// Copyright (c) 1999-2003  INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
//
// 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/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 <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/iterator.h>
#include <CGAL/function_objects.h>
#include <CGAL/Iterator_project.h>
#include <CGAL/Default.h>

#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>

#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;
};
}
#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 NULL;
  }
};

#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::Point_3                 Point;
  typedef typename GT::Segment_3               Segment;
  typedef typename GT::Triangle_3              Triangle;
  typedef typename GT::Tetrahedron_3           Tetrahedron;

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

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 Point                         value_type; // to have a back_inserter
  typedef const value_type&             const_reference;

  //Tag to distinguish triangulations with weighted_points
  typedef Tag_false  Weighted_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

  Comparison_result
  compare_xyz(const Point &p, const Point &q) const
  {
      return geom_traits().compare_xyz_3_object()(p, q);
  }

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

  Orientation
  orientation(const Point &p, const Point &q,
              const Point &r, const Point &s) const
  {
      return geom_traits().orientation_3_object()(p, q, r, s);
  }

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

  Orientation
  coplanar_orientation(const Point &p, const Point &q, const Point &r) const
  {
      return geom_traits().coplanar_orientation_3_object()(p, q, r);
  }

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

  Segment
  construct_segment(const Point &p, const Point &q) const
  {
      return geom_traits().construct_segment_3_object()(p, q);
  }

  Triangle
  construct_triangle(const Point &p, const Point &q, const Point &r) const
  {
      return geom_traits().construct_triangle_3_object()(p, q, r);
  }

  Tetrahedron
  construct_tetrahedron(const Point &p, const Point &q,
                        const Point &r, const Point &s) const
  {
      return geom_traits().construct_tetrahedron_3_object()(p, q, r, 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;
  }

  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 = NULL)
    : 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 = NULL)
    : 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 = NULL)
    : 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)
    {
      // The triangulation passed as argument has been copied,
      // because the parameter tr is passed by value. Then the following
      // swap consumes the *copy*. 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());
    }

  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); }

  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();
  }

  const Point & point(Vertex_handle v) const {
    CGAL_triangulation_precondition( dimension() >= 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 = NULL) 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 = NULL) const;
protected:
  Cell_handle
  exact_locate(const Point& p,
               Locate_type& lt,
               int& li, int & lj,
               Cell_handle start,
               bool *could_lock_zone = NULL
               ) 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 = NULL) 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 = NULL) 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 = NULL
         ) 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 = NULL) 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 = NULL) 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 = NULL) 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 = NULL);

  template < class InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last)
    {
      size_type n = number_of_vertices();

      std::vector<Point> points (first, last);
      spatial_sort (points.begin(), points.end(), geom_traits());

      Vertex_handle hint;
      for (typename std::vector<Point>::const_iterator p = points.begin(), end = points.end();
              p != end; ++p)
          hint = insert(*p, hint);

      return number_of_vertices() - n;
    }

  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 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 = NULL
     , const Facet *this_facet_must_be_in_the_cz = NULL
     , bool *the_facet_is_in_its_cz = NULL
     ) 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 (could_lock_zone)
    {
      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
    std::stack<Cell_handle> cell_stack;
    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 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 = NULL) const;

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

  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();
  }

  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));
  }

  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();
  }

  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));
  }

  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();
  }

  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));
  }

  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();
  }

  Point_iterator points_begin() const
  {
      return Point_iterator(finite_vertices_begin());
  }
  Point_iterator points_end() const
  {
      return Point_iterator(finite_vertices_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
        {
          BOOST_FOREACH(Cell_handle& ch,
            std::make_pair(cells.begin(), cells.end()))
          {
            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);
    BOOST_FOREACH(Cell_handle& ch, std::make_pair(cells.begin(), cells.end()))
    {
      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
        {
          BOOST_FOREACH(Cell_handle& ch,
            std::make_pair(cells.begin(), cells.end()))
          {
            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;
    BOOST_FOREACH(Cell_handle& ch, std::make_pair(cells.begin(), cells.end()))
    {
      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));
  }

  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 give 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 has a meaning only 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
  // supposes 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 has a meaning only 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
  // supposes 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 has a 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 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.
      std::vector<Cell_handle> cells;
      Facet facet;

      cells.reserve(32);

      // Parallel
      if (could_lock_zone)
      {
        std::vector<Facet> facets;
        facets.reserve(32);

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

        if (*could_lock_zone == false)
        {
          BOOST_FOREACH(Cell_handle& ch,
            std::make_pair(cells.begin(), cells.end()))
          {
            ch->tds_data().clear();
          }

          BOOST_FOREACH(Facet& f,
            std::make_pair(facets.begin(), facets.end()))
          {
            f.first->neighbor(f.second)->tds_data().clear();
          }
          return Vertex_handle();
        }

        facet = facets.back();
      }
      // Sequential
      else
      {
        cells.reserve(32);
        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;
    }
  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;
      Facet facet;

      cells.reserve(32);
      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 *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,
                                                   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
  // 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 one also erases a set of cells
// which is useful to the move method
// outputting 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 = NULL; // Initialize to NULL 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 NULL 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);
        }
      }
    }
}

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 = NULL; // Initialize to NULL 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 NULL 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)
  {
    //std::random_shuffle(vertices.begin(), vertices.end());
    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)
  {
    //std::random_shuffle(adj_vertices.begin(), adj_vertices.end());
    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
  // Triangulation_3<Gt,Tds,Lds>::
  // move_if_no_collision(Vertex_handle,Point, VertexRemover, VertexInserter)

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_sort(vps.begin(), vps.end());

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

    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 (t1.geom_traits().compare_xyz_3_object()(vn1->point(),
                                                      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;

    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();

    // 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.
        using namespace boost;
        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(cmp1, _1, _2) == NEGATIVE);
        std::sort(V2.begin(), V2.end(), boost::bind(cmp2, _1, _2) == NEGATIVE);
        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(p2, (*cit)->vertex((inf+1)%(dim+1))->point()))
            Vmap.insert(std::make_pair(v2, (*cit)->vertex((inf+1)%(dim+1))));
        else if (equal(p2, (*cit)->vertex((inf+2)%(dim+1))->point()))
            Vmap.insert(std::make_pair(v2, (*cit)->vertex((inf+2)%(dim+1))));
        else if (dim == 3 &&
                 equal(p2, (*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(p3, (*cit)->vertex((inf+1)%(dim+1))->point()))
            Vmap.insert(std::make_pair(v3, (*cit)->vertex((inf+1)%(dim+1))));
        else if (equal(p3, (*cit)->vertex((inf+2)%(dim+1))->point()))
            Vmap.insert(std::make_pair(v3, (*cit)->vertex((inf+2)%(dim+1))));
        else if (dim == 3 &&
                 equal(p3, (*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(p4, (*cit)->vertex((inf+1)%(dim+1))->point()))
                Vmap.insert(std::make_pair(v4,
                                           (*cit)->vertex((inf+1)%(dim+1))));
            else if (equal(p4, (*cit)->vertex((inf+2)%(dim+1))->point()))
                Vmap.insert(std::make_pair(v4,
                                           (*cit)->vertex((inf+2)%(dim+1))));
            else if (equal(p4, (*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

#endif // CGAL_TRIANGULATION_3_H
