle start,
                             internal::No_structural_filtering_3_tag,
                             bool *could_lock_zone = nullptr) const
  {
    return exact_locate(p, lt, li, lj, start, could_lock_zone);
  }

public:
  Orientation
  inexact_orientation(const Point& p, const Point& q,
                      const Point& r, const Point& s) const
  {
    // So that this code works well with Lazy_kernel
    internal::Static_filters_predicates::Get_approx<Point> get_approx;
    const double px = to_double(get_approx(p).x());
    const double py = to_double(get_approx(p).y());
    const double pz = to_double(get_approx(p).z());
    const double qx = to_double(get_approx(q).x());
    const double qy = to_double(get_approx(q).y());
    const double qz = to_double(get_approx(q).z());
    const double rx = to_double(get_approx(r).x());
    const double ry = to_double(get_approx(r).y());
    const double rz = to_double(get_approx(r).z());
    const double sx = to_double(get_approx(s).x());
    const double sy = to_double(get_approx(s).y());
    const double sz = to_double(get_approx(s).z());

    const double pqx = qx - px;
    const double pqy = qy - py;
    const double pqz = qz - pz;
    const double prx = rx - px;
    const double pry = ry - py;
    const double prz = rz - pz;
    const double psx = sx - px;
    const double psy = sy - py;
    const double psz = sz - pz;

    const double det = determinant(pqx, pqy, pqz,
                                   prx, pry, prz,
                                   psx, psy, psz);
    if(det > 0)
      return POSITIVE;
    if(det < 0)
      return NEGATIVE;
    return ZERO;
  }

public:
  Cell_handle locate(const Point& p,
                     Locate_type& lt, int& li, int& lj,
                     Cell_handle start = Cell_handle(),
                     bool *could_lock_zone = nullptr) const
  {
    typedef Triangulation_structural_filtering_traits<Geom_traits> TSFT;
    typedef typename internal::Structural_filtering_selector_3<
        TSFT::Use_structural_filtering_tag::value >::Tag           Should_filter_tag;

    return generic_locate(p, lt, li, lj, start, Should_filter_tag(), could_lock_zone);
  }
#endif // no CGAL_NO_STRUCTURAL_FILTERING

  Cell_handle locate(const Point& p,
                     Cell_handle start = Cell_handle(),
                     bool *could_lock_zone = nullptr) const
  {
    Locate_type lt;
    int li, lj;
    return locate(p, lt, li, lj, start, could_lock_zone);
  }

  Cell_handle locate(const Point& p,
                     Locate_type& lt, int& li, int& lj, Vertex_handle hint,
                     bool *could_lock_zone = nullptr) const
  {
    return locate(p, lt, li, lj,
                  hint == Vertex_handle() ? infinite_cell() : hint->cell(),
                  could_lock_zone);
  }

  Cell_handle locate(const Point& p, Vertex_handle hint, bool *could_lock_zone = nullptr) const
  {
    return locate(p, hint == Vertex_handle() ? infinite_cell() : hint->cell(),
                  could_lock_zone);
  }

  // PREDICATES ON POINTS ``TEMPLATED'' by the geom traits
  Bounded_side side_of_tetrahedron(const Point& p,
                                   const Point& p0, const Point& p1,
                                   const Point& p2, const Point& p3,
                                   Locate_type& lt, int& i, int& j) const;
  Bounded_side side_of_cell(const Point& p,
                            Cell_handle c,
                            Locate_type& lt, int& i, int& j) const;
  Bounded_side side_of_triangle(const Point& p,
                                const Point& p0, const Point& p1, const Point& p2,
                                Locate_type& lt, int& i, int& j) const;
  Bounded_side side_of_facet(const Point& p,
                             Cell_handle c,
                             Locate_type& lt, int& li, int& lj) const;
  Bounded_side side_of_facet(const Point& p,
                             const Facet& f,
                             Locate_type& lt, int& li, int& lj) const
  {
    CGAL_triangulation_precondition(f.second == 3);
    return side_of_facet(p, f.first, lt, li, lj);
  }
  Bounded_side side_of_segment(const Point& p,
                               const Point& p0, const Point& p1,
                               Locate_type& lt, int& i) const;
  Bounded_side side_of_edge(const Point& p,
                            Cell_handle c,
                            Locate_type& lt, int& li) const;
  Bounded_side side_of_edge(const Point& p,
                            const Edge& e,
                            Locate_type& lt, int& li) const
  {
    CGAL_triangulation_precondition(e.second == 0);
    CGAL_triangulation_precondition(e.third == 1);
    return side_of_edge(p, e.first, lt, li);
  }

  // Functions forwarded from TDS.
  int mirror_index(Cell_handle c, int i) const { return _tds.mirror_index(c, i); }
  Vertex_handle mirror_vertex(Cell_handle c, int i) const { return _tds.mirror_vertex(c, i); }
  Facet mirror_facet(Facet f) const { return _tds.mirror_facet(f);}

  // MODIFIERS

  bool flip(const Facet& f)
  {
    // Returns 'false' if the facet is not flippable
    // true other wise and
    // flips facet i of cell c
    // c will be replaced by one of the new cells
    return flip(f.first, f.second);
  }
  bool flip(Cell_handle c, int i);
  void flip_flippable(const Facet& f) { flip_flippable(f.first, f.second); }
  void flip_flippable(Cell_handle c, int i);
  bool flip(const Edge& e)
  {
    // returns false if the edge is not flippable
    // true otherwise and
    // flips edge i,j of cell c
    // c will be deleted
    return flip(e.first, e.second, e.third);
  }
  bool flip(Cell_handle c, int i, int j);
  void flip_flippable(const Edge& e) { flip_flippable(e.first, e.second, e.third); }
  void flip_flippable(Cell_handle c, int i, int j);

  //INSERTION
  Vertex_handle insert(const Point& p, Vertex_handle hint) {
    return insert(p, hint == Vertex_handle() ? infinite_cell() : hint->cell());
  }
  Vertex_handle insert(const Point& p, Cell_handle start = Cell_handle());
  Vertex_handle insert(const Point& p, Locate_type lt, Cell_handle c,
                       int li, int lj);

  //protected: // internal methods
  template <class OutputItCells>
  Vertex_handle insert_and_give_new_cells(const Point& p,
                                          OutputItCells fit,
                                          Cell_handle start = Cell_handle());

  template <class OutputItCells>
  Vertex_handle insert_and_give_new_cells(const Point& p,
                                          OutputItCells fit,
                                          Vertex_handle hint);

  template <class OutputItCells>
  Vertex_handle insert_and_give_new_cells(const Point& p,
                                          Locate_type lt,
                                          Cell_handle c, int li, int lj,
                                          OutputItCells fit);

  template < class Conflict_tester, class Hidden_points_visitor >
  inline Vertex_handle insert_in_conflict(const Point& p,
                                          Locate_type lt,
                                          Cell_handle c, int li, int lj,
                                          const Conflict_tester& tester,
                                          Hidden_points_visitor& hider,
                                          bool *could_lock_zone = nullptr);

  
#ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
  template < class InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last,
                        typename boost::enable_if<
                          boost::is_convertible<
                              typename std::iterator_traits<InputIterator>::value_type,
                              Point
                          >
                        >::type* = NULL)
#else
  template < class InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last)
#endif //CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
  {
    size_type n = number_of_vertices();

    Vertex_handle hint;
    for(; first != last; ++first)
      hint = insert(*first, hint);

    return number_of_vertices() - n;
  }

#ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
protected:
  //top stands for tuple-or-pair
  template <class Info>
  const Point& top_get_first(const std::pair<Point,Info>& pair) const { return pair.first; }

  template <class Info>
  const Info& top_get_second(const std::pair<Point,Info>& pair) const { return pair.second; }

  template <class Info>
  const Point& top_get_first(const boost::tuple<Point,Info>& tuple) const { return boost::get<0>(tuple); }

  template <class Info>
  const Info& top_get_second(const boost::tuple<Point,Info>& tuple) const { return boost::get<1>(tuple); }

  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<Point> points;
    std::vector<typename Triangulation_data_structure::Vertex::Info> infos;
    for(InputIterator it=first;it!=last;++it)
    {
      Tuple_or_pair value=*it;
      points.push_back(top_get_first(value));
      infos.push_back(top_get_second(value));
    }

    Vertex_handle hint;
    for(std::size_t i=0; i < points.size(); ++i)
      {
        hint = insert(points[i], hint);
        if(hint != Vertex_handle())
          hint->info() = infos[i];
      }

    return number_of_vertices() - n;
  }

public:
  template < class InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last,
                        typename boost::enable_if<
                          boost::is_convertible<
                            typename std::iterator_traits<InputIterator>::value_type,
                            std::pair<Point, typename internal::Info_check<
                                               typename Triangulation_data_structure::Vertex>::type>
                          > >::type* =NULL)
  {
    return insert_with_info< std::pair<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,
          typename boost::enable_if<
            boost::mpl::and_<
              boost::is_convertible< typename std::iterator_traits<InputIterator_1>::value_type, Point >,
              boost::is_convertible< typename std::iterator_traits<InputIterator_2>::value_type, typename internal::Info_check<typename Triangulation_data_structure::Vertex>::type >
            >
          >::type* =NULL)
  {
    return insert_with_info< boost::tuple<Point, typename internal::Info_check<
        typename Triangulation_data_structure::Vertex>::type> >(first,last);
  }
#endif //CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO

  
  
  Vertex_handle insert_in_cell(const Point& p, Cell_handle c);
  Vertex_handle insert_in_facet(const Point& p, Cell_handle c, int i);
  Vertex_handle insert_in_facet(const Point& p, const Facet& f) {
    return insert_in_facet(p, f.first, f.second);
  }
  Vertex_handle insert_in_edge(const Point& p, Cell_handle c, int i, int j);
  Vertex_handle insert_in_edge(const Point& p, const Edge& e) {
    return insert_in_edge(p, e.first, e.second, e.third);
  }

  Vertex_handle insert_outside_convex_hull(const Point& p, Cell_handle c);
  Vertex_handle insert_outside_affine_hull(const Point& p);

  template <class CellIt>
  Vertex_handle insert_in_hole(const Point& p,
                               CellIt cell_begin, CellIt cell_end,
                               Cell_handle begin, int i)
  {
    // Some geometric preconditions should be tested...
    Vertex_handle v = _tds.insert_in_hole(cell_begin, cell_end, begin, i);
    v->set_point(p);
    return v;
  }

  template <class CellIt>
  Vertex_handle insert_in_hole(const Point& p,
                               CellIt cell_begin, CellIt cell_end,
                               Cell_handle begin, int i, Vertex_handle newv)
  {
    // Some geometric preconditions should be tested...
    newv->set_point(p);
    return _tds.insert_in_hole(cell_begin, cell_end, begin, i, newv);
  }

  // Internal function, cells should already be marked.
  template <class CellIt>
  Vertex_handle _insert_in_hole(const Point& p,
                                CellIt cell_begin, CellIt cell_end,
                                Cell_handle begin, int i)
  {
    // Some geometric preconditions should be tested...
    Vertex_handle v = _tds._insert_in_hole(cell_begin, cell_end, begin, i);
    v->set_point(p);
    return v;
  }

 // Internal function, cells should already be marked.
  template <class Cells, class Facets>
  Vertex_handle _insert_in_small_hole(const Point& p,
                                      const Cells& cells,
                                      const Facets& facets )
  {
    Vertex_handle v = _tds._insert_in_small_hole(cells, facets);
    v->set_point(p);
    return v;
  }

  // Internal function, cells should already be marked.
  template <class CellIt>
  Vertex_handle _insert_in_hole(const Point& p,
                                CellIt cell_begin, CellIt cell_end,
                                Cell_handle begin, int i, Vertex_handle newv)
  {
    // Some geometric preconditions should be tested...
    newv->set_point(p);
    return _tds._insert_in_hole(cell_begin, cell_end, begin, i, newv);
  }

protected:
  template < class InputIterator >
  bool does_repeat_in_range(InputIterator first, InputIterator beyond) const;

  template < class InputIterator >
  bool infinite_vertex_in_range(InputIterator first, InputIterator beyond) const;

  // - c is the current cell, which must be in conflict.
  // - tester is the function object that tests if a cell is in conflict.
  template <class Conflict_test,
            class OutputIteratorBoundaryFacets,
            class OutputIteratorCells,
            class OutputIteratorInternalFacets>
  Triple<OutputIteratorBoundaryFacets,
         OutputIteratorCells,
         OutputIteratorInternalFacets>
  find_conflicts(Cell_handle d,
                 const Conflict_test& tester,
                 Triple<OutputIteratorBoundaryFacets,
                 OutputIteratorCells,
                 OutputIteratorInternalFacets> it,
                 bool *could_lock_zone = nullptr,
                 const Facet *this_facet_must_be_in_the_cz = nullptr,
                 bool *the_facet_is_in_its_cz = nullptr) const
  {
    CGAL_triangulation_precondition(dimension()>=2);

    if(the_facet_is_in_its_cz)
      *the_facet_is_in_its_cz = false;

    if(could_lock_zone)
    {
      *could_lock_zone = true;
      if(!this->try_lock_cell(d))
      {
        *could_lock_zone = false;
        return it;
      }
    }

    CGAL_triangulation_precondition(tester(d));

    // To store the boundary cells, in case we need to rollback
    typedef boost::container::small_vector<Cell_handle,64> SV;
    SV sv;
    std::stack<Cell_handle, SV > cell_stack(sv);

    cell_stack.push(d);
    d->tds_data().mark_in_conflict();

    *it.second++ = d;

    do
    {
      Cell_handle c = cell_stack.top();
      cell_stack.pop();

      // For each neighbor cell
      for(int i=0; i<dimension()+1; ++i)
      {
        Cell_handle test = c->neighbor(i);

        // "test" is either in the conflict zone,
        // either facet-adjacent to the CZ

        if(test->tds_data().is_in_conflict())
        {
          Facet f(c, i); // Internal facet.
          // Is it the facet where're looking for?
          if(this_facet_must_be_in_the_cz && the_facet_is_in_its_cz &&
              f == *this_facet_must_be_in_the_cz)
          {
            *the_facet_is_in_its_cz = true;
          }
          if(c < test)
          {
            *it.third++ = f;
          }
          continue; // test was already in conflict.
        }
        if(test->tds_data().is_clear())
        {
          if(tester(test))
          {
            // "test" is in the conflict zone
            if(could_lock_zone)
            {
              if(!this->try_lock_cell(test))
              {
                *could_lock_zone = false;
                // Unlock
                return it;
              }
            }

            Facet f(c, i); // Internal facet.
            // Is it the facet where're looking for?
            if(this_facet_must_be_in_the_cz && the_facet_is_in_its_cz &&
                f == *this_facet_must_be_in_the_cz)
            {
              *the_facet_is_in_its_cz = true;
            }

            if(c < test)
            {
              *it.third++ = f;
            }

            cell_stack.push(test);
            test->tds_data().mark_in_conflict();
            *it.second++ = test;
            continue;
          }

          test->tds_data().mark_on_boundary();
        }

        Facet f(c, i); // Boundary facet.
        // Is it the facet where're looking for?
        if(this_facet_must_be_in_the_cz && the_facet_is_in_its_cz &&
            (mirror_facet(f) == *this_facet_must_be_in_the_cz ||
             f == *this_facet_must_be_in_the_cz))
        {
          *the_facet_is_in_its_cz = true;
        }

        *it.first++ = f;
      }
    }
    while(!cell_stack.empty());

    return it;
  }

  // This one takes a function object to recursively determine the cells in
  // conflict, then calls _tds._insert_in_hole().
  template < class Conflict_test >
  Vertex_handle insert_conflict(Cell_handle c, const Conflict_test& tester)
  {
    CGAL_triangulation_precondition(dimension() >= 2);
    CGAL_triangulation_precondition(c != Cell_handle());
    CGAL_triangulation_precondition(tester(c));

    std::vector<Cell_handle> cells;
    cells.reserve(32);

    Facet facet;

    // Find the cells in conflict
    switch(dimension())
    {
      case 3:
        find_conflicts(c, tester, make_triple(Oneset_iterator<Facet>(facet),
                                              std::back_inserter(cells),
                                              Emptyset_iterator()));
        break;
      case 2:
        find_conflicts(c, tester, make_triple(Oneset_iterator<Facet>(facet),
                                              std::back_inserter(cells),
                                              Emptyset_iterator()));
    }
    // Create the new cells and delete the old.
    return _tds._insert_in_hole(cells.begin(), cells.end(),
                                facet.first, facet.second);
  }

private:
  // Here are the conflit tester function objects passed to
  // insert_conflict_[23]() by insert_outside_convex_hull().
  class Conflict_tester_outside_convex_hull_3
  {
    const Point& p;
    const Self *t;

  public:
    Conflict_tester_outside_convex_hull_3(const Point& pt, const Self *tr)
      : p(pt), t(tr)
    {}

    bool operator()(const Cell_handle c) const
    {
      Locate_type loc;
      int i, j;
      return t->side_of_cell(p, c, loc, i, j) == ON_BOUNDED_SIDE;
    }
  };

  class Conflict_tester_outside_convex_hull_2
  {
    const Point& p;
    const Self *t;

  public:
    Conflict_tester_outside_convex_hull_2(const Point& pt, const Self *tr)
      : p(pt), t(tr)
    {}

    bool operator()(const Cell_handle c) const
    {
      Locate_type loc;
      int i, j;
      return t->side_of_facet(p, c, loc, i, j) == ON_BOUNDED_SIDE;
    }
  };

protected:
  // no point being private, we might need to test whether a displacement
  // decreases the dimension on others inherited triangulations
  bool test_dim_down(Vertex_handle v) const;
  bool test_dim_down_using_incident_cells_3(Vertex_handle v,
                                            std::vector<Cell_handle>& incident_cells,
                                            std::vector<Vertex_handle>& adj_vertices,
                                            bool *could_lock_zone = nullptr) const;

  // REMOVAL
  template < class VertexRemover >
  void remove(Vertex_handle v, VertexRemover& remover);

  // Concurrency-safe version
  // Pre-condition: dimension = 3
  // The return value is only meaningful if *could_lock_zone = true:
  // * returns true if the vertex was removed
  // * returns false if the vertex wasn't removed since it would decrease
  //   the dimension => needs to be done sequentially
  template < class VertexRemover >
  bool remove(Vertex_handle v, VertexRemover& remover, bool *could_lock_zone);

  template < class VertexRemover, class OutputItCells >
  void remove_and_give_new_cells(Vertex_handle v, VertexRemover& remover,
                                 OutputItCells fit);

  // This function removes a batch of points at once.
  // If points are grouped in cluster, the performance is increased
  // compared to removing one by one.
  // For now, this function is only guaranteed for Delaunay triangulations (or Regular as Delaunay).
  // By doing these kind of remove followed by inserting the cluster,
  // we achieve fast relocations for a batch of points (in a Delaunay triangulation).
  template < class InputIterator, class VertexRemover >
  size_type remove(InputIterator first, InputIterator beyond,
                   VertexRemover& remover);
  enum REMOVE_VERTEX_STATE {CLEAR, TO_REMOVE, PROCESSED, EXTREMITY};

  // MOVE
  template < class VertexRemover, class VertexInserter >
  Vertex_handle move_if_no_collision(Vertex_handle v, const Point& p,
                                     VertexRemover& remover,
                                     VertexInserter& inserter);

  template < class VertexRemover, class VertexInserter >
  Vertex_handle move(Vertex_handle v, const Point& p,
                     VertexRemover& remover, VertexInserter& inserter);

  // move and give new cells
  template < class VertexRemover, class VertexInserter, class OutputItCells  >
  Vertex_handle move_if_no_collision_and_give_new_cells(Vertex_handle v,
                                                        const Point& p,
                                                        VertexRemover& remover,
                                                        VertexInserter& inserter,
                                                        OutputItCells fit);

  // This is a function better suited for tds
  // but because it is not required in the model of tds
  // at this time, it should be implemented here.
  void flip_2D(Cell_handle f, int i)
  {
    CGAL_triangulation_precondition(dimension()==2);
    Cell_handle n  = f->neighbor(i);
    int ni = this->_tds.mirror_index(f,i); // ni = n->index(f);

    int cwi = (i+2)%3;
    int ccwi = (i+1)%3;
    int cwni = (ni+2)%3;
    int ccwni = (ni+1)%3;

    Vertex_handle v_cw = f->vertex(cwi);
    Vertex_handle v_ccw = f->vertex(ccwi);

    // bl == bottom left, tr == top right
    Cell_handle tr = f->neighbor(ccwi);
    int tri = this->_tds.mirror_index(f,ccwi);
    Cell_handle bl = n->neighbor(ccwni);
    int bli = this->_tds.mirror_index(n,ccwni);

    f->set_vertex(cwi, n->vertex(ni));
    n->set_vertex(cwni, f->vertex(i));

    // update the neighborhood relations
    this->_tds.set_adjacency(f, i, bl, bli);
    this->_tds.set_adjacency(f, ccwi, n, ccwni);
    this->_tds.set_adjacency(n, ni, tr, tri);

    if(v_cw->cell() == f)
    {
      v_cw->set_cell(n);
    }

    if(v_ccw->cell() == n)
    {
      v_ccw->set_cell(f);
    }
  }

  template < class VertexRemover, class VertexInserter >
  void restore_edges_after_decrease_dimension(Vertex_handle v,
                                              VertexRemover& remover,
                                              VertexInserter& inserter)
  {
    Cell_handle fkstart = v->cell();
    Cell_handle start = fkstart->neighbor(fkstart->index(v));

    std::list<Edge_2D> hole;
    make_hole_2D(v, hole, remover);
    fill_hole_2D(hole, remover);
    // make hole here will work if the link of v is a valid triangulation
    // the aim here is Delaunay triangulations
    // to make it more general one could have an internal function here
    // to remove v without touching its handle

    // This insert must be from Delaunay (or the particular trian.)
    // not the basic Triangulation_3.
    // Here we correct the recent triangulation (with decreased dimension) formed
    // in particular here a 2D (from 3D to 2D displacement)
    Vertex_handle inserted = inserter.insert(v->point(), start);

    // fixing pointer
    Cell_handle fc = inserted->cell(), done(fc);
    std::vector<Cell_handle> faces_pt;
    faces_pt.reserve(16);
    do
    {
      faces_pt.push_back(fc);
      fc = fc->neighbor((fc->index(inserted) + 1)%3);
    }
    while(fc != done);

    std::size_t ss = faces_pt.size();
    for(std::size_t k=0; k<ss; k++)
    {
      Cell_handle f = faces_pt[k];
      int i = f->index(inserted);
      f->set_vertex(i, v);
    }
    v->set_cell(inserted->cell());

    tds().delete_vertex(inserted);
  }

private:
  typedef Facet Edge_2D;
  typedef Triple<Vertex_handle,Vertex_handle,Vertex_handle> Vertex_triple;
  typedef typename Base::template Vertex_triple_Facet_map_generator<
  Vertex_triple, Facet>::type Vertex_triple_Facet_map;
  typedef typename Base::template Vertex_handle_unique_hash_map_generator<
  Vertex_handle>::type Vertex_handle_unique_hash_map;

  Vertex_triple make_vertex_triple(const Facet& f) const;
  void make_canonical(Vertex_triple& t) const;

  template < class VertexRemover >
  VertexRemover& make_hole_2D(Vertex_handle v, std::list<Edge_2D>& hole,
                              VertexRemover& remover);
  template < class VertexRemover >
  VertexRemover& make_hole_2D(Vertex_handle v, std::list<Edge_2D>& hole,
                              VertexRemover& remover,
                              std::set<Cell_handle>& cells_set);

  template < class VertexRemover >
  void fill_hole_2D(std::list<Edge_2D>& hole, VertexRemover& remover);

  void make_hole_3D(Vertex_handle v,
                    Vertex_triple_Facet_map& outer_map,
                    std::vector<Cell_handle>& hole);
  // When the incident cells are already known
  void make_hole_3D(Vertex_handle v,
                    const std::vector<Cell_handle>& incident_cells,
                    Vertex_triple_Facet_map& outer_map);

  template < class VertexRemover >
  VertexRemover& remove_dim_down(Vertex_handle v, VertexRemover& remover);
  template < class VertexRemover >
  VertexRemover& remove_1D(Vertex_handle v, VertexRemover& remover);
  template < class VertexRemover >
  VertexRemover& remove_2D(Vertex_handle v, VertexRemover& remover);
  template < class VertexRemover >
  VertexRemover& remove_3D(Vertex_handle v, VertexRemover& remover);
  // Version of remove_3D if the incident cells and the adjacent vertices
  // are already known
  template < class VertexRemover >
  VertexRemover& remove_3D(Vertex_handle v, VertexRemover& remover,
                           const std::vector<Cell_handle>& inc_cells,
                           std::vector<Vertex_handle>& adj_vertices);

  template < class VertexRemover, class OutputItCells  >
  VertexRemover& remove_dim_down(Vertex_handle v, VertexRemover& remover,
                                 OutputItCells fit);

  template < class VertexRemover, class OutputItCells  >
  VertexRemover& remove_1D(Vertex_handle v, VertexRemover& remover,
                           OutputItCells fit);

  template < class VertexRemover, class OutputItCells  >
  VertexRemover& remove_2D(Vertex_handle v, VertexRemover& remover,
                           OutputItCells fit);

  template < class VertexRemover, class OutputItCells  >
  VertexRemover& remove_3D(Vertex_handle v, VertexRemover& remover,
                           OutputItCells fit);

  template < class VertexRemover, class OutputItCells  >
  void fill_hole_2D(std::list<Edge_2D>& hole, VertexRemover& remover,
                    OutputItCells fit);

  // They access "Self", so need to be friend.
  friend class Conflict_tester_outside_convex_hull_3;
  friend class Conflict_tester_outside_convex_hull_2;
  friend class Infinite_tester;
  friend class Finite_vertices_iterator;
  friend class Finite_cells_iterator;

  // remove cluster
  template < class InputIterator >
  void _mark_vertices_to_remove(InputIterator first, InputIterator beyond,
                                std::map<Vertex_handle,
                                REMOVE_VERTEX_STATE>& vstates) const
  {
    while(first != beyond) vstates[*first++] = TO_REMOVE;
  }

  bool _test_dim_down_cluster(std::map<Vertex_handle,
                              REMOVE_VERTEX_STATE>& vstates) const
  {
  // tests whether removing the cluster of vertices
  // marked as "to remove", decreases the dimension of the triangulation
    CGAL_triangulation_precondition(dimension() == 3);
    int k=0;
    Vertex_handle v[4];
    for(Finite_vertices_iterator fit = finite_vertices_begin();
                                 fit != finite_vertices_end(); ++fit)
    {
      if(vstates[fit] == TO_REMOVE)
        continue;
      v[k++] = fit;
      if(k == 4)
      {
        if(!coplanar(v[0]->point(), v[1]->point(), v[2]->point(), v[3]->point()))
          return false;
        k--;
      }
    }
    return k < 4;
  }

  template < class InputIterator, class VertexRemover >
  bool _remove_cluster_3D(InputIterator first, InputIterator beyond,
                          VertexRemover& remover,
                          std::map<Vertex_handle, REMOVE_VERTEX_STATE>& vstates);

  void _make_big_hole_3D(Vertex_handle v,
                         std::map<Vertex_triple,Facet>& outer_map,
                         std::vector<Cell_handle>& hole,
                         std::vector<Vertex_handle>& vertices,
                         std::map<Vertex_handle, REMOVE_VERTEX_STATE>& vstates);

public:
  //TRAVERSING : ITERATORS AND CIRCULATORS
  Finite_cells_iterator finite_cells_begin() const
  {
    if(dimension() < 3)
      return finite_cells_end();

    return CGAL::filter_iterator(cells_end(), Infinite_tester(this), cells_begin());
  }
  Finite_cells_iterator finite_cells_end() const
  {
    return CGAL::filter_iterator(cells_end(), Infinite_tester(this));
  }

  
  Finite_cell_handles finite_cell_handles() const
  {
    return make_prevent_deref_range(finite_cells_begin(), finite_cells_end()); 
  }

  
  Cell_iterator cells_begin() const { return _tds.cells_begin(); }
  Cell_iterator cells_end() const { return _tds.cells_end(); }

  All_cells_iterator all_cells_begin() const { return _tds.cells_begin(); }
  All_cells_iterator all_cells_end() const { return _tds.cells_end(); }

  All_cell_handles all_cell_handles() const
  {
    return _tds.cell_handles();
  }
  
  Finite_vertices_iterator finite_vertices_begin() const
  {
    if(number_of_vertices() <= 0)
      return finite_vertices_end();

    return CGAL::filter_iterator(vertices_end(), Infinite_tester(this),
                                 vertices_begin());
  }
  Finite_vertices_iterator finite_vertices_end() const
  {
    return CGAL::filter_iterator(vertices_end(), Infinite_tester(this));
  }

  Finite_vertex_handles finite_vertex_handles() const
  {
    return make_prevent_deref_range(finite_vertices_begin(), finite_vertices_end()); 
  }
  
  Vertex_iterator vertices_begin() const { return _tds.vertices_begin(); }
  Vertex_iterator vertices_end() const { return _tds.vertices_end(); }

  All_vertices_iterator all_vertices_begin() const { return _tds.vertices_begin(); }
  All_vertices_iterator all_vertices_end() const { return _tds.vertices_end(); }

  All_vertex_handles all_vertex_handles() const
  {
    return _tds.vertex_handles();
  }
  
  Finite_edges_iterator finite_edges_begin() const
  {
    if(dimension() < 1)
      return finite_edges_end();

    return CGAL::filter_iterator(edges_end(), Infinite_tester(this), edges_begin());
  }
  Finite_edges_iterator finite_edges_end() const
  {
    return CGAL::filter_iterator(edges_end(), Infinite_tester(this));
  }

  Finite_edges finite_edges() const
  {
    return Finite_edges(finite_edges_begin(),finite_edges_end());
  }
  
  Edge_iterator edges_begin() const { return _tds.edges_begin(); }
  Edge_iterator edges_end() const { return _tds.edges_end(); }

  
  All_edges_iterator all_edges_begin() const { return _tds.edges_begin(); }
  All_edges_iterator all_edges_end() const { return _tds.edges_end(); }

  All_edges all_edges() const
  {
    return _tds.edges();
  }
  
  Finite_facets_iterator finite_facets_begin() const
  {
    if(dimension() < 2)
      return finite_facets_end();

    return CGAL::filter_iterator(facets_end(), Infinite_tester(this), facets_begin());
  }
  Finite_facets_iterator finite_facets_end() const
  {
    return CGAL::filter_iterator(facets_end(), Infinite_tester(this));
  }

  Finite_facets finite_facets() const
  {
    return Finite_facets(finite_facets_begin(),finite_facets_end());
  }

  Facet_iterator facets_begin() const { return _tds.facets_begin(); }
  Facet_iterator facets_end() const { return _tds.facets_end(); }

  
  All_facets_iterator all_facets_begin() const { return _tds.facets_begin(); }
  All_facets_iterator all_facets_end() const { return _tds.facets_end(); }

  All_facets all_facets() const
  {
    return _tds.facets();
  }
  
  Point_iterator points_begin() const
  {
    return Point_iterator(finite_vertices_begin());
  }
  Point_iterator points_end() const
  {
    return Point_iterator(finite_vertices_end());
  }

  Points points() const
  {
    return Points(points_begin(),points_end());
  }
  
  // cells around an edge
  Cell_circulator incident_cells(const Edge& e) const
  {
    return _tds.incident_cells(e);
  }
  Cell_circulator incident_cells(Cell_handle c, int i, int j) const
  {
    return _tds.incident_cells(c, i, j);
  }
  Cell_circulator incident_cells(const Edge& e, Cell_handle start) const
  {
    return _tds.incident_cells(e, start);
  }
  Cell_circulator incident_cells(Cell_handle c, int i, int j, Cell_handle start) const
  {
    return _tds.incident_cells(c, i, j, start);
  }

  // facets around an edge
  Facet_circulator incident_facets(const Edge& e) const
  {
    return _tds.incident_facets(e);
  }
  Facet_circulator incident_facets(Cell_handle c, int i, int j) const
  {
    return _tds.incident_facets(c, i, j);
  }
  Facet_circulator incident_facets(const Edge& e, const Facet& start) const
  {
    return _tds.incident_facets(e, start);
  }
  Facet_circulator incident_facets(Cell_handle c, int i, int j, const Facet& start) const
  {
    return _tds.incident_facets(c, i, j, start);
  }
  Facet_circulator incident_facets(const Edge& e, Cell_handle start, int f) const
  {
    return _tds.incident_facets(e, start, f);
  }
  Facet_circulator incident_facets(Cell_handle c, int i, int j, Cell_handle start, int f) const
  {
    return _tds.incident_facets(c, i, j, start, f);
  }

  // around a vertex
  class Finite_filter
  {
    const Self* t;

  public:
    Finite_filter(const Self* _t): t(_t) {}

    template<class T>
    bool operator() (const T& e) const { return t->is_infinite(e); }
  };

  class Finite_filter_2D
  {
    const Self* t;

  public:
    Finite_filter_2D(const Self* _t): t(_t) {}

    template<class T>
    bool operator() (const T& e) const { return t->is_infinite(e); }
    bool operator() (const Cell_handle c) { return t->is_infinite(c, 3); }
  };

  template <typename OutputIterator>
  OutputIterator incident_cells(Vertex_handle v, OutputIterator cells) const
  {
    return _tds.incident_cells(v, cells);
  }

  template <typename OutputIterator>
  void incident_cells_threadsafe(Vertex_handle v, OutputIterator cells) const
  {
    _tds.incident_cells_threadsafe(v, cells);
  }

  template <typename Filter, typename OutputIterator>
  void incident_cells_threadsafe(Vertex_handle v,
                                 OutputIterator cells,
                                 const Filter& filter) const
  {
    _tds.incident_cells_threadsafe(v, cells, filter);
  }

  bool try_lock_and_get_incident_cells(Vertex_handle v, std::vector<Cell_handle>& cells) const
  {
    // We need to lock v individually first, to be sure v->cell() is valid
    if(!this->try_lock_vertex(v))
      return false;

    Cell_handle d = v->cell();
    if(!this->try_lock_cell(d)) // LOCK
      return false;

    cells.push_back(d);
    d->tds_data().mark_in_conflict();
    int head=0;
    int tail=1;
    do
    {
      Cell_handle c = cells[head];

      for(int i=0; i<4; ++i)
      {
        if(c->vertex(i) == v)
          continue;

        Cell_handle next = c->neighbor(i);
        if(!this->try_lock_cell(next)) // LOCK
        {
          for(Cell_handle ch : cells)
          {
            ch->tds_data().clear();
          }
          cells.clear();
          return false;
        }

        if(! next->tds_data().is_clear())
          continue;

        cells.push_back(next);
        ++tail;
        next->tds_data().mark_in_conflict();
      }
      ++head;
    }
    while(head != tail);

    for(Cell_handle ch : cells)
    {
      ch->tds_data().clear();
    }
    return true;
  }

  template <class OutputIterator>
  bool try_lock_and_get_adjacent_vertices_and_cells_3(Vertex_handle v,
                                                      OutputIterator vertices,
                                                      std::vector<Cell_handle>& cells) const
  {
    // We need to lock v individually first, to be sure v->cell() is valid
    if(!this->try_lock_vertex(v))
      return false;

    Cell_handle d = v->cell();
    if(!this->try_lock_cell(d)) // LOCK
      return false;

    cells.push_back(d);
    d->tds_data().mark_in_conflict();
    int head=0;
    int tail=1;
    do
    {
      Cell_handle c = cells[head];

      for(int i=0; i<4; ++i)
      {
        if(c->vertex(i) == v)
          continue;

        Cell_handle next = c->neighbor(i);
        if(!this->try_lock_cell(next)) // LOCK
        {
          for(Cell_handle ch : cells)
          {
            ch->tds_data().clear();
          }
          cells.clear();
          return false;
        }
        if(! next->tds_data().is_clear())
          continue;

        cells.push_back(next);
        ++tail;
        next->tds_data().mark_in_conflict();
      }
      ++head;
    }
    while(head != tail);

    std::set<Vertex_handle> tmp_vertices;
    for(Cell_handle ch : cells)
    {
      ch->tds_data().clear();
      for(int i = 0;  i < 4; ++i)
      {
        Vertex_handle w = ch->vertex(i);
        if(w != v && tmp_vertices.insert(w).second)
        {
          *vertices = w;

        }
      }
    }
    return true;
  }

  template <class OutputIterator>
  OutputIterator finite_incident_cells(Vertex_handle v, OutputIterator cells) const
  {
    if(dimension() == 2)
      return _tds.incident_cells(v, cells, Finite_filter_2D(this));

    return _tds.incident_cells(v, cells, Finite_filter(this));
  }

  template <class OutputIterator>
  OutputIterator incident_facets(Vertex_handle v, OutputIterator facets) const
  {
    return _tds.incident_facets(v, facets);
  }

  template <class OutputIterator>
  OutputIterator finite_incident_facets(Vertex_handle v, OutputIterator facets) const
  {
    return _tds.incident_facets(v, facets, Finite_filter(this));
  }

  template <class OutputIterator>
  OutputIterator incident_facets_threadsafe(Vertex_handle v, OutputIterator facets) const
  {
    return _tds.incident_facets_threadsafe(v, facets);
  }

  template <class OutputIterator>
  OutputIterator finite_incident_facets_threadsafe(Vertex_handle v, OutputIterator facets) const
  {
    return _tds.incident_facets_threadsafe(v, facets, Finite_filter(this));
  }

  // old name (up to CGAL 3.4)
  // kept for backwards compatibility but not documented
  template <class OutputIterator>
  OutputIterator incident_vertices(Vertex_handle v, OutputIterator vertices) const
  {
    return _tds.adjacent_vertices(v, vertices);
  }

  // correct name
  template <class OutputIterator>
  OutputIterator adjacent_vertices(Vertex_handle v, OutputIterator vertices) const
  {
    return _tds.adjacent_vertices(v, vertices);
  }

  template <class OutputIterator>
  OutputIterator adjacent_vertices_and_cells_3(Vertex_handle v, OutputIterator vertices,
                                               std::vector<Cell_handle>& cells) const
  {
    return _tds.adjacent_vertices_and_cells_3(v, vertices, cells);
  }

  // old name (up to CGAL 3.4)
  // kept for backwards compatibility but not documented
  template <class OutputIterator>
  OutputIterator finite_incident_vertices(Vertex_handle v, OutputIterator vertices) const
  {
    return _tds.adjacent_vertices(v, vertices, Finite_filter(this));
  }

  // correct name
  template <class OutputIterator>
  OutputIterator finite_adjacent_vertices(Vertex_handle v, OutputIterator vertices) const
  {
    return _tds.adjacent_vertices(v, vertices, Finite_filter(this));
  }

  template <class OutputIterator>
  OutputIterator incident_edges(Vertex_handle v, OutputIterator edges) const
  {
    return _tds.incident_edges(v, edges);
  }

  template <class OutputIterator>
  OutputIterator finite_incident_edges(Vertex_handle v, OutputIterator edges) const
  {
    return _tds.incident_edges(v, edges, Finite_filter(this));
  }

  template <class OutputIterator>
  OutputIterator incident_edges_threadsafe(Vertex_handle v, OutputIterator edges) const
  {
    return _tds.incident_edges_threadsafe(v, edges);
  }

  template <class OutputIterator>
  OutputIterator finite_incident_edges_threadsafe(Vertex_handle v, OutputIterator edges) const
  {
    return _tds.incident_edges_threadsafe(v, edges, Finite_filter(this));
  }

  size_type degree(Vertex_handle v) const
  {
    return _tds.degree(v);
  }

  // CHECKING
  bool is_valid(bool verbose = false, int level = 0) const;
  bool is_valid(Cell_handle c, bool verbose = false, int level = 0) const;
  bool is_valid_finite(Cell_handle c, bool verbose = false, int level=0) const;
};

template < class GT, class Tds, class Lds >
std::istream& operator>> (std::istream& is, Triangulation_3<GT, Tds, Lds>& tr)
{
  // Reads:
  // - the dimension
  // - the number of finite vertices
  // - the non combinatorial information on vertices (point, etc)
  // - the number of cells
  // - the cells by the indices of their vertices in the preceding list
  //   of vertices, plus the non combinatorial information on each cell
  // - the neighbors of each cell by their index in the preceding list of cells
  // - when dimension < 3 : the same with faces of maximal dimension

  typedef Triangulation_3<GT, Tds>               Triangulation;
  typedef typename Triangulation::Vertex_handle  Vertex_handle;
  typedef typename Triangulation::Cell_handle    Cell_handle;

  tr._tds.clear(); // infinite vertex deleted
  tr.infinite = tr._tds.create_vertex();

  std::size_t n;
  int d;
  if(is_ascii(is))
  {
    is >> d >> n;
  }
  else
  {
    read(is, d);
    read(is, n);
  }
  if(!is)
    return is;

  tr._tds.set_dimension(d);

  std::vector< Vertex_handle > V(n+1);
  V[0] = tr.infinite_vertex(); // the infinite vertex is numbered 0

  for(std::size_t i=1; i <= n; i++)
  {
    V[i] = tr._tds.create_vertex();
    if(!(is >> *V[i]))
      return is;
  }

  std::vector< Cell_handle > C;

  std::size_t m;
  tr._tds.read_cells(is, V, m, C);

  for(std::size_t j=0 ; j < m; j++)
    if(!(is >> *(C[j])))
      return is;

  CGAL_triangulation_assertion(tr.is_valid(false));
  return is;
}

template < class GT, class Tds, class Lds >
std::ostream& operator<< (std::ostream& os, const Triangulation_3<GT, Tds, Lds>& tr)
{
  // Writes:
  // - the dimension
  // - the number of finite vertices
  // - the non combinatorial information on vertices (point, etc)
  // - the number of cells
  // - the cells by the indices of their vertices in the preceding list
  //   of vertices, plus the non combinatorial information on each cell
  // - the neighbors of each cell by their index in the preceding list of cells
  // - when dimension < 3 : the same with faces of maximal dimension

  typedef Triangulation_3<GT, Tds>                 Triangulation;
  typedef typename Triangulation::size_type        size_type;
  typedef typename Triangulation::Vertex_handle    Vertex_handle;
  typedef typename Triangulation::Vertex_iterator  Vertex_iterator;
  typedef typename Triangulation::Cell_iterator    Cell_iterator;
  typedef typename Triangulation::Edge_iterator    Edge_iterator;
  typedef typename Triangulation::Facet_iterator   Facet_iterator;

  // outputs dimension and number of vertices
  size_type n = tr.number_of_vertices();
  if(is_ascii(os))
  {
    os << tr.dimension() << std::endl << n << std::endl;
  }
  else
  {
    write(os, tr.dimension());
    write(os, n);
  }

  if(n == 0)
    return os;

  std::vector<Vertex_handle> TV(n+1);
  size_type i = 0;

  // write the vertices
  for(Vertex_iterator it = tr.vertices_begin(), end = tr.vertices_end(); it != end; ++it)
    TV[i++] = it;

  CGAL_triangulation_assertion(i == n+1);
  CGAL_triangulation_assertion(tr.is_infinite(TV[0]));

  Unique_hash_map<Vertex_handle, std::size_t > V;
  V[tr.infinite_vertex()] = 0;
  for(i=1; i <= n; i++)
  {
    os << *TV[i];
    V[TV[i]] = i;
    if(is_ascii(os))
      os << std::endl;
  }

  // Asks the tds for the combinatorial information
  tr.tds().print_cells(os, V);

  // Write the non combinatorial information on the cells
  // using the << operator of Cell.
  // Works because the iterator of the tds traverses the cells in the
  // same order as the iterator of the triangulation
  switch(tr.dimension())
  {
    case 3:
    {
      for(Cell_iterator it = tr.cells_begin(), end = tr.cells_end(); it != end; ++it)
      {
        os << *it; // other information
        if(is_ascii(os))
          os << std::endl;
      }
      break;
    }
    case 2:
    {
      for(Facet_iterator it = tr.facets_begin(), end = tr.facets_end(); it != end; ++it)
      {
        os << *((*it).first); // other information
        if(is_ascii(os))
          os << std::endl;
      }
      break;
    }
    case 1:
    {
      for(Edge_iterator it = tr.edges_begin(), end = tr.edges_end(); it != end; ++it)
      {
        os << *((*it).first); // other information
        if(is_ascii(os))
          os << std::endl;
      }
      break;
    }
  }
  return os ;
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::size_type
Triangulation_3<GT,Tds,Lds>::
number_of_finite_cells() const
{
  if(dimension() < 3)
    return 0;

  return std::distance(finite_cells_begin(), finite_cells_end());
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::size_type
Triangulation_3<GT,Tds,Lds>::
number_of_cells() const
{
  return _tds.number_of_cells();
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::size_type
Triangulation_3<GT,Tds,Lds>::
number_of_finite_facets() const
{
  if(dimension() < 2)
    return 0;

  return std::distance(finite_facets_begin(), finite_facets_end());
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::size_type
Triangulation_3<GT,Tds,Lds>::
number_of_facets() const
{
  return _tds.number_of_facets();
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::size_type
Triangulation_3<GT,Tds,Lds>::
number_of_finite_edges() const
{
  if(dimension() < 1)
    return 0;

  return std::distance(finite_edges_begin(), finite_edges_end());
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::size_type
Triangulation_3<GT,Tds,Lds>::
number_of_edges() const
{
  return _tds.number_of_edges();
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Triangle
Triangulation_3<GT,Tds,Lds>::
triangle(const Cell_handle c, int i) const
{
  CGAL_triangulation_precondition(dimension() == 2 || dimension() == 3);
  CGAL_triangulation_precondition((dimension() == 2 && i == 3) ||
                                  (dimension() == 3 && i >= 0 && i <= 3));
  CGAL_triangulation_precondition(! is_infinite(Facet(c, i)));
  if((i&1)==0)
    return construct_triangle(c->vertex((i+2)&3)->point(),
                              c->vertex((i+1)&3)->point(),
                              c->vertex((i+3)&3)->point());

  return construct_triangle(c->vertex((i+1)&3)->point(),
                            c->vertex((i+2)&3)->point(),
                            c->vertex((i+3)&3)->point());
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Segment
Triangulation_3<GT,Tds,Lds>::
segment(const Cell_handle c, int i, int j) const
{
  CGAL_triangulation_precondition(i != j);
  CGAL_triangulation_precondition(dimension() >= 1 && dimension() <= 3);
  CGAL_triangulation_precondition(i >= 0 && i <= dimension() &&
                                  j >= 0 && j <= dimension());
  CGAL_triangulation_precondition(! is_infinite(Edge(c, i, j)));
  return construct_segment(c->vertex(i)->point(), c->vertex(j)->point());
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
is_infinite(const Cell_handle c, int i) const
{
  CGAL_triangulation_precondition(dimension() == 2 || dimension() == 3);
  CGAL_triangulation_precondition((dimension() == 2 && i == 3) ||
                                  (dimension() == 3 && i >= 0 && i <= 3));
  return is_infinite(c->vertex(i<=0 ? 1 : 0)) ||
      is_infinite(c->vertex(i<=1 ? 2 : 1)) ||
      is_infinite(c->vertex(i<=2 ? 3 : 2));
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
is_infinite(const Cell_handle c, int i, int j) const
{
  CGAL_triangulation_precondition(i != j);
  CGAL_triangulation_precondition(dimension() >= 1 && dimension() <= 3);
  CGAL_triangulation_precondition(i >= 0 && i <= dimension() &&
                                  j >= 0 && j <= dimension());
  return is_infinite(c->vertex(i)) || is_infinite(c->vertex(j));
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_vertex(const Point& p, Vertex_handle& v) const
{
  Locate_type lt;
  int li, lj;
  Cell_handle c = locate(p, lt, li, lj);
  if(lt != VERTEX)
    return false;

  v = c->vertex(li);
  return true;
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
is_vertex(Vertex_handle v) const
{
  return _tds.is_vertex(v);
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_edge(Vertex_handle u, Vertex_handle v,
        Cell_handle& c, int& i, int& j) const
{
  return _tds.is_edge(u, v, c, i, j);
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_facet(Vertex_handle u, Vertex_handle v, Vertex_handle w,
         Cell_handle& c, int& i, int& j, int& k) const
{
  return _tds.is_facet(u, v, w, c, i, j, k);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
is_cell(Cell_handle c) const
{
  return _tds.is_cell(c);
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_cell(Vertex_handle u, Vertex_handle v,
        Vertex_handle w, Vertex_handle t,
        Cell_handle& c, int& i, int& j, int& k, int& l) const
{
  return _tds.is_cell(u, v, w, t, c, i, j, k, l);
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_cell(Vertex_handle u, Vertex_handle v,
        Vertex_handle w, Vertex_handle t,
        Cell_handle& c) const
{
  int i,j,k,l;
  return _tds.is_cell(u, v, w, t, c, i, j, k, l);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
has_vertex(const Facet& f, Vertex_handle v, int& j) const
{
  return _tds.has_vertex(f.first, f.second, v, j);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
has_vertex(Cell_handle c, int i, Vertex_handle v, int& j) const
{
  return _tds.has_vertex(c, i, v, j);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
has_vertex(const Facet& f, Vertex_handle v) const
{
  return _tds.has_vertex(f.first, f.second, v);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
has_vertex(Cell_handle c, int i, Vertex_handle v) const
{
  return _tds.has_vertex(c, i, v);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
are_equal(Cell_handle c, int i, Cell_handle n, int j) const
{
  return _tds.are_equal(c, i, n, j);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
are_equal(const Facet& f, const Facet& g) const
{
  return _tds.are_equal(f.first, f.second, g.first, g.second);
}

template < class GT, class Tds, class Lds >
inline
bool
Triangulation_3<GT,Tds,Lds>::
are_equal(const Facet& f, Cell_handle n, int j) const
{
  return _tds.are_equal(f.first, f.second, n, j);
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Cell_handle
Triangulation_3<GT,Tds,Lds>::
#ifdef CGAL_NO_STRUCTURAL_FILTERING
locate(const Point& p, Locate_type& lt, int& li, int& lj,
       Cell_handle start, bool *could_lock_zone) const
#else
exact_locate(const Point& p, Locate_type& lt, int& li, int& lj,
             Cell_handle start, bool *could_lock_zone) const
#endif
{
  // Returns the (finite or infinite) cell p lies in.
  // Starts at cell "start".
  // If lt == OUTSIDE_CONVEX_HULL, li is the index of a facet separating p
  //  from the rest of the triangulation
  // In dimension 2 :
  //  returns a facet (Cell_handle,li) if lt == FACET
  //  returns an edge (Cell_handle,li,lj) if lt == EDGE
  //  returns a vertex (Cell_handle,li) if lt == VERTEX
  // If lt == OUTSIDE_CONVEX_HULL, li, lj gives the edge of c separating p
  //  from the rest of the triangulation
  // lt = OUTSIDE_AFFINE_HULL if p is not coplanar with the triangulation

  CGAL_triangulation_expensive_assertion(start == Cell_handle() || tds().is_simplex(start));

  if(could_lock_zone)
    *could_lock_zone = true;

  if(dimension() >= 1)
  {
    // Make sure we continue from here with a finite cell.
    if(start == Cell_handle())
      start = infinite_cell();

    int ind_inf;
    if(start->has_vertex(infinite, ind_inf))
      start = start->neighbor(ind_inf);
  }

  boost::rand48 rng;

  switch(dimension())
  {
    case 3:
    {
      CGAL_triangulation_precondition(start != Cell_handle());
      CGAL_triangulation_precondition(! start->has_vertex(infinite));

      // We implement the remembering visibility/stochastic walk.

      // Remembers the previous cell to avoid useless orientation tests.
      Cell_handle previous = Cell_handle();
      Cell_handle c = start;

      if(could_lock_zone)
      {
        if(!this->try_lock_cell(c))
        {
          *could_lock_zone = false;
          return Cell_handle();
        }
      }

      // Stores the results of the 4 orientation tests.  It will be used
      // at the end to decide if p lies on a face/edge/vertex/interior.
      Orientation o[4];

      boost::uniform_smallint<> four(0, 3);
      boost::variate_generator<boost::rand48&, boost::uniform_smallint<> > die4(rng, four);

      // Now treat the cell c.
      bool try_next_cell = true;
      while(try_next_cell)
      {
        try_next_cell = false;

        // We know that the 4 vertices of c are positively oriented.
        // So, in order to test if p is seen outside from one of c's facets,
        // we just replace the corresponding point by p in the orientation
        // test.  We do this using the array below.
        const Point* pts[4] = { &(c->vertex(0)->point()),
                                &(c->vertex(1)->point()),
                                &(c->vertex(2)->point()),
                                &(c->vertex(3)->point()) };

        // For the remembering stochastic walk,
        // we need to start trying with a random index :
        int i = die4();

        // For the remembering visibility walk (Delaunay and Regular only), we don't :
        // int i = 0;

        // for each vertex
        for(int j=0; !try_next_cell && j != 4; ++j, i = (i+1)&3)
        {
          Cell_handle next = c->neighbor(i);

          if(previous == next)
          {
            o[i] = POSITIVE;
          }
          else
          {
            // We temporarily put p at i's place in pts.
            const Point* backup = pts[i];
            pts[i] = &p;
            o[i] = orientation(*pts[0], *pts[1], *pts[2], *pts[3]);
            if(o[i] != NEGATIVE)
            {
              pts[i] = backup;
            }
            else
            {
              if(next->has_vertex(infinite, li))
              {
                // We are outside the convex hull.
                lt = OUTSIDE_CONVEX_HULL;
                return next;
              }
              previous = c;
              c = next;
              if(could_lock_zone)
              {
                //previous->unlock(); // DON'T do that, "c" may be in
                // the same locking cell as "previous"
                if(!this->try_lock_cell(c))
                {
                  *could_lock_zone = false;
                  return Cell_handle();
                }
              }
              try_next_cell = true;
            }
          }
        } // next vertex
      } // next cell

      // now p is in c or on its boundary
      int sum =(o[0] == COPLANAR)
          +(o[1] == COPLANAR)
          +(o[2] == COPLANAR)
          +(o[3] == COPLANAR);
      switch(sum)
      {
        case 0:
        {
          lt = CELL;
          break;
        }
        case 1:
        {
          lt = FACET;
          li = (o[0] == COPLANAR) ? 0 :
                 (o[1] == COPLANAR) ? 1 :
                   (o[2] == COPLANAR) ? 2 : 3;
          break;
        }
        case 2:
        {
          lt = EDGE;
          li = (o[0] != COPLANAR) ? 0 :
                 (o[1] != COPLANAR) ? 1 : 2;
          lj = (o[li+1] != COPLANAR) ? li+1 :
                 (o[li+2] != COPLANAR) ? li+2 : li+3;
          CGAL_triangulation_assertion(collinear(p,
                                                 c->vertex(li)->point(),
                                                 c->vertex(lj)->point()));
          break;
        }
        case 3:
        {
          lt = VERTEX;
          li = (o[0] != COPLANAR) ? 0 :
                 (o[1] != COPLANAR) ? 1 :
                   (o[2] != COPLANAR) ? 2 : 3;
          break;
        }
      }
      return c;
    }

    case 2:
    {
      CGAL_triangulation_precondition(start != Cell_handle());
      CGAL_triangulation_precondition(! start->has_vertex(infinite));
      Cell_handle c = start;

      boost::uniform_smallint<> three(0, 2);
      boost::variate_generator<boost::rand48&, boost::uniform_smallint<> > die3(rng, three);

      //first tests whether p is coplanar with the current triangulation
      if(orientation(c->vertex(0)->point(),
                     c->vertex(1)->point(),
                     c->vertex(2)->point(),
                     p) != DEGENERATE)
      {
        lt = OUTSIDE_AFFINE_HULL;
        li = 3; // only one facet in dimension 2
        return c;
      }

      // if p is coplanar, location in the triangulation
      // only the facet numbered 3 exists in each cell
      while(1)
      {
        int inf;
        if(c->has_vertex(infinite, inf))
        {
          // c must contain p in its interior
          lt = OUTSIDE_CONVEX_HULL;
          li = cw(inf);
          lj = ccw(inf);
          return c;
        }

        // else c is finite
        // we test its edges in a random order until we find a
        // neighbor to go further
        int i = die3();
        const Point& p0 = c->vertex(i)->point();
        const Point& p1 = c->vertex(ccw(i))->point();
        const Point& p2 = c->vertex(cw(i))->point();
        Orientation o[3];
        CGAL_triangulation_assertion(coplanar_orientation(p0,p1,p2) == POSITIVE);
        o[0] = coplanar_orientation(p0,p1,p);
        if(o[0] == NEGATIVE)
        {
          c = c->neighbor(cw(i));
          continue;
        }

        o[1] = coplanar_orientation(p1,p2,p);
        if(o[1] == NEGATIVE)
        {
          c = c->neighbor(i);
          continue;
        }

        o[2] = coplanar_orientation(p2,p0,p);
        if(o[2] == NEGATIVE)
        {
          c = c->neighbor(ccw(i));
          continue;
        }

        // now p is in c or on its boundary
        int sum =(o[0] == COLLINEAR)
                +(o[1] == COLLINEAR)
                +(o[2] == COLLINEAR);
        switch(sum)
        {
          case 0:
          {
            lt = FACET;
            li = 3; // useless ?
            break;
          }
          case 1:
          {
            lt = EDGE;
            li = (o[0] == COLLINEAR) ? i :
                   (o[1] == COLLINEAR) ? ccw(i) :
                     cw(i);
            lj = ccw(li);
            break;
          }
          case 2:
          {
            lt = VERTEX;
            li = (o[0] != COLLINEAR) ? cw(i) :
                   (o[1] != COLLINEAR) ? i :
                     ccw(i);
            break;
          }
        }
        return c;
      }
    }
    case 1:
    {
      CGAL_triangulation_precondition(start != Cell_handle());
      CGAL_triangulation_precondition(! start->has_vertex(infinite));
      Cell_handle c = start;

      //first tests whether p is collinear with the current triangulation
      if(! collinear(p, c->vertex(0)->point(), c->vertex(1)->point()))
      {
        lt = OUTSIDE_AFFINE_HULL;
        return c;
      }
      // if p is collinear, location :
      while(1)
      {
        if(c->has_vertex(infinite))
        {
          // c must contain p in its interior
          lt = OUTSIDE_CONVEX_HULL;
          return c;
        }

        // else c is finite
        // we test on which direction to continue the traversal
        switch(collinear_position(c->vertex(0)->point(), p, c->vertex(1)->point()))
        {
          case AFTER:
            c = c->neighbor(0);
            continue;
          case BEFORE:
            c = c->neighbor(1);
            continue;
          case MIDDLE:
            lt = EDGE;
            li = 0;
            lj = 1;
            return c;
          case SOURCE:
            lt = VERTEX;
            li = 0;
            return c;
          case TARGET:
            lt = VERTEX;
            li = 1;
            return c;
        }
      }
    }
    case 0:
    {
      Finite_vertices_iterator vit = finite_vertices_begin();
      if(! equal(p, vit->point()))
      {
        lt = OUTSIDE_AFFINE_HULL;
      }
      else
      {
        lt = VERTEX;
        li = 0;
      }
      return vit->cell();
    }
    case -1:
    {
      lt = OUTSIDE_AFFINE_HULL;
      return Cell_handle();
    }
    default:
    {
      CGAL_triangulation_assertion(false);
      return Cell_handle();
    }
  }
}

#ifndef CGAL_NO_STRUCTURAL_FILTERING
template < class Gt, class Tds, class Lds >
inline
typename Triangulation_3<Gt, Tds, Lds>::Cell_handle
Triangulation_3<Gt, Tds, Lds>::
inexact_locate(const Point& t, Cell_handle start, int n_of_turns,
               bool *could_lock_zone) const
{
  CGAL_triangulation_expensive_assertion(start == Cell_handle() ||
                                         tds().is_simplex(start));

  if(could_lock_zone)
    *could_lock_zone = true;

  if(dimension() < 3)
    return start;

  // Make sure we continue from here with a finite cell.
  if(start == Cell_handle())
    start = infinite_cell();

  // CJTODO: useless?
  if(could_lock_zone)
  {
    if(!this->try_lock_cell(start))
    {
      *could_lock_zone = false;
      return Cell_handle();
    }
  }

  int ind_inf;
  if(start->has_vertex(infinite, ind_inf))
    start = start->neighbor(ind_inf);

  CGAL_triangulation_precondition(start != Cell_handle());
  CGAL_triangulation_precondition(! start->has_vertex(infinite));

  // We implement the remembering visibility walk.
  // In this phase, no need to be stochastic

  // Remembers the previous cell to avoid useless orientation tests.
  Cell_handle previous = Cell_handle();
  Cell_handle c = start;

  if(could_lock_zone)
  {
    if(!this->try_lock_cell(c))
    {
      *could_lock_zone = false;
      return Cell_handle();
    }
  }

  // Now treat the cell c.
try_next_cell:
  n_of_turns--;

  // We know that the 4 vertices of c are positively oriented.
  // So, in order to test if p is seen outside from one of c's facets,
  // we just replace the corresponding point by p in the orientation
  // test.  We do this using the array below.
  const Point* pts[4] = { &(c->vertex(0)->point()),
                          &(c->vertex(1)->point()),
                          &(c->vertex(2)->point()),
                          &(c->vertex(3)->point()) };

  // (non-stochastic) visibility walk
  for(int i=0; i != 4; ++i)
  {
    Cell_handle next = c->neighbor(i);
    if(previous == next) continue;

    // We temporarily put p at i's place in pts.
    const Point* backup = pts[i];
    pts[i] = &t;
    if(inexact_orientation(*pts[0], *pts[1], *pts[2], *pts[3]) != NEGATIVE)
    {
      pts[i] = backup;
      continue;
    }

    if(next->has_vertex(infinite))
    {
      // We are outside the convex hull.
      return next;
    }

    previous = c;
    c = next;
    if(could_lock_zone)
    {
      //previous->unlock(); // DON'T do that, "c" may be in
      // the same locking cell as "previous"
      if(!this->try_lock_cell(c))
      {
        *could_lock_zone = false;
        return Cell_handle();
      }
    }

    if(n_of_turns) goto try_next_cell;
  }

  return c;
}
#endif // no CGAL_NO_STRUCTURAL_FILTERING

template < class GT, class Tds, class Lds >
Bounded_side
Triangulation_3<GT,Tds,Lds>::
side_of_tetrahedron(const Point& p,
                    const Point& p0, const Point& p1, const Point& p2, const Point& p3,
                    Locate_type& lt, int& i, int& j) const
{
  // p0,p1,p2,p3 supposed to be non coplanar
  // tetrahedron p0,p1,p2,p3 is supposed to be well oriented
  // returns :
  // - ON_BOUNDED_SIDE if p lies strictly inside the tetrahedron
  // - ON_BOUNDARY if p lies on one of the facets
  // - ON_UNBOUNDED_SIDE if p lies strictly outside the tetrahedron

  CGAL_triangulation_precondition(orientation(p0,p1,p2,p3) == POSITIVE);

  Orientation o0,o1,o2,o3;
  if(((o0 = orientation(p,p1,p2,p3)) == NEGATIVE) ||
     ((o1 = orientation(p0,p,p2,p3)) == NEGATIVE) ||
     ((o2 = orientation(p0,p1,p,p3)) == NEGATIVE) ||
     ((o3 = orientation(p0,p1,p2,p)) == NEGATIVE))
  {
    lt = OUTSIDE_CONVEX_HULL;
    return ON_UNBOUNDED_SIDE;
  }

  // now all the oi's are >=0
  // sum gives the number of facets p lies on
  int sum =  ((o0 == ZERO) ? 1 : 0)
           + ((o1 == ZERO) ? 1 : 0)
           + ((o2 == ZERO) ? 1 : 0)
           + ((o3 == ZERO) ? 1 : 0);

  switch(sum)
  {
    case 0:
    {
      lt = CELL;
      return ON_BOUNDED_SIDE;
    }
    case 1:
    {
      lt = FACET;
      // i = index such that p lies on facet(i)
      i = (o0 == ZERO) ? 0 :
            (o1 == ZERO) ? 1 :
              (o2 == ZERO) ? 2 :
                3;
      return ON_BOUNDARY;
    }
    case 2:
    {
      lt = EDGE;
      // i = smallest index such that p does not lie on facet(i)
      // i must be < 3 since p lies on 2 facets
      i = (o0 == POSITIVE) ? 0 :
            (o1 == POSITIVE) ? 1 :
              2;
      // j = larger index such that p not on facet(j)
      // j must be > 0 since p lies on 2 facets
      j = (o3 == POSITIVE) ? 3 :
            (o2 == POSITIVE) ? 2 :
              1;
      return ON_BOUNDARY;
    }
    case 3:
    {
      lt = VERTEX;
      // i = index such that p does not lie on facet(i)
      i = (o0 == POSITIVE) ? 0 :
            (o1 == POSITIVE) ? 1 :
              (o2 == POSITIVE) ? 2 :
                3;
      return ON_BOUNDARY;
    }
    default:
    {
      // impossible : cannot be on 4 facets for a real tetrahedron
      CGAL_triangulation_assertion(false);
      return ON_BOUNDARY;
    }
  }
}

template < class GT, class Tds, class Lds >
Bounded_side
Triangulation_3<GT,Tds,Lds>::
side_of_cell(const Point& p,
             Cell_handle c,
             Locate_type& lt, int& i, int& j) const
{
  // Returns
  // - ON_BOUNDED_SIDE if p inside the cell
  //  (for an infinite cell this means that p lies strictly in the half space
  //  limited by its finite facet)
  // - ON_BOUNDARY if p on the boundary of the cell
  //  (for an infinite cell this means that p lies on the *finite* facet)
  // - ON_UNBOUNDED_SIDE if p lies outside the cell
  //  (for an infinite cell this means that p is not in the preceding
  //  two cases)
  // lt only has meaning when ON_BOUNDED_SIDE or ON_BOUNDARY

  CGAL_triangulation_precondition(dimension() == 3);
  if(! is_infinite(c))
  {
    return side_of_tetrahedron(p,
                               c->vertex(0)->point(),
                               c->vertex(1)->point(),
                               c->vertex(2)->point(),
                               c->vertex(3)->point(),
                               lt, i, j);
  }
  else
  {
    int inf = c->index(infinite);
    Orientation o;
    Vertex_handle v1 = c->vertex((inf+1)&3),
                  v2 = c->vertex((inf+2)&3),
                  v3 = c->vertex((inf+3)&3);

    if((inf&1) == 0)
      o = orientation(p, v1->point(), v2->point(), v3->point());
    else
      o =  orientation(v3->point(), p, v1->point(), v2->point());

    switch(o)
    {
      case POSITIVE:
      {
        lt = CELL;
        return ON_BOUNDED_SIDE;
      }
      case NEGATIVE:
        return ON_UNBOUNDED_SIDE;
      case ZERO:
      {
        // location in the finite facet
        int i_f, j_f;
        Bounded_side side = side_of_triangle(p,
                                             v1->point(), v2->point(), v3->point(),
                                             lt, i_f, j_f);

        // lt need not be modified in most cases :
        switch(side)
        {
          case ON_BOUNDED_SIDE:
          {
            // lt == FACET ok
            i = inf;
            return ON_BOUNDARY;
          }
          case ON_BOUNDARY:
          {
            // lt == VERTEX OR EDGE ok
            i = (i_f == 0) ? ((inf+1)&3) :
                  (i_f == 1) ? ((inf+2)&3) :
                    ((inf+3)&3);
            if(lt == EDGE)
            {
              j = (j_f == 0) ? ((inf+1)&3) :
                    (j_f == 1) ? ((inf+2)&3) :
                      ((inf+3)&3);
            }
            return ON_BOUNDARY;
          }
          case ON_UNBOUNDED_SIDE:
          {
            // p lies on the plane defined by the finite facet
            // lt must be initialized
            return ON_UNBOUNDED_SIDE;
          }
          default:
          {
            CGAL_triangulation_assertion(false);
            return ON_BOUNDARY;
          }
        } // switch side
      } // case ZERO
      default:
      {
        CGAL_triangulation_assertion(false);
        return ON_BOUNDARY;
      }
    } // switch o
  } // else infinite cell
} // side_of_cell

template < class GT, class Tds, class Lds >
Bounded_side
Triangulation_3<GT,Tds,Lds>::
side_of_triangle(const Point& p,
                 const Point& p0,
                 const Point& p1,
                 const Point& p2,
                 Locate_type& lt, int& i, int& j) const
{
  // p0,p1,p2 supposed to define a plane
  // p supposed to lie on plane p0,p1,p2
  // triangle p0,p1,p2 defines the orientation of the plane.
  // Returns
  // - ON_BOUNDED_SIDE if p lies strictly inside the triangle
  // - ON_BOUNDARY if p lies on one of the edges
  // - ON_UNBOUNDED_SIDE if p lies strictly outside the triangle

  CGAL_triangulation_precondition(coplanar(p,p0,p1,p2));

  Orientation o012 = coplanar_orientation(p0,p1,p2);
  CGAL_triangulation_precondition(o012 != COLLINEAR);

  Orientation o0; // edge p0 p1
  Orientation o1; // edge p1 p2
  Orientation o2; // edge p2 p0

  if((o0 = coplanar_orientation(p0,p1,p)) == opposite(o012) ||
     (o1 = coplanar_orientation(p1,p2,p)) == opposite(o012) ||
     (o2 = coplanar_orientation(p2,p0,p)) == opposite(o012))
  {
    lt = OUTSIDE_CONVEX_HULL;
    return ON_UNBOUNDED_SIDE;
  }

  // now all the oi's are >=0
  // sum gives the number of edges p lies on
  int sum =  ((o0 == ZERO) ? 1 : 0)
           + ((o1 == ZERO) ? 1 : 0)
           + ((o2 == ZERO) ? 1 : 0);

  switch(sum)
  {
    case 0:
    {
      lt = FACET;
      return ON_BOUNDED_SIDE;
    }
    case 1:
    {
      lt = EDGE;
      i = (o0 == ZERO) ? 0 :
            (o1 == ZERO) ? 1 :
              2;
      if(i == 2)
        j=0;
      else
        j = i+1;
      return ON_BOUNDARY;
    }
    case 2:
    {
      lt = VERTEX;
      i = (o0 == o012) ? 2 :
            (o1 == o012) ? 0 :
              1;
      return ON_BOUNDARY;
    }
    default:
    {
      // cannot happen
      CGAL_triangulation_assertion(false);
      return ON_BOUNDARY;
    }
  }
}

template < class GT, class Tds, class Lds >
Bounded_side
Triangulation_3<GT,Tds,Lds>::
side_of_facet(const Point& p,
              Cell_handle c,
              Locate_type& lt, int& li, int& lj) const
{
  // Assumes dimension 2; otherwise does not work for infinite facets.
  // Returns :
  // - ON_BOUNDED_SIDE if p inside the facet
  //  (for an infinite facet this means that p lies strictly in the half plane
  //  limited by its finite edge)
  // - ON_BOUNDARY if p on the boundary of the facet
  //  (for an infinite facet this means that p lies on the *finite* edge)
  // - ON_UNBOUNDED_SIDE if p lies outside the facet
  //  (for an infinite facet this means that p is not in the
  //  preceding two cases)
  // lt only has meaning when ON_BOUNDED_SIDE or ON_BOUNDARY.
  // When they mean anything, li and lj refer to indices in the cell c
  // giving the facet (c,i).

  CGAL_triangulation_precondition(dimension() == 2);
  if(! is_infinite(c,3))
  {
    // The following precondition is useless because it is written
    // in side_of_facet
    // 	CGAL_triangulation_precondition(coplanar (p,
    //                                             c->vertex(0)->point,
    //                                             c->vertex(1)->point,
    //                                             c->vertex(2)->point));
    int i_t, j_t;
    Bounded_side side = side_of_triangle(p,
                                         c->vertex(0)->point(),
                                         c->vertex(1)->point(),
                                         c->vertex(2)->point(),
                                         lt, i_t, j_t);

    // We protect the following code by this test to avoid valgrind messages.
    if(side == ON_BOUNDARY)
    {
      // indices in the original cell :
      li = (i_t == 0) ? 0 :
             (i_t == 1) ? 1 : 2;
      lj = (j_t == 0) ? 0 :
             (j_t == 1) ? 1 : 2;
    }
    return side;
  }

  // else infinite facet
  int inf = c->index(infinite);
  // The following precondition is useless because it is written
  // in side_of_facet
  // 	CGAL_triangulation_precondition(coplanar (p,
  //                                     c->neighbor(inf)->vertex(0)->point(),
  //                                     c->neighbor(inf)->vertex(1)->point(),
  //                                     c->neighbor(inf)->vertex(2)->point()));
  int i2 = next_around_edge(inf,3);
  int i1 = 3 - inf - i2;
  Vertex_handle v1 = c->vertex(i1),
                v2 = c->vertex(i2);

  CGAL_triangulation_assertion(coplanar_orientation(v1->point(), v2->point(),
                                                    mirror_vertex(c, inf)->point()) == POSITIVE);

  switch(coplanar_orientation(v1->point(), v2->point(), p))
  {
    case POSITIVE:
      // p lies on the same side of v1v2 as vn, so not in f
      return ON_UNBOUNDED_SIDE;
    case NEGATIVE:
      // p lies in f
      lt = FACET;
      li = 3;
      return ON_BOUNDED_SIDE;
    default: // case ZERO:
      // p collinear with v1v2
      int i_e;
      switch(side_of_segment(p, v1->point(), v2->point(), lt, i_e))
      {
        // computation of the indices in the original cell
        case ON_BOUNDED_SIDE:
          // lt == EDGE ok
          li = i1;
          lj = i2;
          return ON_BOUNDARY;
        case ON_BOUNDARY:
          // lt == VERTEX ok
          li =(i_e == 0) ? i1 : i2;
          return ON_BOUNDARY;
        default: // case ON_UNBOUNDED_SIDE:
          // p lies on the line defined by the finite edge
          return ON_UNBOUNDED_SIDE;
      }
  }
}

template < class GT, class Tds, class Lds >
Bounded_side
Triangulation_3<GT,Tds,Lds>::
side_of_segment(const Point& p,
                const Point& p0,
                const Point& p1,
                Locate_type& lt, int& i) const
{
  // p0, p1 supposed to be different
  // p supposed to be collinear to p0, p1
  // Returns :
  // - ON_BOUNDED_SIDE if p lies strictly inside the edge
  // - ON_BOUNDARY if p equals p0 or p1
  // - ON_UNBOUNDED_SIDE if p lies strictly outside the edge

  CGAL_triangulation_precondition(! equal(p0, p1));
  CGAL_triangulation_precondition(collinear(p, p0, p1));

  switch(collinear_position(p0, p, p1))
  {
    case MIDDLE:
      lt = EDGE;
      return ON_BOUNDED_SIDE;
    case SOURCE:
      lt = VERTEX;
      i = 0;
      return ON_BOUNDARY;
    case TARGET:
      lt = VERTEX;
      i = 1;
      return ON_BOUNDARY;
    default: // case BEFORE: case AFTER:
      lt = OUTSIDE_CONVEX_HULL;
      return ON_UNBOUNDED_SIDE;
  }
}

template < class GT, class Tds, class Lds >
Bounded_side
Triangulation_3<GT,Tds,Lds>::
side_of_edge(const Point& p,
             Cell_handle c,
             Locate_type& lt, int& li) const
{
  // Assumes dimension 1 otherwise does not work for infinite edges.
  // Returns :
  // - ON_BOUNDED_SIDE if p inside the edge
  //  (for an infinite edge this means that p lies in the half line
  //  defined by the vertex)
  // - ON_BOUNDARY if p equals one of the vertices
  // - ON_UNBOUNDED_SIDE if p lies outside the edge
  //  (for an infinite edge this means that p lies on the other half line)
  // lt only has meaning when ON_BOUNDED_SIDE and ON_BOUNDARY
  // li refer to indices in the cell c

  CGAL_triangulation_precondition(dimension() == 1);
  if(! is_infinite(c,0,1))
    return side_of_segment(p, c->vertex(0)->point(), c->vertex(1)->point(),
                           lt, li);
  // else infinite edge
  int inf = c->index(infinite);
  switch(collinear_position(c->vertex(1-inf)->point(), p,
                             mirror_vertex(c, inf)->point()))
  {
    case SOURCE:
      lt = VERTEX;
      li = 1-inf;
      return ON_BOUNDARY;
    case BEFORE:
      lt = EDGE;
      return ON_BOUNDED_SIDE;
    default: // case MIDDLE: case AFTER: case TARGET:
      return ON_UNBOUNDED_SIDE;
  }
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
flip(Cell_handle c, int i)
{
  CGAL_triangulation_precondition((dimension() == 3) && (0<=i) && (i<4) &&
                                  (number_of_vertices() >= 5));

  Cell_handle n = c->neighbor(i);
  int in = n->index(c);
  if(is_infinite(c) || is_infinite(n))
    return false;

  if(i%2 == 1)
  {
    if(orientation(c->vertex((i+1)&3)->point(),
                   c->vertex((i+2)&3)->point(),
                   n->vertex(in)->point(),
                   c->vertex(i)->point()) != POSITIVE)
      return false;

    if(orientation(c->vertex((i+2)&3)->point(),
                   c->vertex((i+3)&3)->point(),
                   n->vertex(in)->point(),
                   c->vertex(i)->point()) != POSITIVE)
      return false;

    if(orientation(c->vertex((i+3)&3)->point(),
                   c->vertex((i+1)&3)->point(),
                   n->vertex(in)->point(),
                   c->vertex(i)->point()) != POSITIVE)
      return false;
  }
  else
  {
    if(orientation(c->vertex((i+2)&3)->point(),
                   c->vertex((i+1)&3)->point(),
                   n->vertex(in)->point(),
                   c->vertex(i)->point()) != POSITIVE)
      return false;

    if(orientation(c->vertex((i+3)&3)->point(),
                   c->vertex((i+2)&3)->point(),
                   n->vertex(in)->point(),
                   c->vertex(i)->point()) != POSITIVE)
      return false;

    if(orientation(c->vertex((i+1)&3)->point(),
                   c->vertex((i+3)&3)->point(),
                   n->vertex(in)->point(),
                   c->vertex(i)->point()) != POSITIVE)
      return false;
  }

  _tds.flip_flippable(c, i);
  return true;
}

template < class GT, class Tds, class Lds >
void
Triangulation_3<GT,Tds,Lds>::
flip_flippable(Cell_handle c, int i)
{
  CGAL_triangulation_precondition((dimension() == 3) && (0<=i) && (i<4) &&
                                  (number_of_vertices() >= 5));
  CGAL_triangulation_precondition_code(Cell_handle n = c->neighbor(i););
  CGAL_triangulation_precondition_code(int in = n->index(c););
  CGAL_triangulation_precondition((! is_infinite(c)) &&(! is_infinite(n)));

  if(i%2 == 1)
  {
    CGAL_triangulation_precondition(orientation(c->vertex((i+1)&3)->point(),
                                                c->vertex((i+2)&3)->point(),
                                                n->vertex(in)->point(),
                                                c->vertex(i)->point()) == POSITIVE);
    CGAL_triangulation_precondition(orientation(c->vertex((i+2)&3)->point(),
                                                c->vertex((i+3)&3)->point(),
                                                n->vertex(in)->point(),
                                                c->vertex(i)->point()) == POSITIVE);
    CGAL_triangulation_precondition(orientation(c->vertex((i+3)&3)->point(),
                                                c->vertex((i+1)&3)->point(),
                                                n->vertex(in)->point(),
                                                c->vertex(i)->point()) == POSITIVE);
  }
  else
  {
    CGAL_triangulation_precondition(orientation(c->vertex((i+2)&3)->point(),
                                                c->vertex((i+1)&3)->point(),
                                                n->vertex(in)->point(),
                                                c->vertex(i)->point()) == POSITIVE);
    CGAL_triangulation_precondition(orientation(c->vertex((i+3)&3)->point(),
                                                c->vertex((i+2)&3)->point(),
                                                n->vertex(in)->point(),
                                                c->vertex(i)->point()) == POSITIVE);
    CGAL_triangulation_precondition(orientation(c->vertex((i+1)&3)->point(),
                                                c->vertex((i+3)&3)->point(),
                                                n->vertex(in)->point(),
                                                c->vertex(i)->point()) == POSITIVE);
  }

  _tds.flip_flippable(c, i);
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
flip(Cell_handle c, int i, int j)
{
  // Flips the edge (i,j) of cell c

  CGAL_triangulation_precondition((dimension() == 3) &&
                                  (0<=i) && (i<4) && (0<=j) &&
                                  (j<4) &&(i != j) &&
                                  (number_of_vertices() >= 5));

  // Checks that degree 3 and not on the convex hull
  int degree = 0;
  Cell_circulator ccir = incident_cells(c,i,j);
  Cell_circulator cdone = ccir;
  do
  {
    if(is_infinite(ccir))
      return false;

    ++degree;
    ++ccir;
  }
  while(ccir != cdone);

  if(degree != 3)
    return false;

  // Checks that future tetrahedra are well oriented
  Cell_handle n = c->neighbor(next_around_edge(i,j));
  int in = n->index(c->vertex(i));
  int jn = n->index(c->vertex(j));
  if(orientation(c->vertex(next_around_edge(i,j))->point(),
                 c->vertex(next_around_edge(j,i))->point(),
                 n->vertex(next_around_edge(jn,in))->point(),
                 c->vertex(j)->point()) != POSITIVE)
    return false;

  if(orientation(c->vertex(i)->point(),
                 c->vertex(next_around_edge(j,i))->point(),
                 n->vertex(next_around_edge(jn,in))->point(),
                 c->vertex(next_around_edge(i,j))->point()) != POSITIVE)
    return false;

  _tds.flip_flippable(c, i, j);
  return true;
}

template < class GT, class Tds, class Lds >
void
Triangulation_3<GT,Tds,Lds>::
flip_flippable(Cell_handle c, int i, int j)
{
  // flips edge i,j of cell c

#if !defined CGAL_TRIANGULATION_NO_PRECONDITIONS && \
  !defined CGAL_NO_PRECONDITIONS && !defined NDEBUG
  CGAL_triangulation_precondition((dimension() == 3) &&
                                  (0<=i) && (i<4) && (0<=j) && (j<4) &&
                                  (i != j) && (number_of_vertices() >= 5));
  int degree = 0;
  Cell_circulator ccir = incident_cells(c,i,j);
  Cell_circulator cdone = ccir;
  do
  {
    CGAL_triangulation_precondition(! is_infinite(ccir));
    ++degree;
    ++ccir;
  }
  while(ccir != cdone);
  CGAL_triangulation_precondition(degree == 3);

  Cell_handle n = c->neighbor(next_around_edge(i, j));
  int in = n->index(c->vertex(i));
  int jn = n->index(c->vertex(j));
  CGAL_triangulation_precondition(orientation(c->vertex(next_around_edge(i,j))->point(),
                                              c->vertex(next_around_edge(j,i))->point(),
                                              n->vertex(next_around_edge(jn,in))->point(),
                                              c->vertex(j)->point()) == POSITIVE);
  CGAL_triangulation_precondition(orientation(c->vertex(i)->point(),
                                              c->vertex(next_around_edge(j,i))->point(),
                                              n->vertex(next_around_edge(jn,in))->point(),
                                              c->vertex(next_around_edge(i,j))->point()) == POSITIVE);
#endif

  _tds.flip_flippable(c, i, j);
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert(const Point& p, Cell_handle start)
{
  Locate_type lt;
  int li, lj;
  Cell_handle c = locate(p, lt, li, lj, start);
  return insert(p, lt, c, li, lj);
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert(const Point& p, Locate_type lt, Cell_handle c, int li, int lj)
{
  switch(lt)
  {
    case VERTEX:
      return c->vertex(li);
    case EDGE:
      return insert_in_edge(p, c, li, lj);
    case FACET:
      return insert_in_facet(p, c, li);
    case CELL:
      return insert_in_cell(p, c);
    case OUTSIDE_CONVEX_HULL:
      return insert_outside_convex_hull(p, c);
    case OUTSIDE_AFFINE_HULL:
    default:
      return insert_outside_affine_hull(p);
  }
}

template < class GT, class Tds, class Lds >
template < class Conflict_tester, class Hidden_points_visitor >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert_in_conflict(const Point& p,
                   Locate_type lt, Cell_handle c, int li, int /*lj*/,
                   const Conflict_tester& tester,
                   Hidden_points_visitor& hider,
                   bool *could_lock_zone)
{
  if(could_lock_zone)
    *could_lock_zone = true;

  switch(dimension())
  {
    case 3:
    {
      if((lt == VERTEX) && (tester.compare_weight(c->vertex(li)->point(), p)==0))
      {
        return c->vertex(li);
      }

      // If the new point is not in conflict with its cell, it is hidden.
      if(!tester.test_initial_cell(c))
      {
        hider.hide_point(c,p);
        return Vertex_handle();
      }

      // Ok, we really insert the point now.
      // First, find the conflict region.
      boost::container::small_vector<Cell_handle,32> cells;
      Facet facet;

      boost::container::small_vector<Facet,32> facets;

      // Parallel
      if(could_lock_zone)
      {

        find_conflicts(c,
                       tester,
                       make_triple(std::back_inserter(facets),
                                   std::back_inserter(cells),
                                   Emptyset_iterator()),
                       could_lock_zone);

        if(*could_lock_zone == false)
        {
          for(Cell_handle ch : cells)
          {
            ch->tds_data().clear();
          }

          for(Facet& f : facets)
          {
            f.first->neighbor(f.second)->tds_data().clear();
          }
          return Vertex_handle();
        }
      }
      // Sequential
      else
      {
        find_conflicts(c,
                       tester,
                       make_triple(
                         std::back_inserter(facets),
                         std::back_inserter(cells),
                         Emptyset_iterator()));
      }

      facet = facets.back();

      // Remember the points that are hidden by the conflicting cells,
      // as they will be deleted during the insertion.
      hider.process_cells_in_conflict(cells.begin(), cells.end());

      Vertex_handle v =
        tds().is_small_hole(facets.size()) ?
        _insert_in_small_hole(p, cells, facets) :
        _insert_in_hole(p,
                        cells.begin(), cells.end(),
                        facet.first, facet.second);

      // Store the hidden points in their new cells.
      hider.reinsert_vertices(v);
      return v;
    }
    case 2:
    {
      // This check is added compared to the 3D case
      if(lt == OUTSIDE_AFFINE_HULL)
        return insert_outside_affine_hull (p);

      if((lt == VERTEX) && (tester.compare_weight(c->vertex(li)->point(), p)==0))
      {
        return c->vertex(li);
      }
      // If the new point is not in conflict with its cell, it is hidden.
      if(!tester.test_initial_cell(c))
      {
        hider.hide_point(c,p);
        return Vertex_handle();
      }

      // Ok, we really insert the point now.
      // First, find the conflict region.
      std::vector<Cell_handle> cells;
      cells.reserve(32);
      Facet facet;

      find_conflicts(c, tester, make_triple(Oneset_iterator<Facet>(facet),
                                            std::back_inserter(cells),
                                            Emptyset_iterator()));

      // Remember the points that are hidden by the conflicting cells,
      // as they will be deleted during the insertion.
      hider.process_cells_in_conflict(cells.begin(), cells.end());

      Vertex_handle v = _insert_in_hole(p, cells.begin(), cells.end(),
                                        facet.first, facet.second);

      // Store the hidden points in their new cells.
      hider.reinsert_vertices(v);
      return v;
    }
    default:
    {
      // dimension() <= 1
      if(lt == OUTSIDE_AFFINE_HULL)
        return insert_outside_affine_hull (p);

      if(lt == VERTEX && tester.compare_weight(c->vertex(li)->point(), p) == 0)
        return c->vertex(li);

      // If the new point is not in conflict with its cell, it is hidden.
      if(! tester.test_initial_cell(c))
      {
        hider.hide_point(c,p);
        return Vertex_handle();
      }

      if(dimension() == 0)
        return hider.replace_vertex(c, li, p);

      // dimension() == 1;

      // Ok, we really insert the point now.
      // First, find the conflict region.
      std::vector<Cell_handle> cells;
      Facet facet;
      Cell_handle bound[2];
      // corresponding index: bound[j]->neighbor(1-j) is in conflict.

      // We get all cells in conflict,
      // and remember the 2 external boundaries.
      cells.push_back(c);

      for(int j = 0; j<2; ++j)
      {
        Cell_handle n = c->neighbor(j);
        while(tester(n))
        {
          cells.push_back(n);
          n = n->neighbor(j);
        }
        bound[j] = n;
      }

      // Insertion.
      hider.process_cells_in_conflict(cells.begin(), cells.end());
      tds().delete_cells(cells.begin(), cells.end());

      // We preserve the order (like the orientation in 2D-3D).
      Vertex_handle v = tds().create_vertex();
      Cell_handle c0 = tds().create_face(v, bound[0]->vertex(0), Vertex_handle());
      Cell_handle c1 = tds().create_face(bound[1]->vertex(1), v, Vertex_handle());
      tds().set_adjacency(c0, 1, c1, 0);
      tds().set_adjacency(bound[0], 1, c0, 0);
      tds().set_adjacency(c1, 1, bound[1], 0);
      bound[0]->vertex(0)->set_cell(bound[0]);
      bound[1]->vertex(1)->set_cell(bound[1]);
      v->set_cell(c0);
      v->set_point (p);

      hider.reinsert_vertices(v);

      return v;
    }
  }
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert_in_cell(const Point& p, Cell_handle c)
{
  CGAL_triangulation_precondition(dimension() == 3);
  CGAL_triangulation_precondition_code(
    Locate_type lt;
    int i; int j;
  );
  CGAL_triangulation_precondition(side_of_tetrahedron(p,
                                                      c->vertex(0)->point(),
                                                      c->vertex(1)->point(),
                                                      c->vertex(2)->point(),
                                                      c->vertex(3)->point(),
                                                      lt,i,j) == ON_BOUNDED_SIDE);

  Vertex_handle v = _tds.insert_in_cell(c);
  v->set_point(p);
  return v;
}

template < class GT, class Tds, class Lds >
inline
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert_in_facet(const Point& p, Cell_handle c, int i)
{
  CGAL_triangulation_precondition(dimension() == 2 || dimension() == 3);
  CGAL_triangulation_precondition((dimension() == 2 && i == 3) ||
                                  (dimension() == 3 && i >= 0 && i <= 3));
  CGAL_triangulation_exactness_precondition_code(
    Locate_type lt;
    int li; int lj;
  );
  CGAL_triangulation_exactness_precondition(coplanar(p, c->vertex((i+1)&3)->point(),
                                                     c->vertex((i+2)&3)->point(),
                                                     c->vertex((i+3)&3)->point()) &&
                                            side_of_triangle(p,
                                                             c->vertex((i+1)&3)->point(),
                                                             c->vertex((i+2)&3)->point(),
                                                             c->vertex((i+3)&3)->point(),
                                                             lt, li, lj) == ON_BOUNDED_SIDE);

  Vertex_handle v = _tds.insert_in_facet(c, i);
  v->set_point(p);
  return v;
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert_in_edge(const Point& p, Cell_handle c, int i, int j)
{
  CGAL_triangulation_precondition(i != j);
  CGAL_triangulation_precondition(dimension() >= 1 && dimension() <= 3);
  CGAL_triangulation_precondition(i >= 0 && i <= dimension() &&
                                  j >= 0 && j <= dimension());
  CGAL_triangulation_exactness_precondition_code(
    Locate_type lt;
    int li;
  );

  switch(dimension())
  {
    case 3:
    case 2:
    {
      CGAL_triangulation_precondition(! is_infinite(c, i, j));
      CGAL_triangulation_exactness_precondition(collinear(c->vertex(i)->point(),
                                                          p,
                                                          c->vertex(j)->point())
                                                && side_of_segment(p,
                                                                   c->vertex(i)->point(),
                                                                   c->vertex(j)->point(),
                                                                   lt, li) == ON_BOUNDED_SIDE);
      break;
    }
    case 1:
    {
      CGAL_triangulation_exactness_precondition(side_of_edge(p, c, lt, li) == ON_BOUNDED_SIDE);
      break;
    }
  }

  Vertex_handle v = _tds.insert_in_edge(c, i, j);
  v->set_point(p);
  return v;
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert_outside_convex_hull(const Point& p, Cell_handle c)
{
  // c is an infinite cell containing p
  // p is strictly outside the convex hull
  // dimension 0 not allowed, use outside-affine-hull

  CGAL_triangulation_precondition(dimension() > 0);
  CGAL_triangulation_precondition(c->has_vertex(infinite));
  // the precondition that p is in c is tested in each of the
  // insertion methods called from this method

  switch(dimension())
  {
    case 1:
    {
      // 	// p lies in the infinite edge neighboring c
      // 	// on the other side of li
      // 	return insert_in_edge(p,c->neighbor(1-li),0,1);
      return insert_in_edge(p,c,0,1);
    }
    case 2:
    {
      Conflict_tester_outside_convex_hull_2 tester(p, this);
      Vertex_handle v = insert_conflict(c, tester);
      v->set_point(p);
      return v;
    }
    default: // case 3:
    {
      Conflict_tester_outside_convex_hull_3 tester(p, this);
      Vertex_handle v = insert_conflict(c, tester);
      v->set_point(p);
      return v;
    }
  }
}

template < class GT, class Tds, class Lds >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::
insert_outside_affine_hull(const Point& p)
{
  CGAL_triangulation_precondition(dimension() < 3);
  bool reorient;
  switch(dimension())
  {
    case 1:
    {
      Cell_handle c = infinite_cell();
      Cell_handle n = c->neighbor(c->index(infinite_vertex()));
      Orientation o = coplanar_orientation(n->vertex(0)->point(),
                                           n->vertex(1)->point(), p);
      CGAL_triangulation_precondition(o != COLLINEAR);
      reorient = o == NEGATIVE;
      break;
    }
    case 2:
    {
      Cell_handle c = infinite_cell();
      Cell_handle n = c->neighbor(c->index(infinite_vertex()));
      Orientation o = orientation(n->vertex(0)->point(),
                                  n->vertex(1)->point(),
                                  n->vertex(2)->point(), p);
      CGAL_triangulation_precondition(o != COPLANAR);
      reorient = o == NEGATIVE;
      break;
    }
    default:
      reorient = false;
  }

  Vertex_handle v = _tds.insert_increase_dimension(infinite_vertex());
  v->set_point(p);

  if(reorient)
    _tds.reorient();

  return v;
}

template < class GT, class Tds, class Lds >
template < class OutputItCells >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::insert_and_give_new_cells(const Point& p,
                                                       OutputItCells fit,
                                                       Cell_handle start)
{
  Vertex_handle v = insert(p, start);
  int dimension = this->dimension();
  if(dimension == 3) this->incident_cells(v, fit);
  else if(dimension == 2)
  {
    Cell_handle c = v->cell(), end = c;
    do
    {
      *fit++ = c;
      int i = c->index(v);
      c = c->neighbor((i+1)%3);
    }
    while(c != end);
  }
  else if(dimension == 1)
  {
    Cell_handle c = v->cell();
    *fit++ = c;
    *fit++ = c->neighbor((~(c->index(v)))&1);
  }
  else *fit++ = v->cell(); // dimension = 0
  return v;
}

template < class GT, class Tds, class Lds >
template < class OutputItCells >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::insert_and_give_new_cells(const Point& p,
                                                       OutputItCells fit,
                                                       Vertex_handle hint)
{
  Vertex_handle v = insert(p, hint);
  int dimension = this->dimension();
  if(dimension == 3)
  {
    this->incident_cells(v, fit);
  }
  else if(dimension == 2)
  {
    Cell_handle c = v->cell(), end = c;
    do
    {
      *fit++ = c;
      int i = c->index(v);
      c = c->neighbor((i+1)%3);
    }
    while(c != end);
  }
  else if(dimension == 1)
  {
    Cell_handle c = v->cell();
    *fit++ = c;
    *fit++ = c->neighbor((~(c->index(v)))&1);
  }
  else // dimension = 0
  {
    *fit++ = v->cell();
  }

  return v;
}

template < class GT, class Tds, class Lds >
template < class OutputItCells >
typename Triangulation_3<GT,Tds,Lds>::Vertex_handle
Triangulation_3<GT,Tds,Lds>::insert_and_give_new_cells(const Point& p,
                                                       Locate_type lt,
                                                       Cell_handle c, int li, int lj,
                                                       OutputItCells fit)
{
  Vertex_handle v = insert(p, lt, c, li, lj);
  int dimension = this->dimension();
  if(dimension == 3) this->incident_cells(v, fit);
  else if(dimension == 2)
  {
    Cell_handle c = v->cell(), end = c;
    do {
      *fit++ = c;
      int i = c->index(v);
      c = c->neighbor((i+1)%3);
    }
    while(c != end);
  }
  else if(dimension == 1)
  {
    Cell_handle c = v->cell();
    *fit++ = c;
    *fit++ = c->neighbor((~(c->index(v)))&1);
  }
  else *fit++ = v->cell(); // dimension = 0
  return v;
}

template < class Gt, class Tds, class Lds >
typename Triangulation_3<Gt,Tds,Lds>::Vertex_triple
Triangulation_3<Gt,Tds,Lds>::
make_vertex_triple(const Facet& f) const
{
  Cell_handle ch = f.first;
  int i = f.second;

  return Vertex_triple(ch->vertex(vertex_triple_index(i,0)),
                       ch->vertex(vertex_triple_index(i,1)),
                       ch->vertex(vertex_triple_index(i,2)));
}

template < class Gt, class Tds, class Lds >
void
Triangulation_3<Gt,Tds,Lds>::
make_canonical(Vertex_triple& t) const
{
  int i = (t.first < t.second) ? 0 : 1;
  if(i==0)
  {
    i = (t.first < t.third) ? 0 : 2;
  } else
  {
    i = (t.second < t.third) ? 1 : 2;
  }

  Vertex_handle tmp;
  switch(i)
  {
    case 0: return;
    case 1:
      tmp = t.first;
      t.first = t.second;
      t.second = t.third;
      t.third = tmp;
      return;
    default:
      tmp = t.first;
      t.first = t.third;
      t.third = t.second;
      t.second = tmp;
  }
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
test_dim_down(Vertex_handle v) const
{
  // Tests whether removing v decreases the dimension of the triangulation.
  // Returns true iff v is incident to all finite cells/facets
  // and all the other vertices are coplanar/collinear in dim3/2.

  CGAL_triangulation_precondition(dimension() >= 0);
  CGAL_triangulation_precondition(! is_infinite(v));

  if(dimension() == 3)
  {
    Finite_cells_iterator cit = finite_cells_begin();

    int iv;
    if(! cit->has_vertex(v,iv))
      return false;

    const Point& p1=cit->vertex((iv+1)&3)->point();
    const Point& p2=cit->vertex((iv+2)&3)->point();
    const Point& p3=cit->vertex((iv+3)&3)->point();
    ++cit;

    for(; cit != finite_cells_end(); ++cit)
    {
      if(! cit->has_vertex(v,iv))
        return false;

      for(int i=1; i<4; i++)
        if(!coplanar(p1,p2,p3,cit->vertex((iv+i)&3)->point()))
          return false;
    }
  }
  else if(dimension() == 2)
  {
    Finite_facets_iterator cit = finite_facets_begin();

    int iv;
    if(! cit->first->has_vertex(v,iv))
      return false;

    const Point& p1 = cit->first->vertex(cw(iv))->point();
    const Point& p2 = cit->first->vertex(ccw(iv))->point();
    ++cit;

    for(; cit != finite_facets_end(); ++cit)
    {
      if(! cit->first->has_vertex(v,iv))
        return false;

      if(!collinear(p1, p2, cit->first->vertex(cw(iv))->point()) ||
         !collinear(p1, p2, cit->first->vertex(ccw(iv))->point()))
        return false;
    }
  }
  else // dimension() == 1 or 0
  {
    return number_of_vertices() == (size_type) dimension() + 1;
  }

  return true;
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
test_dim_down_using_incident_cells_3(Vertex_handle v,
                                     std::vector<Cell_handle>& incident_cells,
                                     std::vector<Vertex_handle>& adj_vertices,
                                     bool *could_lock_zone) const
{
  CGAL_triangulation_precondition(dimension() == 3);
  CGAL_triangulation_precondition(! is_infinite(v));

  // Collect all vertices on the boundary
  // and all incident cells
  if(could_lock_zone)
  {
    *could_lock_zone = try_lock_and_get_adjacent_vertices_and_cells_3(
                         v, std::back_inserter(adj_vertices), incident_cells);
    if(*could_lock_zone == false)
      return false;
  }
  else
  {
    adjacent_vertices_and_cells_3(v, std::back_inserter(adj_vertices),
                                  incident_cells);
  }

  typedef Filter_iterator< typename std::vector<Vertex_handle>::const_iterator,
                           Infinite_tester> Finite_vertex_iterator;

  Finite_vertex_iterator vit(adj_vertices.end(),
                             Infinite_tester(this),
                             adj_vertices.begin());
  Finite_vertex_iterator vit_end(adj_vertices.end(),
                                 Infinite_tester(this));

  const Point& p1 = (*vit++)->point();
  const Point& p2 = (*vit++)->point();
  const Point& p3 = (*vit++)->point();

  for(; vit != vit_end ; ++vit)
  {
    if(!coplanar(p1, p2, p3, (*vit)->point()))
      return false;
  }

  for(typename std::vector<Cell_handle>::const_iterator it_inc_cell
       = incident_cells.begin() ;
       it_inc_cell != incident_cells.end() ;
       ++it_inc_cell)
  {
    if(!is_infinite(*it_inc_cell))
      return is_infinite(mirror_vertex(*it_inc_cell, (*it_inc_cell)->index(v)));
  }

  return true;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt, Tds, Lds>::
make_hole_2D(Vertex_handle v, std::list<Edge_2D>& hole, VertexRemover& remover)
{
  std::vector<Cell_handle> to_delete;
  to_delete.reserve(32);

  Face_circulator fc = tds().incident_faces(v);
  Face_circulator done(fc);

  // We prepare for deleting all interior cells.
  // We ->set_cell() pointers to cells outside the hole.
  // We push the Edges_2D of the boundary (seen from outside) in "hole".
  do
  {
    Cell_handle f = fc;
    int i = f->index(v);
    Cell_handle fn = f->neighbor(i);
    int in = fn->index(f);

    f->vertex(cw(i))->set_cell(fn);
    fn->set_neighbor(in, Cell_handle());

    hole.push_back(Edge_2D(fn, in));
    remover.add_hidden_points(f);
    to_delete.push_back(f);

    ++fc;
  }
  while(fc != done);

  tds().delete_cells(to_delete.begin(), to_delete.end());
  return remover;
}

// This method also erases a set of cells, which is useful to the move method
// that outputs newly created cells
template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt, Tds, Lds>::
make_hole_2D(Vertex_handle v, std::list<Edge_2D>& hole, VertexRemover& remover,
             std::set<Cell_handle>& cells_set)
{
  std::vector<Cell_handle> to_delete;
  to_delete.reserve(32);

  Face_circulator fc = tds().incident_faces(v);
  Face_circulator done(fc);

  // We prepare for deleting all interior cells.
  // We ->set_cell() pointers to cells outside the hole.
  // We push the Edges_2D of the boundary (seen from outside) in "hole".
  do
  {
    Cell_handle f = fc;
    int i = f->index(v);
    Cell_handle fn = f->neighbor(i);
    int in = fn->index(f);

    f->vertex(cw(i))->set_cell(fn);
    fn->set_neighbor(in, Cell_handle());

    hole.push_back(Edge_2D(fn, in));
    remover.add_hidden_points(f);
    to_delete.push_back(f);

    ++fc;
  }
  while(fc != done);

  for(typename std::vector<Cell_handle>::const_iterator ib = to_delete.begin(),
                                                        iend = to_delete.end();
      ib != iend; ib++)
  {
    cells_set.erase(*ib);
  }

  tds().delete_cells(to_delete.begin(), to_delete.end());
  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
void
Triangulation_3<Gt, Tds, Lds>::
fill_hole_2D(std::list<Edge_2D>& first_hole, VertexRemover& remover)
{
  typedef std::list<Edge_2D> Hole;

  std::vector<Hole> hole_list;
  Cell_handle f, ff, fn;
  int i, ii, in;

  hole_list.push_back(first_hole);

  while(! hole_list.empty())
  {
    Hole hole = hole_list.back();
    hole_list.pop_back();

    // If the hole has only three edges, create the triangle
    if(hole.size() == 3)
    {
      typename Hole::iterator hit = hole.begin();
      f = (*hit).first;        i = (*hit).second;
      ff = (* ++hit).first;    ii = (*hit).second;
      fn = (* ++hit).first;    in = (*hit).second;
      tds().create_face(f, i, ff, ii, fn, in);
      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
    while(1)
    {
      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
      {
        break;
      }
    }

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

    Vertex_handle v0 = ff->vertex(cw(ii));
    Vertex_handle v1 = ff->vertex(ccw(ii));
    Vertex_handle v2 = infinite_vertex();
    const Point& p0 = v0->point();
    const Point& p1 = v1->point();
    const Point *p2 = nullptr; // Initialize to nullptr to avoid warning.

    typename Hole::iterator hdone = hole.end();
    typename Hole::iterator 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--;
    for(; hit != hdone; ++hit)
    {
      fn = hit->first;
      in = hit->second;
      Vertex_handle vv = fn->vertex(ccw(in));
      if(is_infinite(vv))
      {
        if(is_infinite(v2))
          cut_after = hit;
      }
      else // vv is a finite vertex
      {
        const Point& p = vv->point();
        if(coplanar_orientation(p0, p1, p) == COUNTERCLOCKWISE)
        {
          if(is_infinite(v2) ||
             remover.side_of_bounded_circle(p0, p1, *p2, p, true) == ON_BOUNDED_SIDE)
          {
            v2 = vv;
            p2 = &p;
            cut_after = hit;
          }
        }
      }
    }

    // Create new triangle and update adjacency relations
    Cell_handle newf;

    // Update the hole and push back in the Hole_List stack.
    // If v2 belongs to the neighbor following or preceding *f
    // the hole remains 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 == ccw(in))
    {
      newf = tds().create_face(ff, ii, fn, in);
      hole.pop_front();
      hole.push_front(Edge_2D(newf, 1));
      hole_list.push_back(hole);
    }
    else
    {
      fn = (hole.back()).first;
      in = (hole.back()).second;
      if(fn->has_vertex(v2, i) && i == cw(in))
      {
        newf = tds().create_face(fn, in, ff, ii);
        hole.pop_back();
        hole.push_back(Edge_2D(newf, 1));
        hole_list.push_back(hole);
      }
      else
      {
        // split the hole in two holes
        newf = tds().create_face(ff, ii, v2);
        Hole new_hole;
        ++cut_after;
        while(hole.begin() != cut_after)
        {
          new_hole.push_back(hole.front());
          hole.pop_front();
        }

        hole.push_front(Edge_2D(newf, 1));
        new_hole.push_front(Edge_2D(newf, 0));
        hole_list.push_back(hole);
        hole_list.push_back(new_hole);
      }
    }
  }
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover, class OutputItCells >
void
Triangulation_3<Gt, Tds, Lds>::
fill_hole_2D(std::list<Edge_2D>& first_hole, VertexRemover& remover, OutputItCells fit)
{
  typedef std::list<Edge_2D> Hole;

  std::vector<Hole> hole_list;
  Cell_handle f, ff, fn;
  int i, ii, in;

  hole_list.push_back(first_hole);

  while(! hole_list.empty())
  {
    Hole hole = hole_list.back();
    hole_list.pop_back();

    // If the hole has only three edges, create the triangle
    if(hole.size() == 3)
    {
      typename Hole::iterator hit = hole.begin();
      f = (*hit).first;
      i = (*hit).second;

      ff = (* ++hit).first;
      ii = (*hit).second;

      fn = (* ++hit).first;
      in = (*hit).second;

      *fit++ = tds().create_face(f, i, ff, ii, fn, in);
      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
    while(1)
    {
      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
      {
        break;
      }
    }

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

    Vertex_handle v0 = ff->vertex(cw(ii));
    Vertex_handle v1 = ff->vertex(ccw(ii));
    Vertex_handle v2 = infinite_vertex();
    const Point& p0 = v0->point();
    const Point& p1 = v1->point();
    const Point *p2 = nullptr; // Initialize to nullptr to avoid warning.

    typename Hole::iterator hdone = hole.end();
    typename Hole::iterator 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--;
    for(; hit != hdone; ++hit)
    {
      fn = hit->first;
      in = hit->second;
      Vertex_handle vv = fn->vertex(ccw(in));
      if(is_infinite(vv))
      {
        if(is_infinite(v2))
          cut_after = hit;
      }
      else // vv is a finite vertex
      {
        const Point& p = vv->point();
        if(coplanar_orientation(p0, p1, p) == COUNTERCLOCKWISE)
        {
          if(is_infinite(v2) ||
              remover.side_of_bounded_circle(p0, p1, *p2, p, true) == ON_BOUNDED_SIDE)
          {
            v2 = vv;
            p2 = &p;
            cut_after = hit;
          }
        }
      }
    }

    // create new triangle and update adjacency relations
    Cell_handle 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 == ccw(in))
    {
      newf = tds().create_face(ff, ii, fn, in);
      hole.pop_front();
      hole.push_front(Edge_2D(newf, 1));
      hole_list.push_back(hole);
    }
    else
    {
      fn = (hole.back()).first;
      in = (hole.back()).second;
      if(fn->has_vertex(v2, i) && i == cw(in))
      {
        newf = tds().create_face(fn, in, ff, ii);
        hole.pop_back();
        hole.push_back(Edge_2D(newf, 1));
        hole_list.push_back(hole);
      }
      else
      {
        // split the hole in two holes
        newf = tds().create_face(ff, ii, v2);
        Hole new_hole;
        ++cut_after;
        while(hole.begin() != cut_after)
        {
          new_hole.push_back(hole.front());
          hole.pop_front();
        }

        hole.push_front(Edge_2D(newf, 1));
        new_hole.push_front(Edge_2D(newf, 0));
        hole_list.push_back(hole);
        hole_list.push_back(new_hole);
      }
    }

    *fit++ = newf;
  }
}

template < class Gt, class Tds, class Lds >
void
Triangulation_3<Gt,Tds,Lds>::
make_hole_3D(Vertex_handle v,
             Vertex_triple_Facet_map& outer_map,
             std::vector<Cell_handle>& hole)
{
  CGAL_triangulation_expensive_precondition(! test_dim_down(v));

  incident_cells(v, std::back_inserter(hole));

  for(typename std::vector<Cell_handle>::iterator cit = hole.begin(),
                                                  end = hole.end();
      cit != end; ++cit)
  {
    int indv = (*cit)->index(v);
    Cell_handle opp_cit = (*cit)->neighbor(indv);
    Facet f(opp_cit, opp_cit->index(*cit));
    Vertex_triple vt = make_vertex_triple(f);
    make_canonical(vt);
    outer_map[vt] = f;
    for(int i=0; i<4; i++)
    {
      if(i != indv)
        (*cit)->vertex(i)->set_cell(opp_cit);
    }
  }
}

// When the incident cells are already known
template < class Gt, class Tds, class Lds >
void
Triangulation_3<Gt,Tds,Lds>::
make_hole_3D(Vertex_handle v,
             const std::vector<Cell_handle>& incident_cells,
             Vertex_triple_Facet_map& outer_map)
{
  CGAL_triangulation_expensive_precondition(! test_dim_down(v));

  for(typename std::vector<Cell_handle>::const_iterator cit = incident_cells.begin(),
       end = incident_cells.end(); cit != end; ++cit)
  {
    int indv = (*cit)->index(v);
    Cell_handle opp_cit = (*cit)->neighbor(indv);
    Facet f(opp_cit, opp_cit->index(*cit));
    Vertex_triple vt = make_vertex_triple(f);
    make_canonical(vt);
    outer_map[vt] = f;
    for(int i=0; i<4; i++)
    {
      if(i != indv)
        (*cit)->vertex(i)->set_cell(opp_cit);
    }
  }
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt,Tds,Lds>::
remove_dim_down(Vertex_handle v, VertexRemover& remover)
{
  CGAL_triangulation_precondition (dimension() >= 0);

  // Collect all the hidden points.
  for(All_cells_iterator ci = tds().raw_cells_begin(),
                         end = tds().raw_cells_end(); ci != end; ++ci)
    remover.add_hidden_points(ci);

  tds().remove_decrease_dimension(v, infinite_vertex());

  // Now try to see if we need to re-orient.
  if(dimension() == 2)
  {
    Facet f = *finite_facets_begin();
    if(coplanar_orientation(f.first->vertex(0)->point(),
                            f.first->vertex(1)->point(),
                            f.first->vertex(2)->point()) == NEGATIVE)
    {
      tds().reorient();
    }
  }

  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt,Tds,Lds>::
remove_1D(Vertex_handle v, VertexRemover& remover)
{
  CGAL_triangulation_precondition (dimension() == 1);

  Cell_handle c1 = v->cell();
  Cell_handle c2 = c1->neighbor(c1->index(v) == 0 ? 1 : 0);
  remover.add_hidden_points(c1);
  remover.add_hidden_points(c2);

  tds().remove_from_maximal_dimension_simplex (v);

  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt,Tds,Lds>::
remove_2D(Vertex_handle v, VertexRemover& remover)
{
  CGAL_triangulation_precondition(dimension() == 2);
  std::list<Edge_2D> hole;
  make_hole_2D(v, hole, remover);
  fill_hole_2D(hole, remover);
  tds().delete_vertex(v);
  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt,Tds,Lds>::
remove_3D(Vertex_handle v, VertexRemover& remover)
{
  std::vector<Cell_handle> hole;
  hole.reserve(64);

  // Construct the set of vertex triples on the boundary
  // with the facet just behind
  Vertex_triple_Facet_map outer_map;
  Vertex_triple_Facet_map inner_map;

  make_hole_3D(v, outer_map, hole);
  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());

  // Output the hidden points.
  for(typename std::vector<Cell_handle>::iterator hi = hole.begin(),
                                                  hend = hole.end();
      hi != hend; ++hi)
  {
    remover.add_hidden_points(*hi);
  }

  bool inf = false;

  // collect all vertices on the boundary
  std::vector<Vertex_handle> vertices;
  vertices.reserve(64);
  adjacent_vertices(v, std::back_inserter(vertices));

  // create a Delaunay triangulation of the points on the boundary
  // and make a map from the vertices in remover.tmp towards the vertices
  // in *this

  unsigned int i = 0;
  Vertex_handle_unique_hash_map vmap;
  Cell_handle ch = Cell_handle();
#ifdef CGAL_TRIANGULATION_3_USE_THE_4_POINTS_CONSTRUCTOR
  size_t num_vertices = vertices.size();
  if(num_vertices >= 5)
  {
    for(int j = 0 ; j < 4 ; ++j)
    {
      if(is_infinite(vertices[j]))
      {
        std::swap(vertices[j], vertices[4]);
        break;
      }
    }

    Orientation o = orientation(vertices[0]->point(),
                                vertices[1]->point(),
                                vertices[2]->point(),
                                vertices[3]->point());

    if(o == NEGATIVE)
      std::swap(vertices[0], vertices[1]);

    if(o != ZERO)
    {
      Vertex_handle vh1, vh2, vh3, vh4;
      remover.tmp.init_tds(vertices[0]->point(), vertices[1]->point(),
                           vertices[2]->point(), vertices[3]->point(),
                           vh1, vh2, vh3, vh4);
      ch = vh1->cell();
      vmap[vh1] = vertices[0];
      vmap[vh2] = vertices[1];
      vmap[vh3] = vertices[2];
      vmap[vh4] = vertices[3];
      i = 4;
    }
  }
#endif

  for(; i < vertices.size(); i++)
  {
    if(! is_infinite(vertices[i]))
    {
      Vertex_handle vh = remover.tmp.insert(vertices[i]->point(), ch);
      ch = vh->cell();
      vmap[vh] = vertices[i];
    }
    else
    {
      inf = true;
    }
  }

  if(remover.tmp.dimension() == 2)
  {
    Vertex_handle fake_inf = remover.tmp.insert(v->point());
    vmap[fake_inf] = infinite_vertex();
  }
  else
  {
    vmap[remover.tmp.infinite_vertex()] = infinite_vertex();
  }

  CGAL_triangulation_assertion(remover.tmp.dimension() == 3);

  // Construct the set of vertex triples of remover.tmp
  // We reorient the vertex triple so that it matches those from outer_map
  // Also note that we use the vertices of *this, not of remover.tmp

  if(inf)
  {
    for(All_cells_iterator it = remover.tmp.all_cells_begin(),
                           end = remover.tmp.all_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }
  else
  {
    for(Finite_cells_iterator it = remover.tmp.finite_cells_begin(),
                              end = remover.tmp.finite_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }
  // Grow inside the hole, by extending the surface
  while(! outer_map.empty())
  {
    typename Vertex_triple_Facet_map::iterator oit = outer_map.begin();
    while(is_infinite(oit->first.first) ||
          is_infinite(oit->first.second) ||
          is_infinite(oit->first.third))
    {
      ++oit;
      // Otherwise the lookup in the inner_map fails
      // because the infinite vertices are different
    }

    typename Vertex_triple_Facet_map::value_type o_vt_f_pair = *oit;
    Cell_handle o_ch = o_vt_f_pair.second.first;
    unsigned int o_i = o_vt_f_pair.second.second;

    typename Vertex_triple_Facet_map::iterator iit = inner_map.find(o_vt_f_pair.first);
    CGAL_triangulation_assertion(iit != inner_map.end());
    typename Vertex_triple_Facet_map::value_type i_vt_f_pair = *iit;
    Cell_handle i_ch = i_vt_f_pair.second.first;
    unsigned int i_i = i_vt_f_pair.second.second;

    // Create a new cell and glue it to the outer surface
    Cell_handle new_ch = tds().create_cell();
    new_ch->set_vertices(vmap[i_ch->vertex(0)], vmap[i_ch->vertex(1)],
                         vmap[i_ch->vertex(2)], vmap[i_ch->vertex(3)]);

    o_ch->set_neighbor(o_i,new_ch);
    new_ch->set_neighbor(i_i, o_ch);

    // For the other faces check, if they can also be glued
    for(i = 0; i < 4; i++)
    {
      if(i != i_i)
      {
        Facet f = std::pair<Cell_handle,int>(new_ch,i);
        Vertex_triple vt = make_vertex_triple(f);
        make_canonical(vt);
        std::swap(vt.second,vt.third);

        typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
        if(oit2 == outer_map.end())
        {
          std::swap(vt.second,vt.third);
          outer_map[vt]= f;
        }
        else
        {
          // glue the faces
          typename Vertex_triple_Facet_map::value_type o_vt_f_pair2 = *oit2;
          Cell_handle o_ch2 = o_vt_f_pair2.second.first;
          int o_i2 = o_vt_f_pair2.second.second;
          o_ch2->set_neighbor(o_i2,new_ch);
          new_ch->set_neighbor(i, o_ch2);
          outer_map.erase(oit2);
        }
      }
    }
    outer_map.erase(oit);
  }
  tds().delete_vertex(v);
  tds().delete_cells(hole.begin(), hole.end());

  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
VertexRemover&
Triangulation_3<Gt,Tds,Lds>::
remove_3D(Vertex_handle v, VertexRemover& remover,
          const std::vector<Cell_handle>& inc_cells,
          std::vector<Vertex_handle>& adj_vertices)
{
  // Construct the set of vertex triples on the boundary with the facet just behind
  Vertex_triple_Facet_map outer_map;
  Vertex_triple_Facet_map inner_map;

  make_hole_3D(v, inc_cells, outer_map);

  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());

  // Output the hidden points.
  for(typename std::vector<Cell_handle>::const_iterator hi = inc_cells.begin(),
                                                        hend = inc_cells.end();
      hi != hend; ++hi)
  {
    remover.add_hidden_points(*hi);
  }

  bool inf = false;

  // Create a Delaunay triangulation of the points on the boundary
  // and make a map from the vertices in remover.tmp towards the vertices
  // in *this
  unsigned int i = 0;
  Vertex_handle_unique_hash_map vmap;
  Cell_handle ch = Cell_handle();
#ifdef CGAL_TRIANGULATION_3_USE_THE_4_POINTS_CONSTRUCTOR
  size_t num_vertices = adj_vertices.size();
  if(num_vertices >= 5)
  {
    for(int j = 0 ; j < 4 ; ++j)
    {
      if(is_infinite(adj_vertices[j]))
      {
        std::swap(adj_vertices[j], adj_vertices[4]);
        break;
      }
    }

    Orientation o = orientation(adj_vertices[0]->point(),
                                adj_vertices[1]->point(),
                                adj_vertices[2]->point(),
                                adj_vertices[3]->point());

    if(o == NEGATIVE)
      std::swap(adj_vertices[0], adj_vertices[1]);

    if(o != ZERO)
    {
      Vertex_handle vh1, vh2, vh3, vh4;
      remover.tmp.init_tds(adj_vertices[0]->point(), adj_vertices[1]->point(),
                           adj_vertices[2]->point(), adj_vertices[3]->point(),
                           vh1, vh2, vh3, vh4);

      ch = vh1->cell();
      vmap[vh1] = adj_vertices[0];
      vmap[vh2] = adj_vertices[1];
      vmap[vh3] = adj_vertices[2];
      vmap[vh4] = adj_vertices[3];
      i = 4;
    }
  }
#endif

  for(; i < adj_vertices.size(); i++)
  {
    if(! is_infinite(adj_vertices[i]))
    {
      Vertex_handle vh = remover.tmp.insert(adj_vertices[i]->point(), ch);
      ch = vh->cell();
      vmap[vh] = adj_vertices[i];
    }
    else
    {
      inf = true;
    }
  }

  if(remover.tmp.dimension()==2)
  {
    Vertex_handle fake_inf = remover.tmp.insert(v->point());
    vmap[fake_inf] = infinite_vertex();
  }
  else
  {
    vmap[remover.tmp.infinite_vertex()] = infinite_vertex();
  }

  CGAL_triangulation_assertion(remover.tmp.dimension() == 3);

  // Construct the set of vertex triples of remover.tmp
  // We reorient the vertex triple so that it matches those from outer_map
  // Also note that we use the vertices of *this, not of remover.tmp
  if(inf)
  {
    for(All_cells_iterator it = remover.tmp.all_cells_begin(),
                           end = remover.tmp.all_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }
  else
  {
    for(Finite_cells_iterator it = remover.tmp.finite_cells_begin(),
                              end = remover.tmp.finite_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }

  // Grow inside the hole, by extending the surface
  while(! outer_map.empty())
  {
    typename Vertex_triple_Facet_map::iterator oit = outer_map.begin();
    while(is_infinite(oit->first.first) ||
          is_infinite(oit->first.second) ||
          is_infinite(oit->first.third))
    {
      ++oit;
      // otherwise the lookup in the inner_map fails
      // because the infinite vertices are different
    }

    typename Vertex_triple_Facet_map::value_type o_vt_f_pair = *oit;
    Cell_handle o_ch = o_vt_f_pair.second.first;
    unsigned int o_i = o_vt_f_pair.second.second;

    typename Vertex_triple_Facet_map::iterator iit =
      inner_map.find(o_vt_f_pair.first);
    CGAL_triangulation_assertion(iit != inner_map.end());
    typename Vertex_triple_Facet_map::value_type i_vt_f_pair = *iit;
    Cell_handle i_ch = i_vt_f_pair.second.first;
    unsigned int i_i = i_vt_f_pair.second.second;

    // create a new cell and glue it to the outer surface
    Cell_handle new_ch = tds().create_cell();
    new_ch->set_vertices(vmap[i_ch->vertex(0)], vmap[i_ch->vertex(1)],
                         vmap[i_ch->vertex(2)], vmap[i_ch->vertex(3)]);

    o_ch->set_neighbor(o_i,new_ch);
    new_ch->set_neighbor(i_i, o_ch);

    // for the other faces check, if they can also be glued
    for(i = 0; i < 4; i++)
    {
      if(i != i_i)
      {
        Facet f = std::pair<Cell_handle,int>(new_ch,i);
        Vertex_triple vt = make_vertex_triple(f);
        make_canonical(vt);
        std::swap(vt.second,vt.third);

        typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
        if(oit2 == outer_map.end())
        {
          std::swap(vt.second,vt.third);
          outer_map[vt]= f;
        }
        else
        {
          // glue the faces
          typename Vertex_triple_Facet_map::value_type o_vt_f_pair2 = *oit2;
          Cell_handle o_ch2 = o_vt_f_pair2.second.first;
          int o_i2 = o_vt_f_pair2.second.second;
          o_ch2->set_neighbor(o_i2,new_ch);
          new_ch->set_neighbor(i, o_ch2);
          outer_map.erase(oit2);
        }
      }
    }
    outer_map.erase(oit);
  }
  tds().delete_vertex(v);
  tds().delete_cells(inc_cells.begin(), inc_cells.end());

  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
void
Triangulation_3<Gt, Tds, Lds>::
remove(Vertex_handle v, VertexRemover& remover)
{
  CGAL_triangulation_precondition(v != Vertex_handle());
  CGAL_triangulation_precondition(!is_infinite(v));
  CGAL_triangulation_expensive_precondition(tds().is_vertex(v));

  if(test_dim_down (v))
  {
    remove_dim_down (v, remover);
  }
  else
  {
    switch(dimension())
    {
      case 1: remove_1D (v, remover);
        break;
      case 2: remove_2D (v, remover);
        break;
      case 3: remove_3D (v, remover);
        break;
      default:
        CGAL_triangulation_assertion (false);
    }
  }
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover >
bool
Triangulation_3<Gt, Tds, Lds>::
remove(Vertex_handle v, VertexRemover& remover, bool *could_lock_zone)
{
  // N.B.: dimension doesn't need to be atomic since the parallel removal
  //       will never decrease the dimension (the last few removes are done
  //       sequentially)
  CGAL_triangulation_precondition(v != Vertex_handle());
  CGAL_triangulation_precondition(!is_infinite(v));
  CGAL_triangulation_precondition(dimension() == 3);
  CGAL_triangulation_expensive_precondition(tds().is_vertex(v));

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
  static Profile_branch_counter_3 bcounter(
        "early withdrawals / late withdrawals / successes [Delaunay_tri_3::remove]");
#endif

  bool removed = true;

  // Locking vertex v is a good start
  if(!this->try_lock_vertex(v))
  {
    *could_lock_zone = false;
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
    bcounter.increment_branch_2(); // THIS is an early withdrawal!
#endif
  }
  else
  {
    std::vector<Cell_handle> incident_cells;
    incident_cells.reserve(64);
    std::vector<Vertex_handle> adj_vertices;
    adj_vertices.reserve(64);
    bool dim_down = test_dim_down_using_incident_cells_3(
                      v, incident_cells, adj_vertices, could_lock_zone);

    if(*could_lock_zone)
    {
      if(dim_down)
        removed = false;
      else
        remove_3D (v, remover, incident_cells, adj_vertices);
    }
  }

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
  if(could_lock_zone)
  {
    if(*could_lock_zone)
      ++bcounter;
    else
      bcounter.increment_branch_1(); // THIS is a late withdrawal!
  }
#endif

  return removed;
}

// The remove here uses the remover, but
// no function envolving hidden points
// will be used in this internal version
template < class Gt, class Tds, class Lds >
template < class VertexRemover, class OutputItCells >
VertexRemover&
Triangulation_3<Gt, Tds, Lds>::
remove_dim_down(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
{
  remove_dim_down(v, remover);
  for(All_cells_iterator afi = tds().raw_cells_begin();
                         afi != tds().raw_cells_end(); afi++)
  {
    *fit++ = afi;
  }
  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover, class OutputItCells >
VertexRemover&
Triangulation_3<Gt, Tds, Lds>::
remove_1D(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
{
  Point p = v->point();
  remove_1D(v, remover);
  *fit++ = locate(p);
  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover, class OutputItCells >
VertexRemover&
Triangulation_3<Gt, Tds, Lds>::
remove_2D(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
{
  CGAL_triangulation_precondition(dimension() == 2);
  std::list<Edge_2D> hole;
  make_hole_2D(v, hole, remover);
  fill_hole_2D(hole, remover, fit);
  tds().delete_vertex(v);
  return remover;
}

template < class Gt, class Tds, class Lds >
template < class VertexRemover, class OutputItCells >
VertexRemover&
Triangulation_3<Gt, Tds, Lds>::
remove_3D(Vertex_handle v, VertexRemover& remover, OutputItCells fit)
{
  CGAL_triangulation_precondition(dimension() == 3);

  std::vector<Cell_handle> hole;
  hole.reserve(64);

  // Construct the set of vertex triples on the boundary
  // with the facet just behind
  Vertex_triple_Facet_map outer_map;
  Vertex_triple_Facet_map inner_map;

  make_hole_3D(v, outer_map, hole);

  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());

  // Output the hidden points.
  for(typename std::vector<Cell_handle>::iterator hi = hole.begin(),
                                                  hend = hole.end();
      hi != hend; ++hi)
  {
    remover.add_hidden_points(*hi);
  }

  bool inf = false;
  unsigned int i;

  // collect all vertices on the boundary
  std::vector<Vertex_handle> vertices;
  vertices.reserve(64);
  adjacent_vertices(v, std::back_inserter(vertices));

  // create a Delaunay triangulation of the points on the boundary
  // and make a map from the vertices in remover.tmp towards the vertices
  // in *this
  Vertex_handle_unique_hash_map vmap;
  Cell_handle ch = Cell_handle();
  for(i=0; i<vertices.size(); i++)
  {
    if(! is_infinite(vertices[i]))
    {
      Vertex_handle vh = remover.tmp.insert(vertices[i]->point(), ch);
      ch = vh->cell();
      vmap[vh] = vertices[i];
    }
    else
    {
      inf = true;
    }
  }

  if(remover.tmp.dimension()==2)
  {
    Vertex_handle fake_inf = remover.tmp.insert(v->point());
    vmap[fake_inf] = infinite_vertex();
  }
  else
  {
    vmap[remover.tmp.infinite_vertex()] = infinite_vertex();
  }

  CGAL_triangulation_assertion(remover.tmp.dimension() == 3);

  // Construct the set of vertex triples of remover.tmp
  // We reorient the vertex triple so that it matches those from outer_map
  // Also note that we use the vertices of *this, not of remover.tmp

  if(inf)
  {
    for(All_cells_iterator it = remover.tmp.all_cells_begin(),
                           end = remover.tmp.all_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt] = f;
      }
    }
  } else
  {
    for(Finite_cells_iterator it = remover.tmp.finite_cells_begin(),
                              end = remover.tmp.finite_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt] = f;
      }
    }
  }

  // Grow inside the hole, by extending the surface
  while(! outer_map.empty())
  {
    typename Vertex_triple_Facet_map::iterator oit = outer_map.begin();
    while(is_infinite(oit->first.first) ||
          is_infinite(oit->first.second) ||
          is_infinite(oit->first.third))
    {
      ++oit;
      // otherwise the lookup in the inner_map fails
      // because the infinite vertices are different
    }

    typename Vertex_triple_Facet_map::value_type o_vt_f_pair = *oit;
    Cell_handle o_ch = o_vt_f_pair.second.first;
    unsigned int o_i = o_vt_f_pair.second.second;

    typename Vertex_triple_Facet_map::iterator iit =
        inner_map.find(o_vt_f_pair.first);
    CGAL_triangulation_assertion(iit != inner_map.end());
    typename Vertex_triple_Facet_map::value_type i_vt_f_pair = *iit;
    Cell_handle i_ch = i_vt_f_pair.second.first;
    unsigned int i_i = i_vt_f_pair.second.second;

    // create a new cell and glue it to the outer surface
    Cell_handle new_ch = tds().create_cell();
    *fit++ = new_ch;

    new_ch->set_vertices(vmap[i_ch->vertex(0)], vmap[i_ch->vertex(1)],
                         vmap[i_ch->vertex(2)], vmap[i_ch->vertex(3)]);

    o_ch->set_neighbor(o_i,new_ch);
    new_ch->set_neighbor(i_i, o_ch);

    // for the other faces check, if they can also be glued
    for(i = 0; i < 4; i++)
    {
      if(i != i_i)
      {
        Facet f = std::pair<Cell_handle,int>(new_ch,i);
        Vertex_triple vt = make_vertex_triple(f);
        make_canonical(vt);
        std::swap(vt.second, vt.third);
        typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
        if(oit2 == outer_map.end())
        {
          std::swap(vt.second, vt.third);
          outer_map[vt]= f;
        }
        else
        {
          // glue the faces
          typename Vertex_triple_Facet_map::value_type o_vt_f_pair2 = *oit2;
          Cell_handle o_ch2 = o_vt_f_pair2.second.first;
          int o_i2 = o_vt_f_pair2.second.second;
          o_ch2->set_neighbor(o_i2, new_ch);
          new_ch->set_neighbor(i, o_ch2);
          outer_map.erase(oit2);
        }
      }
    }
    outer_map.erase(oit);
  }
  tds().delete_vertex(v);
  tds().delete_cells(hole.begin(), hole.end());

  return remover;
}


template < class Gt, class Tds, class Lds >
template < class VertexRemover, class OutputItCells >
void
Triangulation_3<Gt, Tds, Lds>::
remove_and_give_new_cells(Vertex_handle v, VertexRemover& remover,
                          OutputItCells fit)
{
  CGAL_triangulation_precondition(v != Vertex_handle());
  CGAL_triangulation_precondition(!is_infinite(v));
  CGAL_triangulation_expensive_precondition(tds().is_vertex(v));

  if(test_dim_down (v))
  {
    remove_dim_down (v, remover, fit);
  }
  else
  {
    switch(dimension())
    {
      case 1: remove_1D (v, remover, fit);
        break;
      case 2: remove_2D (v, remover, fit);
        break;
      case 3: remove_3D (v, remover, fit);
        break;
      default:
        CGAL_triangulation_assertion (false);
    }
  }
}

// The VertexInserter is needed so as to
// allow us the usage of the insertion method
// from the particular triangulation
template < class Gt, class Tds, class Lds >
template < class VertexRemover, class VertexInserter >
typename Triangulation_3<Gt,Tds,Lds>::Vertex_handle
Triangulation_3<Gt,Tds,Lds>::
move_if_no_collision(Vertex_handle v, const Point& p,
                     VertexRemover& remover, VertexInserter& inserter)
{
  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());
  CGAL_triangulation_precondition(!is_infinite(v));
  if(v->point() == p)
    return v;
  const int dim = dimension();

  // If displacements are known to be small, we might want to optimize by checking
  // whether there is a topological change or not before.
  // In this version, this will not be put inside this method because
  // it is for general purposes, and remaining Delaunay after motion
  // is a bit too restrictive.
  //
  // In the filtered version optimized for displacements, it will be used
  // as an a priori. However, a non-fully optimized but good version of
  // is_delaunay_after_displacement is provided as an internal method of
  // Delaunay_triangulation_3 (see the class for more details).

  Locate_type lt;
  int li, lj;
  Cell_handle loc = locate(p, lt, li, lj, v->cell());

  if(lt == VERTEX)
    return loc->vertex(li);

  if(dim == 0)
  {
    v->set_point(p);
    return v;
  }

  size_type n_vertices = tds().number_of_vertices();

  if((lt == OUTSIDE_AFFINE_HULL) && (dim == 1) && (n_vertices == 3))
  {
    v->set_point(p);
    return v;
  }

  if((lt == OUTSIDE_AFFINE_HULL) && (dim == 2) && (n_vertices == 4))
  {
    v->set_point(p);
    return v;
  }

  if((lt != OUTSIDE_AFFINE_HULL) && (dim == 1))
  {
    if(loc->has_vertex(v))
    {
      v->set_point(p);
    }
    else
    {
      Vertex_handle inserted = insert(p, lt, loc, li, lj);
      Cell_handle f = v->cell();
      int i = f->index(v);
      if(i == 0)
        f = f->neighbor(1);

      CGAL_triangulation_assertion(f->index(v) == 1);
      Cell_handle g= f->neighbor(0);
      f->set_vertex(1, g->vertex(1));
      f->set_neighbor(0,g->neighbor(0));
      g->neighbor(0)->set_neighbor(1,f);
      g->vertex(1)->set_cell(f);
      tds().delete_cell(g);
      Cell_handle f_ins = inserted->cell();
      i = f_ins->index(inserted);
      if(i == 0)
        f_ins = f_ins->neighbor(1);

      CGAL_triangulation_assertion(f_ins->index(inserted) == 1);
      Cell_handle g_ins = f_ins->neighbor(0);
      f_ins->set_vertex(1, v);
      g_ins->set_vertex(0, v);
      v->set_point(p);
      v->set_cell(inserted->cell());
      tds().delete_vertex(inserted);
    }
    return v;
  }

  bool dim_down = test_dim_down(v);
  if((lt != OUTSIDE_AFFINE_HULL) && dim_down && (dim == 2))
  {
    // verify if p and two static vertices are collinear in this case
    int iinf;
    Cell_handle finf = infinite_vertex()->cell(), fdone;
    fdone = finf;
    do
    {
      iinf = finf->index(infinite_vertex());
      if(!finf->has_vertex(v))
        break;

      finf = finf->neighbor((iinf+1)%3);
    }
    while(finf != fdone);

    iinf = ~iinf;
    if(this->collinear(finf->vertex(iinf&1)->point(),
                       finf->vertex(iinf&2)->point(),
                       p))
    {
      v->set_point(p);
      _tds.decrease_dimension(loc, loc->index(v));
      return v;
    }
  }

  if(((dim == 2) && (lt != OUTSIDE_AFFINE_HULL)) ||
     ((lt == OUTSIDE_AFFINE_HULL) && (dim == 1)))
  {
    // This is insert must be from Delaunay (or the particular trian.)
    // not Triangulation_3 !
    Vertex_handle inserted = inserter.insert(p, lt, loc, li, lj);

    std::list<Edge_2D> hole;
    make_hole_2D(v, hole, remover);
    fill_hole_2D(hole, remover);

    // fixing pointer
    Cell_handle fc = inserted->cell(), done(fc);
    std::vector<Cell_handle> faces_pt;
    faces_pt.reserve(16);
    do
    {
      faces_pt.push_back(fc);
      fc = fc->neighbor((fc->index(inserted) + 1)%3);
    }
    while(fc != done);

    std::size_t ss = faces_pt.size();
    for(std::size_t k=0; k<ss; k++)
    {
      Cell_handle f = faces_pt[k];
      int i = f->index(inserted);
      f->set_vertex(i, v);
    }
    v->set_point(p);
    v->set_cell(inserted->cell());

    tds().delete_vertex(inserted);

    return v;
  }

  if((lt != OUTSIDE_AFFINE_HULL) && dim_down && (dim == 3))
  {
    // verify if p and two static vertices are collinear in this case
    std::vector<Cell_handle> ics;
    incident_cells(infinite_vertex(), std::back_inserter(ics));
    std::size_t size = ics.size();
    Cell_handle finf;
    for(std::size_t i=0; i<size; i++)
    {
      finf = ics[i];
      if(!finf->has_vertex(v))
        break;
    }

    int iinf = finf->index(infinite_vertex());
    if(remover.tmp.coplanar(finf->vertex((iinf+1)&3)->point(),
                            finf->vertex((iinf+2)&3)->point(),
                            finf->vertex((iinf+3)&3)->point(),
                            p))
    {
      v->set_point(p);
      _tds.decrease_dimension(loc, loc->index(v));
      Facet f = *finite_facets_begin();
      if(coplanar_orientation(f.first->vertex(0)->point(),
                               f.first->vertex(1)->point(),
                               f.first->vertex(2)->point()) == NEGATIVE)
      {
        tds().reorient();
      }

      restore_edges_after_decrease_dimension(v, remover,inserter);
      return v;
    }
  }

  // This is insert must be from Delaunay (or the particular trian.)
  // not Triangulation_3 !
  Vertex_handle inserted = inserter.insert(p, lt, loc, li, lj);

  std::vector<Cell_handle> hole;
  hole.reserve(64);

  // Construct the set of vertex triples on the boundary
  // with the facet just behind
  Vertex_triple_Facet_map outer_map;
  Vertex_triple_Facet_map inner_map;

  make_hole_3D(v, outer_map, hole);

  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());

  // Output the hidden points.
  for(typename std::vector<Cell_handle>::iterator hi = hole.begin(),
                                                  hend = hole.end(); hi != hend; ++hi)
  {
    remover.add_hidden_points(*hi);
  }

  bool inf = false;
  unsigned int i;

  // collect all vertices on the boundary
  std::vector<Vertex_handle> vertices;
  vertices.reserve(64);
  adjacent_vertices(v, std::back_inserter(vertices));

  // create a Delaunay triangulation of the points on the boundary
  // and make a map from the vertices in remover.tmp towards the vertices
  // in *this
  Vertex_handle_unique_hash_map vmap;
  Cell_handle ch = Cell_handle();
  for(i=0; i < vertices.size(); i++)
  {
    if(! is_infinite(vertices[i]))
    {
      Vertex_handle vh = remover.tmp.insert(vertices[i]->point(), ch);
      ch = vh->cell();
      vmap[vh] = vertices[i];
    }
    else
    {
      inf = true;
    }
  }

  if(remover.tmp.dimension() == 2)
  {
    Vertex_handle fake_inf = remover.tmp.insert(v->point());
    vmap[fake_inf] = infinite_vertex();
  }
  else
  {
    vmap[remover.tmp.infinite_vertex()] = infinite_vertex();
  }

  CGAL_triangulation_assertion(remover.tmp.dimension() == 3);

  // Construct the set of vertex triples of remover.tmp
  // We reorient the vertex triple so that it matches those from outer_map
  // Also note that we use the vertices of *this, not of remover.tmp

  if(inf)
  {
    for(All_cells_iterator it = remover.tmp.all_cells_begin(),
                           end = remover.tmp.all_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }
  else
  {
    for(Finite_cells_iterator it = remover.tmp.finite_cells_begin(),
                              end = remover.tmp.finite_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first],vmap[vt_aux.third],vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }

  // Grow inside the hole, by extending the surface
  while(! outer_map.empty())
  {
    typename Vertex_triple_Facet_map::iterator oit = outer_map.begin();
    while(is_infinite(oit->first.first) ||
          is_infinite(oit->first.second) ||
          is_infinite(oit->first.third))
    {
      ++oit;
      // otherwise the lookup in the inner_map fails
      // because the infinite vertices are different
    }

    typename Vertex_triple_Facet_map::value_type o_vt_f_pair = *oit;
    Cell_handle o_ch = o_vt_f_pair.second.first;
    unsigned int o_i = o_vt_f_pair.second.second;

    typename Vertex_triple_Facet_map::iterator iit =
        inner_map.find(o_vt_f_pair.first);
    CGAL_triangulation_assertion(iit != inner_map.end());
    typename Vertex_triple_Facet_map::value_type i_vt_f_pair = *iit;
    Cell_handle i_ch = i_vt_f_pair.second.first;
    unsigned int i_i = i_vt_f_pair.second.second;

    // create a new cell and glue it to the outer surface
    Cell_handle new_ch = tds().create_cell();

    new_ch->set_vertices(vmap[i_ch->vertex(0)], vmap[i_ch->vertex(1)],
                         vmap[i_ch->vertex(2)], vmap[i_ch->vertex(3)]);

    o_ch->set_neighbor(o_i,new_ch);
    new_ch->set_neighbor(i_i, o_ch);

    // for the other faces check, if they can also be glued
    for(i = 0; i < 4; i++)
    {
      if(i != i_i)
      {
        Facet f = std::pair<Cell_handle,int>(new_ch,i);
        Vertex_triple vt = make_vertex_triple(f);
        make_canonical(vt);
        std::swap(vt.second,vt.third);
        typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
        if(oit2 == outer_map.end())
        {
          std::swap(vt.second,vt.third);
          outer_map[vt]= f;
        }
        else
        {
          // glue the faces
          typename Vertex_triple_Facet_map::value_type o_vt_f_pair2 = *oit2;
          Cell_handle o_ch2 = o_vt_f_pair2.second.first;
          int o_i2 = o_vt_f_pair2.second.second;
          o_ch2->set_neighbor(o_i2,new_ch);
          new_ch->set_neighbor(i, o_ch2);
          outer_map.erase(oit2);
        }
      }
    }
    outer_map.erase(oit);
  }

  // fixing pointer
  std::vector<Cell_handle> cells_pt;
  cells_pt.reserve(64);
  incident_cells(inserted, std::back_inserter(cells_pt));
  std::size_t size = cells_pt.size();
  for(std::size_t i=0; i<size; i++)
  {
    Cell_handle c = cells_pt[i];
    c->set_vertex(c->index(inserted), v);
  }

  v->set_point(p);
  v->set_cell(inserted->cell());
  tds().delete_vertex(inserted);
  tds().delete_cells(hole.begin(), hole.end());
  return v;
} // end of Vertex_handle

template < class Gt, class Tds, class Lds >
template < class VertexRemover, class VertexInserter >
typename Triangulation_3<Gt,Tds,Lds>::Vertex_handle
Triangulation_3<Gt,Tds,Lds>::
move(Vertex_handle v, const Point& p,
     VertexRemover& remover, VertexInserter& inserter)
{
  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());
  CGAL_triangulation_precondition(!is_infinite(v));

  if(v->point() == p)
    return v;

  Vertex_handle w = move_if_no_collision(v,p,remover,inserter);
  if(w != v)
  {
    remove(v, remover);
    return w;
  }

  return v;
}

// The VertexInserter is needed so as to allow us the usage of the insertion method
// from the particular triangulation
template < class Gt, class Tds, class Lds >
template < class VertexRemover, class VertexInserter, class OutputItCells >
typename Triangulation_3<Gt,Tds,Lds>::Vertex_handle
Triangulation_3<Gt,Tds,Lds>::
move_if_no_collision_and_give_new_cells(Vertex_handle v, const Point& p,
                                        VertexRemover& remover, VertexInserter& inserter,
                                        OutputItCells fit)
{
  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());
  CGAL_triangulation_precondition(!is_infinite(v));

  if(v->point() == p)
    return v;

  const int dim = dimension();

  // If displacements are known to be small, we might want to optimize by checking
  // whether there is a topological change or not before.
  //
  // In this version this will not be put inside this method because it is
  // for general purposes, and remaining Delaunay after motion is a bit
  // too restrictive.
  //
  // In the filtered version optimized for displacements, it will be used
  // as an a priori. However, a non-fully optimized but good version of
  // is_delaunay_after_displacement is provided as an internal method of
  // Delaunay_triangulation_3 (see the class for more details).

  Locate_type lt;
  int li, lj;
  Cell_handle loc = locate(p, lt, li, lj, v->cell());

  if(lt == VERTEX) return loc->vertex(li);

  if(dim == 0)
  {
    v->set_point(p);
    return v;
  }

  int n_vertices = tds().number_of_vertices();

  if((lt == OUTSIDE_AFFINE_HULL) && (dim == 1) && (n_vertices == 3))
  {
    v->set_point(p);
    for(All_cells_iterator afi = tds().raw_cells_begin();
                           afi != tds().raw_cells_end(); afi++)
    {
      *fit++ = afi;
    }
    return v;
  }

  if((lt == OUTSIDE_AFFINE_HULL) && (dim == 2) && (n_vertices == 4))
  {
    v->set_point(p);
    for(All_cells_iterator afi = tds().raw_cells_begin();
                           afi != tds().raw_cells_end(); afi++)
    {
      *fit++ = afi;
    }
    return v;
  }

  if((lt != OUTSIDE_AFFINE_HULL) && (dim == 1))
  {
    if(loc->has_vertex(v))
    {
      v->set_point(p);
    }
    else
    {
      Vertex_handle inserted = insert(p, lt, loc, li, lj);
      Cell_handle f = v->cell();
      int i = f->index(v);
      if(i==0)
        f = f->neighbor(1);

      CGAL_triangulation_assertion(f->index(v) == 1);
      Cell_handle g = f->neighbor(0);
      f->set_vertex(1, g->vertex(1));
      f->set_neighbor(0,g->neighbor(0));
      g->neighbor(0)->set_neighbor(1,f);
      g->vertex(1)->set_cell(f);
      tds().delete_cell(g);
      *fit++ = f;
      Cell_handle f_ins = inserted->cell();
      i = f_ins->index(inserted);
      if(i==0)
        f_ins = f_ins->neighbor(1);

      CGAL_triangulation_assertion(f_ins->index(inserted) == 1);
      Cell_handle g_ins = f_ins->neighbor(0);
      f_ins->set_vertex(1, v);
      g_ins->set_vertex(0, v);
      v->set_point(p);
      v->set_cell(inserted->cell());
      tds().delete_vertex(inserted);
    }

    *fit++ = v->cell();
    if(v->cell()->neighbor(0)->has_vertex(v))
      *fit++ = v->cell()->neighbor(0);

    if(v->cell()->neighbor(1)->has_vertex(v))
      *fit++ = v->cell()->neighbor(1);

    return v;
  }

  bool dim_down = test_dim_down(v);

  if((lt != OUTSIDE_AFFINE_HULL) && dim_down && (dim == 2))
  {
    // Verify if p and two static vertices are collinear in this case
    int iinf;
    Cell_handle finf = infinite_vertex()->cell(), fdone;
    fdone = finf;
    do
    {
      iinf = finf->index(infinite_vertex());
      if(!finf->has_vertex(v)) break;
      finf = finf->neighbor((iinf+1)%3);
    }
    while(finf != fdone);

    iinf = ~iinf;
    if(this->collinear(finf->vertex(iinf&1)->point(),
                       finf->vertex(iinf&2)->point(),
                       p))
    {
      v->set_point(p);
      _tds.decrease_dimension(loc, loc->index(v));
      for(All_cells_iterator afi = tds().raw_cells_begin();
                             afi != tds().raw_cells_end(); afi++)
      {
        *fit++ = afi;
      }
      return v;
    }
  }

  if(((dim == 2) && (lt != OUTSIDE_AFFINE_HULL)) ||
     ((lt == OUTSIDE_AFFINE_HULL) && (dim == 1)))
  {
    std::set<Cell_handle> cells_set;
    // This is insert must be from Delaunay (or the particular trian.)
    // not Triangulation_3 !
    Vertex_handle inserted = inserter.insert(p, lt, loc, li, lj);
    Cell_handle c = inserted->cell(), end = c;
    do
    {
      cells_set.insert(c);
      int i = c->index(inserted);
      c = c->neighbor((i+1)%3);
    }
    while(c != end);

    std::list<Edge_2D> hole;
    make_hole_2D(v, hole, remover, cells_set);
    fill_hole_2D(hole, remover, fit);

    // fixing pointer
    Cell_handle fc = inserted->cell(), done(fc);
    std::vector<Cell_handle> faces_pt;
    faces_pt.reserve(16);
    do
    {
      faces_pt.push_back(fc);
      fc = fc->neighbor((fc->index(inserted) + 1)%3);
    }
    while(fc != done);

    int ss = faces_pt.size();
    for(int k=0; k<ss; k++)
    {
      Cell_handle f = faces_pt[k];
      int i = f->index(inserted);
      f->set_vertex(i, v);
    }
    v->set_point(p);
    v->set_cell(inserted->cell());

    tds().delete_vertex(inserted);

    for(typename std::set<Cell_handle>::const_iterator ib = cells_set.begin(),
                                                       iend = cells_set.end();
        ib != iend; ib++)
    {
      *fit++ = *ib;
    }
    return v;
  }

  if((lt != OUTSIDE_AFFINE_HULL) && dim_down && (dim == 3))
  {
    // verify if p and two static vertices are collinear in this case
    std::vector<Cell_handle> ics;
    incident_cells(infinite_vertex(), std::back_inserter(ics));
    int size = ics.size();
    Cell_handle finf;
    for(int i=0; i<size; i++)
    {
      finf = ics[i];
      if(!finf->has_vertex(v)) break;
    }

    int iinf = finf->index(infinite_vertex());
    if(remover.tmp.coplanar(finf->vertex((iinf+1)&3)->point(),
                            finf->vertex((iinf+2)&3)->point(),
                            finf->vertex((iinf+3)&3)->point(),
                            p))
    {
      v->set_point(p);
      _tds.decrease_dimension(loc, loc->index(v));
      Facet f = *finite_facets_begin();
      if(coplanar_orientation(f.first->vertex(0)->point(),
                               f.first->vertex(1)->point(),
                               f.first->vertex(2)->point()) == NEGATIVE)
      {
        tds().reorient();
      }

      restore_edges_after_decrease_dimension(v, remover,inserter);
      for(All_cells_iterator afi = tds().raw_cells_begin();
                             afi != tds().raw_cells_end(); afi++)
      {
        *fit++ = afi;
      }
      return v;
    }
  }

  std::set<Cell_handle> cells_set;

  // This is insert must be from Delaunay (or the particular trian.)
  // not Triangulation_3 !
  Vertex_handle inserted = inserter.insert(p, lt, loc, li, lj);

  std::vector<Cell_handle> cells_tmp;
  cells_tmp.reserve(64);
  incident_cells(inserted, std::back_inserter(cells_tmp));
  int size = cells_tmp.size();
  for(int i=0; i<size; i++)
  {
    cells_set.insert(cells_tmp[i]);
  }

  std::vector<Cell_handle> hole;
  hole.reserve(64);

  // Construct the set of vertex triples on the boundary
  // with the facet just behind
  Vertex_triple_Facet_map outer_map;
  Vertex_triple_Facet_map inner_map;

  make_hole_3D(v, outer_map, hole);

  for(typename std::vector<Cell_handle>::const_iterator ib = hole.begin(),
                                                        iend = hole.end();
      ib != iend; ib++)
  {
    cells_set.erase(*ib);
  }

  CGAL_assertion(remover.hidden_points_begin() == remover.hidden_points_end());

  // Output the hidden points.
  for(typename std::vector<Cell_handle>::iterator hi = hole.begin(),
                                                  hend = hole.end();
      hi != hend; ++hi)
  {
    remover.add_hidden_points(*hi);
  }

  bool inf = false;
  unsigned int i;

  // Collect all vertices on the boundary
  std::vector<Vertex_handle> vertices;
  vertices.reserve(64);
  adjacent_vertices(v, std::back_inserter(vertices));

  // Create a Delaunay triangulation of the points on the boundary
  // and make a map from the vertices in remover.tmp towards the vertices
  // in *this
  Vertex_handle_unique_hash_map vmap;
  Cell_handle ch = Cell_handle();
  for(i=0; i < vertices.size(); i++)
  {
    if(! is_infinite(vertices[i]))
    {
      Vertex_handle vh = remover.tmp.insert(vertices[i]->point(), ch);
      ch = vh->cell();
      vmap[vh] = vertices[i];
    }else {
      inf = true;
    }
  }

  if(remover.tmp.dimension()==2)
  {
    Vertex_handle fake_inf = remover.tmp.insert(v->point());
    vmap[fake_inf] = infinite_vertex();
  }
  else
  {
    vmap[remover.tmp.infinite_vertex()] = infinite_vertex();
  }

  CGAL_triangulation_assertion(remover.tmp.dimension() == 3);

  // Construct the set of vertex triples of remover.tmp
  // We reorient the vertex triple so that it matches those from outer_map
  // Also note that we use the vertices of *this, not of remover.tmp
  if(inf)
  {
    for(All_cells_iterator it = remover.tmp.all_cells_begin(),
                           end = remover.tmp.all_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }
  else
  {
    for(Finite_cells_iterator it = remover.tmp.finite_cells_begin(),
                              end = remover.tmp.finite_cells_end(); it != end; ++it)
    {
      for(i=0; i < 4; i++)
      {
        Facet f = std::pair<Cell_handle,int>(it,i);
        Vertex_triple vt_aux = make_vertex_triple(f);
        Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
        make_canonical(vt);
        inner_map[vt]= f;
      }
    }
  }

  // Grow inside the hole, by extending the surface
  while(! outer_map.empty())
  {
    typename Vertex_triple_Facet_map::iterator oit = outer_map.begin();
    while(is_infinite(oit->first.first) ||
          is_infinite(oit->first.second) ||
          is_infinite(oit->first.third))
    {
      ++oit;
      // otherwise the lookup in the inner_map fails
      // because the infinite vertices are different
    }

    typename Vertex_triple_Facet_map::value_type o_vt_f_pair = *oit;
    Cell_handle o_ch = o_vt_f_pair.second.first;
    unsigned int o_i = o_vt_f_pair.second.second;

    typename Vertex_triple_Facet_map::iterator iit =
        inner_map.find(o_vt_f_pair.first);
    CGAL_triangulation_assertion(iit != inner_map.end());
    typename Vertex_triple_Facet_map::value_type i_vt_f_pair = *iit;
    Cell_handle i_ch = i_vt_f_pair.second.first;
    unsigned int i_i = i_vt_f_pair.second.second;

    // create a new cell and glue it to the outer surface
    Cell_handle new_ch = tds().create_cell();
    *fit++ = new_ch;

    new_ch->set_vertices(vmap[i_ch->vertex(0)], vmap[i_ch->vertex(1)],
                         vmap[i_ch->vertex(2)], vmap[i_ch->vertex(3)]);

    o_ch->set_neighbor(o_i,new_ch);
    new_ch->set_neighbor(i_i, o_ch);

    // for the other faces check, if they can also be glued
    for(i = 0; i < 4; i++)
    {
      if(i != i_i)
      {
        Facet f = std::pair<Cell_handle, int>(new_ch, i);
        Vertex_triple vt = make_vertex_triple(f);
        make_canonical(vt);
        std::swap(vt.second,vt.third);
        typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
        if(oit2 == outer_map.end())
        {
          std::swap(vt.second,vt.third);
          outer_map[vt] = f;
        }
        else
        {
          // glue the faces
          typename Vertex_triple_Facet_map::value_type o_vt_f_pair2 = *oit2;
          Cell_handle o_ch2 = o_vt_f_pair2.second.first;
          int o_i2 = o_vt_f_pair2.second.second;
          o_ch2->set_neighbor(o_i2,new_ch);
          new_ch->set_neighbor(i, o_ch2);
          outer_map.erase(oit2);
        }
      }
    }
    outer_map.erase(oit);
  }

  // fixing pointer
  std::vector<Cell_handle> cells_pt;
  cells_pt.reserve(64);
  incident_cells(inserted, std::back_inserter(cells_pt));
  size = cells_pt.size();
  for(int i=0; i<size; i++)
  {
    Cell_handle c = cells_pt[i];
    c->set_vertex(c->index(inserted), v);
  }

  v->set_point(p);
  v->set_cell(inserted->cell());
  tds().delete_vertex(inserted);
  tds().delete_cells(hole.begin(), hole.end());

  for(typename std::set<Cell_handle>::const_iterator ib = cells_set.begin(),
                                                     iend = cells_set.end();
      ib != iend; ib++)
  {
    *fit++ = *ib;
  }
  return v;
}

template < class Gt, class Tds, class Lds >
void
Triangulation_3<Gt,Tds,Lds>::
_make_big_hole_3D(Vertex_handle v,
                  std::map<Vertex_triple,Facet>& outer_map,
                  std::vector<Cell_handle>& hole,
                  std::vector<Vertex_handle>& vertices,
                  std::map<Vertex_handle, REMOVE_VERTEX_STATE>& vstates)
{
  Cell_handle start = v->cell();
  start->tds_data().mark_processed();
  hole.push_back(start);
  std::size_t i=0, n=1;
  while(i < n)
  {
    Cell_handle c = hole[i++];
    for(int k=0; k<4; k++)
    {
      Vertex_handle v0 = c->vertex(k);
      const REMOVE_VERTEX_STATE vst = vstates[v0];

      if(vst == CLEAR)
      {
        vstates[v0] = EXTREMITY;
        vertices.push_back(v0);
      }
      else if(vst == TO_REMOVE)
      {
        // we mark the vertices, so all the vertices
        // from the same cluster will be skipped
        // in the remove_cluster_3D function
        vstates[v0] = PROCESSED;
      }

      int i1 = vertex_triple_index(k, 0);
      int i2 = vertex_triple_index(k, 1);
      int i3 = vertex_triple_index(k, 2);

      Vertex_handle v1 = c->vertex(i1);
      Vertex_handle v2 = c->vertex(i2);
      Vertex_handle v3 = c->vertex(i3);

      Cell_handle opp_cit = c->neighbor(k);
      int opp_i = tds().mirror_index(c, k);
      Vertex_handle vm = opp_cit->vertex(opp_i);

      bool pb1 = false, pb2 = false, pb3 = false, pbm = false;

      const REMOVE_VERTEX_STATE vst1 = vstates[v1];
      pb1 = vst1 == TO_REMOVE || vst1 == PROCESSED;

      if(!pb1)
      {
        const REMOVE_VERTEX_STATE vst2 = vstates[v2];
        pb2 = vst2 == TO_REMOVE || vst2 == PROCESSED;
        if(!pb2)
        {
          const REMOVE_VERTEX_STATE vst3 = vstates[v3];
          pb3 = vst3 == TO_REMOVE || vst3 == PROCESSED;
          if(!pb3)
          {
            const REMOVE_VERTEX_STATE vstm = vstates[vm];
            pbm = vstm == TO_REMOVE || vstm == PROCESSED;
          }
        }
      }

      bool bad_opposite_cell = pb1 || pb2 || pb3 || pbm;

      // update the hole if needed
      // when the vertex is not to be removed
      if(bad_opposite_cell)
      {
        if(opp_cit->tds_data().is_clear())
        {
          hole.push_back(opp_cit);
          opp_cit->tds_data().mark_processed();
          n++;
        }
        continue;
      }

      Facet f(opp_cit, opp_i);
      Vertex_triple vt = make_vertex_triple(f);
      make_canonical(vt);
      outer_map[vt] = f;
      v1->set_cell(opp_cit);
      v2->set_cell(opp_cit);
      v3->set_cell(opp_cit);
      vm->set_cell(opp_cit);
    }
  }

  std::size_t vsize = vertices.size();
  for(std::size_t i=0; i<vsize; i++)
    vstates[vertices[i]] = CLEAR;
}

template < class Gt, class Tds, class Lds >
template < class InputIterator, class VertexRemover >
bool
Triangulation_3<Gt, Tds, Lds>::
_remove_cluster_3D(InputIterator first, InputIterator beyond, VertexRemover& remover,
                   std::map<Vertex_handle, REMOVE_VERTEX_STATE>& vstates)
{
  InputIterator init = first;
  while(first != beyond)
  {
    Vertex_handle v = *first++;

    if(vstates[v] == PROCESSED)
      continue;

    // _make_big_hole_3D and we fill the hole for each cluster
    vstates[v] = PROCESSED;

    // here, we make the hole for the cluster with v inside
    typedef std::map<Vertex_triple,Facet> Vertex_triple_Facet_map;
    std::vector<Cell_handle> hole;
    std::vector<Vertex_handle> vertices;
    hole.reserve(64);
    vertices.reserve(32);
    Vertex_triple_Facet_map outer_map;
    _make_big_hole_3D(v, outer_map, hole, vertices, vstates);

    // the connectivity is totally lost, we need to rebuild
    if(!outer_map.size())
    {
      std::size_t nh = hole.size();
      for(std::size_t i=0; i<nh; i++) hole[i]->tds_data().clear();
      return false;
    }

    std::size_t vsi = vertices.size();

    bool inf = false;
    std::size_t i;
    Vertex_handle_unique_hash_map vmap;
    Cell_handle ch = Cell_handle();

    if(vsi > 100)
    {
      // spatial sort if too many points
      std::vector<Point> vps;
      std::map<Point, Vertex_handle> mp_vps;
      for(i=0; i<vsi;i++)
      {
        Vertex_handle vv = vertices[i];
        if(! this->is_infinite(vv))
        {
          vps.push_back(vv->point());
          mp_vps[vv->point()] = vv;
        }
        else
        {
          inf = true;
        }
      }

      // Spatial sorting can only be applied to bare points, so we need an adaptor
      typedef typename Geom_traits::Construct_point_3 Construct_point_3;
      typedef typename boost::result_of<const Construct_point_3(const Point&)>::type Ret;
      typedef CGAL::internal::boost_::function_property_map<Construct_point_3, Point, Ret> fpmap;
      typedef CGAL::Spatial_sort_traits_adapter_3<Geom_traits, fpmap> Search_traits_3;

      spatial_sort(vps.begin(), vps.end(),
                   Search_traits_3(
                     CGAL::internal::boost_::make_function_property_map<Point, Ret, Construct_point_3>(
                       geom_traits().construct_point_3_object()), geom_traits()));

      std::size_t svps = vps.size();

      for(i=0; i < svps; i++)
      {
        Vertex_handle vv = mp_vps[vps[i]];
        Vertex_handle vh = remover.tmp.insert(vv->point(), ch);
        ch = vh->cell();
        vmap[vh] = vv;
      }

      if(remover.tmp.dimension()==2)
      {
        Vertex_handle fake_inf = remover.tmp.insert(v->point());
        vmap[fake_inf] = this->infinite_vertex();
      }
      else
      {
        vmap[remover.tmp.infinite_vertex()] = this->infinite_vertex();
      }
    }
    else
    {
      for(i=0; i < vsi; i++)
      {
        if(!this->is_infinite(vertices[i]))
        {
          Vertex_handle vh = remover.tmp.insert(vertices[i]->point(), ch);
          ch = vh->cell();
          vmap[vh] = vertices[i];
        }
        else
        {
          inf = true;
        }
      }

      if(remover.tmp.dimension()==2)
      {
        Vertex_handle fake_inf = remover.tmp.insert(v->point());
        vmap[fake_inf] = this->infinite_vertex();
      }
      else
      {
        vmap[remover.tmp.infinite_vertex()] = this->infinite_vertex();
      }
    }

    Vertex_triple_Facet_map inner_map;

    if(inf)
    {
      for(All_cells_iterator it = remover.tmp.all_cells_begin(),
                             end = remover.tmp.all_cells_end(); it != end; ++it)
      {
        for(unsigned int index=0; index < 4; index++)
        {
          Facet f = std::pair<Cell_handle,int>(it,index);
          Vertex_triple vt_aux = this->make_vertex_triple(f);
          Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
          this->make_canonical(vt);
          inner_map[vt]= f;
        }
      }
    }
    else
    {
      for(Finite_cells_iterator it = remover.tmp.finite_cells_begin(),
                                end = remover.tmp.finite_cells_end(); it != end; ++it)
      {
        for(unsigned int index=0; index < 4; index++)
        {
          Facet f = std::pair<Cell_handle,int>(it,index);
          Vertex_triple vt_aux = this->make_vertex_triple(f);
          Vertex_triple vt(vmap[vt_aux.first], vmap[vt_aux.third], vmap[vt_aux.second]);
          this->make_canonical(vt);
          inner_map[vt]= f;
        }
      }
    }

    // Grow inside the hole, by extending the surface
    while(! outer_map.empty())
    {
      typename Vertex_triple_Facet_map::iterator oit = outer_map.begin();

      while(this->is_infinite(oit->first.first) ||
            this->is_infinite(oit->first.second) ||
            this->is_infinite(oit->first.third))
      {
        ++oit;
        // otherwise the lookup in the inner_map fails
        // because the infinite vertices are different
      }

      typename Vertex_triple_Facet_map::value_type o_vt_f_pair = *oit;
      Cell_handle o_ch = o_vt_f_pair.second.first;
      unsigned int o_i = o_vt_f_pair.second.second;

      typename Vertex_triple_Facet_map::iterator iit =
          inner_map.find(o_vt_f_pair.first);
      CGAL_triangulation_assertion(iit != inner_map.end());
      typename Vertex_triple_Facet_map::value_type i_vt_f_pair = *iit;
      Cell_handle i_ch = i_vt_f_pair.second.first;
      unsigned int i_i = i_vt_f_pair.second.second;

      // create a new cell and glue it to the outer surface
      Cell_handle new_ch = tds().create_cell();
      new_ch->set_vertices(vmap[i_ch->vertex(0)], vmap[i_ch->vertex(1)],
                           vmap[i_ch->vertex(2)], vmap[i_ch->vertex(3)]);

      o_ch->set_neighbor(o_i,new_ch);
      new_ch->set_neighbor(i_i, o_ch);

      for(int j=0; j<4; j++)
        new_ch->vertex(j)->set_cell(new_ch);

      // for the other faces check, if they can also be glued
      for(unsigned int index = 0; index < 4; index++)
      {
        if(index != i_i)
        {
          Facet f = std::pair<Cell_handle,int>(new_ch,index);
          Vertex_triple vt = this->make_vertex_triple(f);
          this->make_canonical(vt);
          std::swap(vt.second,vt.third);
          typename Vertex_triple_Facet_map::iterator oit2 = outer_map.find(vt);
          if(oit2 == outer_map.end())
          {
            std::swap(vt.second,vt.third);
            outer_map[vt]= f;
          }
          else
          {
            // glue the faces
            typename Vertex_triple_Facet_map::value_type o_vt_f_pair2 = *oit2;
            Cell_handle o_ch2 = o_vt_f_pair2.second.first;
            int o_i2 = o_vt_f_pair2.second.second;
            o_ch2->set_neighbor(o_i2,new_ch);
            new_ch->set_neighbor(index, o_ch2);
            outer_map.erase(oit2);
          }
        }
      }

      outer_map.erase(oit);
    }

    this->tds().delete_cells(hole.begin(), hole.end());
    remover.tmp.clear();
  }

  this->tds().delete_vertices(init, beyond);

  return true;
}

template < class Gt, class Tds, class Lds >
template < class InputIterator >
bool
Triangulation_3<Gt, Tds, Lds>::
does_repeat_in_range(InputIterator first, InputIterator beyond) const
{
  std::set<Vertex_handle> s;
  while(first != beyond)
    if(! s.insert(*first++).second)
      return true;

  return false;
}

template < class Gt, class Tds, class Lds >
template < class InputIterator >
bool
Triangulation_3<Gt, Tds, Lds>::
infinite_vertex_in_range(InputIterator first, InputIterator beyond) const
{
  while(first != beyond)
    if(is_infinite(*first++))
      return true;

  return false;
}

template < class Gt, class Tds, class Lds >
template < class InputIterator, class VertexRemover >
typename Triangulation_3<Gt, Tds, Lds>::size_type
Triangulation_3<Gt, Tds, Lds>::
remove(InputIterator first, InputIterator beyond, VertexRemover& remover)
{
  CGAL_triangulation_precondition(!does_repeat_in_range(first, beyond));
  CGAL_triangulation_precondition(!infinite_vertex_in_range(first, beyond));

  size_type n = number_of_vertices();
  InputIterator init = first, init2 = first;
  if(dimension() == 3 && n > 4)
  {
    // If we could add states on a vertex base as it is done
    // for cells, it would improve the performance.
    std::map<Vertex_handle, REMOVE_VERTEX_STATE> vstates;
    _mark_vertices_to_remove(first, beyond, vstates);
    if(!_test_dim_down_cluster(vstates))
    {
      if(_remove_cluster_3D(init, beyond, remover, vstates))
        return n - number_of_vertices();
    }
  }

  // dimension() < 3 or no connectivity of the remaining vertices
  // we remove one by one
  while(init2 != beyond)
  {
    Vertex_handle v = *init2++;
    remover.tmp.clear();
    remove(v, remover);
  }
  return n - number_of_vertices();
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_valid(bool verbose, int level) const
{
  if(! _tds.is_valid(verbose,level))
  {
    if(verbose)
      std::cerr << "invalid data structure" << std::endl;

    CGAL_triangulation_assertion(false);
    return false;
  }

  if(infinite_vertex() == Vertex_handle())
  {
    if(verbose)
      std::cerr << "no infinite vertex" << std::endl;

    CGAL_triangulation_assertion(false);
    return false;
  }

  switch(dimension())
  {
    case 3:
    {
      for(Finite_cells_iterator it = finite_cells_begin(),
                                end = finite_cells_end(); it != end; ++it)
        is_valid_finite(it, verbose, level);
      break;
    }
    case 2:
    {
      for(Finite_facets_iterator it = finite_facets_begin(),
                                 end = finite_facets_end(); it != end; ++it)
        is_valid_finite(it->first,verbose,level);
      break;
    }
    case 1:
    {
      for(Finite_edges_iterator it = finite_edges_begin(),
                                end = finite_edges_end(); it != end; ++it)
        is_valid_finite(it->first,verbose,level);
      break;
    }
  }

  if(verbose)
    std::cerr << "valid triangulation" << std::endl;
  return true;
}

template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_valid(Cell_handle c, bool verbose, int level) const
{
  if(! _tds.is_valid(c,verbose,level))
  {
    if(verbose)
    {
      std::cerr << "combinatorially invalid cell";
      for(int i=0; i <= dimension(); i++)
        std::cerr << c->vertex(i)->point() << ", ";

      std::cerr << std::endl;
    }

    CGAL_triangulation_assertion(false);
    return false;
  }

  if(! is_infinite(c))
    is_valid_finite(c, verbose, level);

  if(verbose)
    std::cerr << "geometrically valid cell" << std::endl;

  return true;
}


template < class GT, class Tds, class Lds >
bool
Triangulation_3<GT,Tds,Lds>::
is_valid_finite(Cell_handle c, bool verbose, int) const
{
  switch(dimension())
  {
    case 3:
    {
      if(orientation(c->vertex(0)->point(),
                     c->vertex(1)->point(),
                     c->vertex(2)->point(),
                     c->vertex(3)->point()) != POSITIVE)
      {
        if(verbose)
        {
          std::cerr << "badly oriented cell "
                    << c->vertex(0)->point() << ", "
                    << c->vertex(1)->point() << ", "
                    << c->vertex(2)->point() << ", "
                    << c->vertex(3)->point() << std::endl;
        }
        CGAL_triangulation_assertion(false);
        return false;
      }
      break;
    }
    case 2:
    {
      if(coplanar_orientation(c->vertex(0)->point(),
                              c->vertex(1)->point(),
                              c->vertex(2)->point()) != POSITIVE)
      {
        if(verbose)
        {
          std::cerr << "badly oriented face "
                    << c->vertex(0)->point() << ", "
                    << c->vertex(1)->point() << ", "
                    << c->vertex(2)->point() << std::endl;
        }
        CGAL_triangulation_assertion(false);
        return false;
      }
      break;
    }
    case 1:
    {
      const Point& p0 = c->vertex(0)->point();
      const Point& p1 = c->vertex(1)->point();

      Vertex_handle v = c->neighbor(0)->vertex(c->neighbor(0)->index(c));
      if(! is_infinite(v))
      {
        if(collinear_position(p0, p1, v->point()) != MIDDLE)
        {
          if(verbose)
          {
            std::cerr << "badly oriented edge " << p0 << ", " << p1 << std::endl
                      << "with neighbor 0"
                      << c->neighbor(0)->vertex(1-c->neighbor(0)->index(c))->point()
                      << ", " << v->point() << std::endl;
          }
          CGAL_triangulation_assertion(false);
          return false;
        }
      }

      v = c->neighbor(1)->vertex(c->neighbor(1)->index(c));
      if(! is_infinite(v))
      {
        if(collinear_position(p1, p0, v->point()) != MIDDLE)
        {
          if(verbose)
          {
            std::cerr << "badly oriented edge " << p0 << ", " << p1 << std::endl
                      << "with neighbor 1"
                      << c->neighbor(1)->vertex(1-c->neighbor(1)->index(c))->point()
                      << ", " << v->point() << std::endl;
          }
          CGAL_triangulation_assertion(false);
          return false;
        }
      }
      break;
    }
  }
  return true;
}


namespace internal {

// Internal function used by operator==.
template < class GT, class Tds1, class Tds2, class Lds >
bool
test_next(const Triangulation_3<GT, Tds1, Lds>& t1,
          const Triangulation_3<GT, Tds2, Lds>& t2,
          typename Triangulation_3<GT, Tds1, Lds>::Cell_handle c1,
          typename Triangulation_3<GT, Tds2, Lds>::Cell_handle c2,
          std::map<typename Triangulation_3<GT, Tds1, Lds>::Cell_handle,
          typename Triangulation_3<GT, Tds2, Lds>::Cell_handle>& Cmap,
          std::map<typename Triangulation_3<GT, Tds1, Lds>::Vertex_handle,
          typename Triangulation_3<GT, Tds2, Lds>::Vertex_handle>& Vmap)
{
  // This function tests and registers the 4 neighbors of c1/c2,
  // and recursively calls itself over them.
  // We don't use the call stack as it may overflow
  // Returns false if an inequality has been found.

  // Precondition: c1, c2 have been registered as well as their 4 vertices.
  CGAL_triangulation_precondition(t1.dimension() >= 2);
  CGAL_triangulation_precondition(Cmap[c1] == c2);
  CGAL_triangulation_precondition(Vmap.find(c1->vertex(0)) != Vmap.end());
  CGAL_triangulation_precondition(Vmap.find(c1->vertex(1)) != Vmap.end());
  CGAL_triangulation_precondition(Vmap.find(c1->vertex(2)) != Vmap.end());
  CGAL_triangulation_precondition(t1.dimension() == 2 ||
                                  Vmap.find(c1->vertex(3)) != Vmap.end());

  typedef Triangulation_3<GT, Tds1, Lds>          Tr1;
  typedef Triangulation_3<GT, Tds2, Lds>          Tr2;
  typedef typename Tr1::Vertex_handle             Vertex_handle1;
  typedef typename Tr1::Cell_handle               Cell_handle1;
  typedef typename Tr2::Vertex_handle             Vertex_handle2;
  typedef typename Tr2::Cell_handle               Cell_handle2;

  typedef typename std::map<Vertex_handle1, Vertex_handle2>::const_iterator Vit;
  typedef typename std::map<Cell_handle1, Cell_handle2>::const_iterator     Cit;

  typedef typename Tr1::Geom_traits::Construct_point_3    Construct_point_3;
  typedef typename Tr1::Geom_traits::Compare_xyz_3        Compare_xyz_3;

  Compare_xyz_3 cmp1 = t1.geom_traits().compare_xyz_3_object();
  Construct_point_3 cp = t1.geom_traits().construct_point_3_object();

  std::vector<std::pair<Cell_handle1, Cell_handle2> > cell_stack;
  cell_stack.push_back(std::make_pair(c1, c2));

  while(! cell_stack.empty())
  {
    Cell_handle1 c1 = cell_stack.back().first;
    Cell_handle2 c2 = cell_stack.back().second;
    cell_stack.pop_back();

    for(int i=0; i <= t1.dimension(); ++i)
    {
      Cell_handle1 n1 = c1->neighbor(i);
      Cit cit = Cmap.find(n1);
      Vertex_handle1 v1 = c1->vertex(i);
      Vertex_handle2 v2 = Vmap[v1];
      Cell_handle2 n2 = c2->neighbor(c2->index(v2));
      if(cit != Cmap.end())
      {
        // n1 was already registered.
        if(cit->second != n2)
          return false;

        continue;
      }

      // n1 has not yet been registered.
      // We check that the new vertices match geometrically.
      // And we register them.
      Vertex_handle1 vn1 = n1->vertex(n1->index(c1));
      Vertex_handle2 vn2 = n2->vertex(n2->index(c2));
      Vit vit = Vmap.find(vn1);
      if(vit != Vmap.end())
      {
        // vn1 already registered
        if(vit->second != vn2)
          return false;
      }
      else
      {
        if(t2.is_infinite(vn2))
          return false; // vn1 can't be infinite,

        // since it would have been registered.
        if(cmp1(cp(vn1->point()), cp(vn2->point())) != 0)
          return false;

        // We register vn1/vn2.
        Vmap.insert(std::make_pair(vn1, vn2));
      }

      // We register n1/n2.
      Cmap.insert(std::make_pair(n1, n2));
      cell_stack.push_back(std::make_pair(n1, n2));
    }
  }

  return true;
}

} // namespace internal

template < class GT, class Tds1, class Tds2, class Lds >
bool
operator==(const Triangulation_3<GT, Tds1, Lds>& t1,
           const Triangulation_3<GT, Tds2, Lds>& t2)
{
  typedef typename Triangulation_3<GT, Tds1>::Vertex_handle Vertex_handle1;
  typedef typename Triangulation_3<GT, Tds1>::Cell_handle   Cell_handle1;
  typedef typename Triangulation_3<GT, Tds2>::Vertex_handle Vertex_handle2;
  typedef typename Triangulation_3<GT, Tds2>::Cell_handle   Cell_handle2;

  typedef typename Triangulation_3<GT, Tds1>::Point                          Point;

  typedef typename Triangulation_3<GT, Tds1>::Geom_traits::Equal_3           Equal_3;
  typedef typename Triangulation_3<GT, Tds1>::Geom_traits::Compare_xyz_3     Compare_xyz_3;
  typedef typename Triangulation_3<GT, Tds1>::Geom_traits::Construct_point_3 Construct_point_3;

  Equal_3 equal = t1.geom_traits().equal_3_object();
  Compare_xyz_3 cmp1 = t1.geom_traits().compare_xyz_3_object();
  Compare_xyz_3 cmp2 = t2.geom_traits().compare_xyz_3_object();
  Construct_point_3 cp = t1.geom_traits().construct_point_3_object();

  // Some quick checks.
  if(t1.dimension() != t2.dimension() ||
     t1.number_of_vertices() != t2.number_of_vertices() ||
     t1.number_of_cells() != t2.number_of_cells())
    return false;

  int dim = t1.dimension();
  // Special case for dimension < 1.
  // The triangulation is uniquely defined in these cases.
  if(dim < 1)
    return true;

  // Special case for dimension == 1.
  if(dim == 1)
  {
    // It's enough to test that the points are the same,
    // since the triangulation is uniquely defined in this case.
    std::vector<Point> V1 (t1.points_begin(), t1.points_end());
    std::vector<Point> V2 (t2.points_begin(), t2.points_end());

    std::sort(V1.begin(), V1.end(),
              boost::bind<Comparison_result>(
                cmp1,
                boost::bind<
                typename boost::result_of<const Construct_point_3(const Point&)>::type>(cp, _1),
                boost::bind<
                typename boost::result_of<const Construct_point_3(const Point&)>::type>(cp, _2))
              == SMALLER);

    std::sort(V2.begin(), V2.end(),
              boost::bind<Comparison_result>(
                cmp2,
                boost::bind<
                typename boost::result_of<const Construct_point_3(const Point&)>::type>(cp, _1),
                boost::bind<
                typename boost::result_of<const Construct_point_3(const Point&)>::type>(cp, _2))
              == SMALLER);

    return V1 == V2;
  }

  // We will store the mapping between the 2 triangulations vertices and
  // cells in 2 maps.
  std::map<Vertex_handle1, Vertex_handle2> Vmap;
  std::map<Cell_handle1, Cell_handle2> Cmap;

  // Handle the infinite vertex.
  Vertex_handle1 v1 = t1.infinite_vertex();
  Vertex_handle2 iv2 = t2.infinite_vertex();
  Vmap.insert(std::make_pair(v1, iv2));

  // We pick one infinite cell of t1, and try to match it against the
  // infinite cells of t2.
  Cell_handle1 c = v1->cell();
  Vertex_handle1 v2 = c->vertex((c->index(v1)+1)%(dim+1));
  Vertex_handle1 v3 = c->vertex((c->index(v1)+2)%(dim+1));
  Vertex_handle1 v4 = c->vertex((c->index(v1)+3)%(dim+1));
  const Point& p2 = v2->point();
  const Point& p3 = v3->point();
  const Point& p4 = v4->point();

  std::vector<Cell_handle2> ics;
  t2.incident_cells(iv2, std::back_inserter(ics));
  for(typename std::vector<Cell_handle2>::const_iterator cit = ics.begin();
                                                         cit != ics.end(); ++cit)
  {
    int inf = (*cit)->index(iv2);

    if(equal(cp(p2), cp((*cit)->vertex((inf+1)%(dim+1))->point())))
      Vmap.insert(std::make_pair(v2, (*cit)->vertex((inf+1)%(dim+1))));
    else if(equal(cp(p2), cp((*cit)->vertex((inf+2)%(dim+1))->point())))
      Vmap.insert(std::make_pair(v2, (*cit)->vertex((inf+2)%(dim+1))));
    else if(dim == 3 && equal(cp(p2), cp((*cit)->vertex((inf+3)%(dim+1))->point())))
      Vmap.insert(std::make_pair(v2, (*cit)->vertex((inf+3)%(dim+1))));
    else
      continue; // None matched v2.

    if(equal(cp(p3), cp((*cit)->vertex((inf+1)%(dim+1))->point())))
      Vmap.insert(std::make_pair(v3, (*cit)->vertex((inf+1)%(dim+1))));
    else if(equal(cp(p3), cp((*cit)->vertex((inf+2)%(dim+1))->point())))
      Vmap.insert(std::make_pair(v3, (*cit)->vertex((inf+2)%(dim+1))));
    else if(dim == 3 && equal(cp(p3), cp((*cit)->vertex((inf+3)%(dim+1))->point())))
      Vmap.insert(std::make_pair(v3, (*cit)->vertex((inf+3)%(dim+1))));
    else
      continue; // None matched v3.

    if(dim == 3)
    {
      if(equal(cp(p4), cp((*cit)->vertex((inf+1)%(dim+1))->point())))
        Vmap.insert(std::make_pair(v4, (*cit)->vertex((inf+1)%(dim+1))));
      else if(equal(cp(p4), cp((*cit)->vertex((inf+2)%(dim+1))->point())))
        Vmap.insert(std::make_pair(v4, (*cit)->vertex((inf+2)%(dim+1))));
      else if(equal(cp(p4), cp((*cit)->vertex((inf+3)%(dim+1))->point())))
        Vmap.insert(std::make_pair(v4, (*cit)->vertex((inf+3)%(dim+1))));
      else
        continue; // None matched v4.
    }

    // Found it !
    Cmap.insert(std::make_pair(c, *cit));
    break;
  }

  if(Cmap.size() == 0)
    return false;

  // We now have one cell, we need to propagate recursively.
  return internal::test_next(t1, t2,
                             Cmap.begin()->first, Cmap.begin()->second, Cmap, Vmap);
}

template < class GT, class Tds1, class Tds2 >
inline
bool
operator!=(const Triangulation_3<GT, Tds1>& t1,
           const Triangulation_3<GT, Tds2>& t2)
{
  return ! (t1 == t2);
}

} //namespace CGAL

#include <CGAL/enable_warnings.h>

#endif // CGAL_TRIANGULATION_3_H
