#include <Rcpp.h>
// Copyright (c) 1999-2004   INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v6.0.1/Triangulation_3/include/CGAL/Delaunay_triangulation_3.h $
// $Id: include/CGAL/Delaunay_triangulation_3.h 50cfbde3b84 $
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s)     : Monique Teillaud <Monique.Teillaud@sophia.inria.fr>
//                 Sylvain Pion
//                 Andreas Fabri <Andreas.Fabri@sophia.inria.fr>
//                 Clement Jamin

#ifndef CGAL_DELAUNAY_TRIANGULATION_3_H
#define CGAL_DELAUNAY_TRIANGULATION_3_H

#include <CGAL/license/Triangulation_3.h>

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

#include <CGAL/Delaunay_triangulation_cell_base_3.h>
#include <CGAL/Triangulation_3.h>

#include <CGAL/iterator.h>
#include <CGAL/Location_policy.h>

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
# define CGAL_PROFILE
# include <CGAL/Profile_counter.h>
#endif
#ifdef CGAL_TRIANGULATION_3_PROFILING
# include <CGAL/Mesh_3/Profiling_tools.h>
#endif

#ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
#include <CGAL/Spatial_sort_traits_adapter_3.h>

#include <boost/tuple/tuple.hpp>
#include <boost/mpl/and.hpp>
#endif //CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO

#ifdef CGAL_LINKED_WITH_TBB
# include <CGAL/point_generators_3.h>
# include <tbb/parallel_for.h>
# include <thread>
# include <tbb/enumerable_thread_specific.h>
# include <tbb/concurrent_vector.h>
#endif

#ifdef CGAL_DELAUNAY_3_OLD_REMOVE
#  error "The old remove() code has been removed.  Please report any issue you may have with the current one."
#endif

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
#include <CGAL/point_generators_3.h>
#endif

#include <utility>
#include <vector>

namespace CGAL {

// Here is the declaration of a class template with three arguments, one
// having a default value. There is no definition of that class template.
template < class Gt,
           class Tds_ = Default,
           class Location_policy = Default,
           class Lock_data_structure_ = Default >
class Delaunay_triangulation_3;

// There is a specialization Delaunay_triangulation_3<Gt, Tds, Fast_location>
// defined in <CGAL/Triangulation_3/internal/Delaunay_triangulation_hierarchy_3.h>.

// Here is the specialization Delaunay_triangulation_3<Gt, Tds>, with three
// arguments, that is if Location_policy being the default value 'Default'.
template < class Gt, class Tds_, class Lock_data_structure_ >
class Delaunay_triangulation_3<Gt, Tds_, Default, Lock_data_structure_>
  : public Triangulation_3<Gt,
                           typename Default::Get<Tds_,
                                                 Triangulation_data_structure_3<
                                                   Triangulation_vertex_base_3<Gt>,
                                                   Delaunay_triangulation_cell_base_3<Gt> > >::type,
                           Lock_data_structure_>
{
  typedef typename Default::Get<Tds_,
                     Triangulation_data_structure_3 <
                       Triangulation_vertex_base_3<Gt>,
                       Delaunay_triangulation_cell_base_3<Gt> > >::type     Tds;

  typedef Delaunay_triangulation_3<Gt, Tds_, Default, Lock_data_structure_> Self;

public:
  typedef typename Tds::Concurrency_tag                     Concurrency_tag;
  typedef Triangulation_3<Gt, Tds, Lock_data_structure_>    Tr_Base;

  typedef Tds                                               Triangulation_data_structure;
  typedef Gt                                                Geom_traits;
  typedef Compact_location                                  Location_policy;

  typedef typename Tr_Base::Lock_data_structure             Lock_data_structure;

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

  // types for dual:
  typedef typename Gt::Line_3        Line;
  typedef typename Gt::Ray_3         Ray;
  //typedef typename Gt::Plane_3       Plane;
  typedef typename Gt::Object_3      Object;

  typedef typename Tr_Base::Cell_handle   Cell_handle;
  typedef typename Tr_Base::Vertex_handle Vertex_handle;

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

  typedef typename Tr_Base::Cell_circulator  Cell_circulator;
  typedef typename Tr_Base::Facet_circulator Facet_circulator;
  typedef typename Tr_Base::Cell_iterator    Cell_iterator;
  typedef typename Tr_Base::Facet_iterator   Facet_iterator;
  typedef typename Tr_Base::Edge_iterator    Edge_iterator;
  typedef typename Tr_Base::Vertex_iterator  Vertex_iterator;

  typedef typename Tr_Base::Finite_vertices_iterator Finite_vertices_iterator;
  typedef typename Tr_Base::Finite_cells_iterator    Finite_cells_iterator;
  typedef typename Tr_Base::Finite_facets_iterator   Finite_facets_iterator;
  typedef typename Tr_Base::Finite_edges_iterator    Finite_edges_iterator;

  typedef typename Tr_Base::All_cells_iterator       All_cells_iterator;

  typedef typename Tr_Base::size_type                size_type;
  typedef typename Tr_Base::Locate_type              Locate_type;

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

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

#ifndef CGAL_CFG_USING_BASE_MEMBER_BUG_2
  using Tr_Base::cw;
  using Tr_Base::ccw;
  using Tr_Base::geom_traits;
  using Tr_Base::number_of_vertices;
  using Tr_Base::dimension;
  using Tr_Base::finite_facets_begin;
  using Tr_Base::finite_facets_end;
  using Tr_Base::finite_vertices_begin;
  using Tr_Base::finite_vertices_end;
  using Tr_Base::finite_cells_begin;
  using Tr_Base::finite_cells_end;
  using Tr_Base::finite_edges_begin;
  using Tr_Base::finite_edges_end;
  using Tr_Base::tds;
  using Tr_Base::infinite_vertex;
  using Tr_Base::next_around_edge;
  using Tr_Base::vertex_triple_index;
  using Tr_Base::mirror_vertex;
  using Tr_Base::coplanar;
  using Tr_Base::coplanar_orientation;
  using Tr_Base::orientation;
  using Tr_Base::adjacent_vertices;
  using Tr_Base::construct_segment;
  using Tr_Base::incident_facets;
  using Tr_Base::insert_in_conflict;
  using Tr_Base::is_infinite;
  using Tr_Base::is_valid_finite;
  using Tr_Base::locate;
  using Tr_Base::side_of_edge;
  using Tr_Base::side_of_segment;
  using Tr_Base::find_conflicts;
#endif

protected:

  Oriented_side side_of_oriented_sphere(const Point& p0, const Point& p1,
                                        const Point& p2, const Point& p3,
                                        const Point& t, bool perturb = false) const;

  Bounded_side coplanar_side_of_bounded_circle(const Point& p, const Point& q,
                                               const Point& r, const Point& s, bool perturb = false) const;

  // for dual:
  Point construct_circumcenter(const Point& p, const Point& q, const Point& r) const
  {
    return geom_traits().construct_circumcenter_3_object()(p, q, r);
  }

  Line construct_equidistant_line(const Point& p1, const Point& p2, const Point& p3) const
  {
    return geom_traits().construct_equidistant_line_3_object()(p1, p2, p3);
  }

  Ray construct_ray(const Point& p, const Line& l) const
  {
    return geom_traits().construct_ray_3_object()(p, l);
  }

  Object construct_object(const Point& p) const
  {
    return geom_traits().construct_object_3_object()(p);
  }

  Object construct_object(const Segment& s) const
  {
    return geom_traits().construct_object_3_object()(s);
  }

  Object construct_object(const Ray& r) const
  {
    return geom_traits().construct_object_3_object()(r);
  }

  bool less_distance(const Point& p, const Point& q, const Point& r) const
  {
    return geom_traits().compare_distance_3_object()(p, q, r) == SMALLER;
  }

public:
  Delaunay_triangulation_3(const Gt& gt = Gt(), Lock_data_structure *lock_ds = nullptr)
    : Tr_Base(gt, lock_ds)
  {}

  Delaunay_triangulation_3(Lock_data_structure *lock_ds, const Gt& gt = Gt())
    : Tr_Base(lock_ds, gt)
  {}

  // Create a 3D triangulation from 4 points which must be well-oriented
  // AND non-coplanar
  Delaunay_triangulation_3(const Point& p0, const Point& p1,
                           const Point& p2, const Point& p3,
                           const Gt& gt = Gt(),
                           Lock_data_structure *lock_ds = nullptr)
    : Tr_Base(p0, p1, p2, p3, gt, lock_ds)
  {}

  template < typename InputIterator >
  Delaunay_triangulation_3(InputIterator first, InputIterator last,
                           const Gt& gt = Gt(), Lock_data_structure *lock_ds = nullptr)
    : Tr_Base(gt, lock_ds)
  {
    insert(first, last);
  }

  template < typename InputIterator >
  Delaunay_triangulation_3(InputIterator first, InputIterator last,
                           Lock_data_structure *lock_ds,
                           const Gt& gt = Gt())
    : Tr_Base(gt, lock_ds)
  {
    insert(first, last);
  }


private:
  #ifdef CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
  std::vector<Vertex_handle>
  add_temporary_points_on_far_sphere(const size_t num_points)
  {
    std::vector<Vertex_handle> far_sphere_vertices;

    const size_t MIN_NUM_POINTS_FOR_FAR_SPHERE_POINTS = 1000000;
    if(num_points >= MIN_NUM_POINTS_FOR_FAR_SPHERE_POINTS)
    {
      // Add temporary vertices on a "far sphere" to reduce contention on
      // the infinite vertex

      // Get bbox
      const Bbox_3& bbox = *this->get_bbox();

      // Compute radius for far sphere
      const double& xdelta = bbox.xmax() - bbox.xmin();
      const double& ydelta = bbox.ymax() - bbox.ymin();
      const double& zdelta = bbox.zmax() - bbox.zmin();
      const double radius = 1.3 * 0.5 * std::sqrt(xdelta*xdelta +
                                                  ydelta*ydelta +
                                                  zdelta*zdelta);

      // WARNING - TODO: this code has to be fixed because Vector_3 is not
      // required by the traits concept
      const typename Gt::Vector_3 center(bbox.xmin() + 0.5*xdelta,
                                         bbox.ymin() + 0.5*ydelta,
                                         bbox.zmin() + 0.5*zdelta);
      Random_points_on_sphere_3<Point> random_point(radius);
      const int NUM_PSEUDO_INFINITE_VERTICES = static_cast<int>(
                                                 std::thread::hardware_concurrency() * 3.5);
      std::vector<Point> points_on_far_sphere;

      points_on_far_sphere.reserve(NUM_PSEUDO_INFINITE_VERTICES);
      far_sphere_vertices.reserve(NUM_PSEUDO_INFINITE_VERTICES);

      for(int i = 0 ; i < NUM_PSEUDO_INFINITE_VERTICES ; ++i, ++random_point)
        points_on_far_sphere.push_back(*random_point + center);

      spatial_sort(points_on_far_sphere.begin(),
                   points_on_far_sphere.end(),
                   geom_traits());

      typename std::vector<Point>::const_iterator it_p = points_on_far_sphere.begin();
      typename std::vector<Point>::const_iterator it_p_end = points_on_far_sphere.end();

      Vertex_handle hint;
      for(; it_p != it_p_end ; ++it_p)
      {
        hint = insert(*it_p, hint);
        far_sphere_vertices.push_back(hint);
      }
    }

    return far_sphere_vertices;
  }

  void remove_temporary_points_on_far_sphere(
    const std::vector<Vertex_handle>& far_sphere_vertices)
  {
      if(!far_sphere_vertices.empty())
      {
          // Remove the temporary vertices on far sphere
          remove(far_sphere_vertices.begin(), far_sphere_vertices.end());
      }
  }
  #endif

public:

#ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
  template < class InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last,
                        std::enable_if_t<
                          std::is_convertible<
                              typename std::iterator_traits<InputIterator>::value_type,
                              Point
                          >::value
                        >* = nullptr)
#else
  template < class InputIterator >
  std::ptrdiff_t insert(InputIterator first, InputIterator last)
#endif //CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
  {
#ifdef CGAL_TRIANGULATION_3_PROFILING
    WallClockTimer t;
#endif

    size_type n = number_of_vertices();
    std::vector<Point> points(first, last);
    spatial_sort<Concurrency_tag>(points.begin(), points.end(), geom_traits());

    // Parallel
#ifdef CGAL_LINKED_WITH_TBB
    if(this->is_parallel())
    {
      size_t num_points = points.size();

      Vertex_handle hint;

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
      std::vector<Vertex_handle> far_sphere_vertices =
        add_temporary_points_on_far_sphere(num_points);
#endif // CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE

      size_t i = 0;
      // Insert "num_points_seq" points sequentially
      // (or more if dim < 3 after that)
      size_t num_points_seq = (std::min)(num_points, (size_t) 100);
      while (i < num_points_seq || (dimension() < 3 && i < num_points))
      {
        hint = insert(points[i], hint);
        ++i;
      }

      tbb::enumerable_thread_specific<Vertex_handle> tls_hint(hint);
      tbb::parallel_for(tbb::blocked_range<size_t>(i, num_points),
                        Insert_point<Self>(*this, points, tls_hint));

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
      remove_temporary_points_on_far_sphere(far_sphere_vertices);
#endif // CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
    }
    // Sequential
    else
#endif // CGAL_LINKED_WITH_TBB
    {
      Vertex_handle hint;
      for(typename std::vector<Point>::const_iterator p = points.begin(),
                                                      end = points.end(); p != end; ++p)
      {
          hint = insert(*p, hint);
      }
    }

#ifdef CGAL_TRIANGULATION_3_PROFILING
    Rcpp::Rcerr << "Triangulation computed in " << t.elapsed() << " seconds." << std::endl;
#endif

    return number_of_vertices() - n;
  }

#ifndef CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO
private:
  using Tr_Base::top_get_first;
  using Tr_Base::top_get_second;

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

    typedef typename Pointer_property_map<Point>::type Pmap;
    typedef Spatial_sort_traits_adapter_3<Geom_traits,Pmap> Search_traits;

    spatial_sort<Concurrency_tag>(indices.begin(), indices.end(),
                                  Search_traits(make_property_map(points),geom_traits()));

#ifdef CGAL_LINKED_WITH_TBB
    if(this->is_parallel())
    {
      size_t num_points = points.size();
      Vertex_handle hint;

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
      std::vector<Vertex_handle> far_sphere_vertices =
          add_temporary_points_on_far_sphere(num_points);
#endif // CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE

      size_t i = 0;
      // Insert "num_points_seq" points sequentially
      // (or more if dim < 3 after that)
      size_t num_points_seq = (std::min)(num_points, (size_t)100);
      while (i < num_points_seq || (dimension() < 3 && i < num_points))
      {
        hint = insert(points[indices[i]], hint);
        if(hint != Vertex_handle()) hint->info() = infos[indices[i]];
        ++i;
      }

      tbb::enumerable_thread_specific<Vertex_handle> tls_hint(hint);
      tbb::parallel_for(tbb::blocked_range<size_t>(i, num_points),
                        Insert_point_with_info<Self>(*this, points, infos, indices, tls_hint));

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
      remove_temporary_points_on_far_sphere(far_sphere_vertices);
#endif // CGAL_CONCURRENT_TRIANGULATION_3_ADD_TEMPORARY_POINTS_ON_FAR_SPHERE
    }
    // Sequential
    else
#endif
    {
      Vertex_handle hint;
      for(typename std::vector<std::size_t>::const_iterator
          it = indices.begin(), end = indices.end(); it != end; ++it)
      {
        hint = insert(points[*it], hint);
        if(hint != Vertex_handle())
          hint->info() = infos[*it];
      }
    }

    return number_of_vertices() - n;
  }

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

  template <class  InputIterator_1,class InputIterator_2>
  std::ptrdiff_t
  insert(boost::zip_iterator< boost::tuple<InputIterator_1,InputIterator_2> > first,
          boost::zip_iterator< boost::tuple<InputIterator_1,InputIterator_2> > last,
          std::enable_if_t<
              std::is_convertible_v< typename std::iterator_traits<InputIterator_1>::value_type, Point > &&
              std::is_convertible_v< typename std::iterator_traits<InputIterator_2>::value_type, typename internal::Info_check<typename Triangulation_data_structure::Vertex>::type >
          >* =nullptr)
  {
    return insert_with_info< boost::tuple<Point, typename internal::Info_check<
        typename Triangulation_data_structure::Vertex>::type> >(first,last);
  }
#endif //CGAL_TRIANGULATION_3_DONT_INSERT_RANGE_OF_POINTS_WITH_INFO

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

  Vertex_handle insert(const Point& p, Cell_handle start = Cell_handle(),
                       bool *could_lock_zone = nullptr);

  Vertex_handle insert(const Point& p, Locate_type lt,
                       Cell_handle c, int li, int,
                       bool *could_lock_zone = nullptr);

public: // 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);

public:
  template <class OutputIteratorBoundaryFacets,
            class OutputIteratorCells,
            class OutputIteratorInternalFacets>
  Triple<OutputIteratorBoundaryFacets,
         OutputIteratorCells,
         OutputIteratorInternalFacets>
  find_conflicts(const Point& p, Cell_handle c,
                 OutputIteratorBoundaryFacets bfit,
                 OutputIteratorCells cit,
                 OutputIteratorInternalFacets ifit,
                 bool *could_lock_zone = nullptr) const
  {
    CGAL_precondition(dimension() >= 2);

    std::vector<Cell_handle> cells;
    cells.reserve(32);
    std::vector<Facet> facets;
    facets.reserve(64);

    if(dimension() == 2)
    {
      Conflict_tester_2 tester(p, this);
      ifit = Tr_Base::find_conflicts(c, tester,
                                     make_triple(std::back_inserter(facets),
                                                 std::back_inserter(cells),
                                                 ifit), could_lock_zone).third;
    }
    else
    {
      Conflict_tester_3 tester(p, this);
      ifit = Tr_Base::find_conflicts(c, tester,
                                     make_triple(std::back_inserter(facets),
                                                 std::back_inserter(cells),
                                                 ifit), could_lock_zone).third;
    }

    // Reset the conflict flag on the boundary.
    for(typename std::vector<Facet>::iterator fit=facets.begin();
                                              fit != facets.end(); ++fit)
    {
      fit->first->neighbor(fit->second)->tds_data().clear();
      *bfit++ = *fit;
    }

    // Reset the conflict flag in the conflict cells.
    for(typename std::vector<Cell_handle>::iterator ccit=cells.begin();
                                                    ccit != cells.end(); ++ccit)
    {
      (*ccit)->tds_data().clear();
      *cit++ = *ccit;
    }
    return make_triple(bfit, cit, ifit);
  }

  template <class OutputIteratorBoundaryFacets, class OutputIteratorCells>
  std::pair<OutputIteratorBoundaryFacets, OutputIteratorCells>
  find_conflicts(const Point& p, Cell_handle c,
                 OutputIteratorBoundaryFacets bfit,
                 OutputIteratorCells cit,
                 bool *could_lock_zone = nullptr) const
  {
      Triple<OutputIteratorBoundaryFacets,
             OutputIteratorCells,
             Emptyset_iterator> t = find_conflicts(p, c, bfit, cit,
                                                   Emptyset_iterator(),
                                                   could_lock_zone);
      return std::make_pair(t.first, t.second);
  }

#ifndef CGAL_NO_DEPRECATED_CODE
  // Returns the vertices on the boundary of the conflict hole.
  template <class OutputIterator>
  OutputIterator
  vertices_in_conflict(const Point&p, Cell_handle c, OutputIterator res) const
  {
    return vertices_on_conflict_zone_boundary(p, c, res);
  }
#endif // CGAL_NO_DEPRECATED_CODE

  // Returns the vertices on the boundary of the conflict hole.
  template <class OutputIterator>
  OutputIterator
  vertices_on_conflict_zone_boundary(const Point&p, Cell_handle c,
                                     OutputIterator res) const
  {
    CGAL_precondition(dimension() >= 2);

    // Get the facets on the boundary of the hole.
    std::vector<Facet> facets;
    find_conflicts(p, c, std::back_inserter(facets),
                   Emptyset_iterator(), Emptyset_iterator());

    // Then extract uniquely the vertices.
    std::set<Vertex_handle> vertices;
    if(dimension() == 3)
    {
      for(typename std::vector<Facet>::const_iterator i = facets.begin();
          i != facets.end(); ++i)
      {
        vertices.insert(i->first->vertex((i->second+1)&3));
        vertices.insert(i->first->vertex((i->second+2)&3));
        vertices.insert(i->first->vertex((i->second+3)&3));
      }
    }
    else
    {
      for(typename std::vector<Facet>::const_iterator i = facets.begin();
          i != facets.end(); ++i)
      {
        vertices.insert(i->first->vertex(cw(i->second)));
        vertices.insert(i->first->vertex(ccw(i->second)));
      }
    }

    return std::copy(vertices.begin(), vertices.end(), res);
  }

  // REMOVE
  void remove(Vertex_handle v);
  // Concurrency-safe
  // See Triangulation_3::remove for more information
  bool remove(Vertex_handle v, bool *could_lock_zone);

  // return new cells (internal)
  template <class OutputItCells>
  void remove_and_give_new_cells(Vertex_handle v,
                                 OutputItCells fit);

  template < typename InputIterator >
  size_type remove(InputIterator first, InputIterator beyond)
  {
    CGAL_precondition(!this->does_repeat_in_range(first, beyond));
    size_type n = number_of_vertices();

#ifdef CGAL_TRIANGULATION_3_PROFILING
    WallClockTimer t;
#endif

    // Parallel
#ifdef CGAL_LINKED_WITH_TBB
    if(this->is_parallel())
    {
      // TODO: avoid that by asking for random-access iterators?
      std::vector<Vertex_handle> vertices(first, beyond);
      tbb::concurrent_vector<Vertex_handle> vertices_to_remove_sequentially;

      tbb::parallel_for(tbb::blocked_range<size_t>(0, vertices.size()),
                        Remove_point<Self>(*this, vertices, vertices_to_remove_sequentially));

      // Do the rest sequentially
      for(typename tbb::concurrent_vector<Vertex_handle>::const_iterator
              it = vertices_to_remove_sequentially.begin(),
              it_end = vertices_to_remove_sequentially.end()
          ; it != it_end
          ; ++it)
      {
        remove(*it);
      }
    }
    // Sequential
    else
#endif // CGAL_LINKED_WITH_TBB
    {
      while(first != beyond)
      {
        remove(*first);
        ++first;
      }
    }

#ifdef CGAL_TRIANGULATION_3_PROFILING
    double elapsed = t.elapsed();
    Rcpp::Rcerr << "Points removed in " << elapsed << " seconds." << std::endl;
#endif
    return n - number_of_vertices();
  }

  template < typename InputIterator >
  size_type remove_cluster(InputIterator first, InputIterator beyond)
  {
    Self tmp;
    Vertex_remover<Self> remover(tmp);
    return Tr_Base::remove(first, beyond, remover);
  }

  // MOVE
  Vertex_handle move_if_no_collision(Vertex_handle v, const Point& p);
  Vertex_handle move(Vertex_handle v, const Point& p);

  // return new cells(internal)
  template <class OutputItCells>
  Vertex_handle move_if_no_collision_and_give_new_cells(Vertex_handle v,
                                                        const Point& p,
                                                        OutputItCells fit);

private:
  Bounded_side
  side_of_sphere(Vertex_handle v0, Vertex_handle v1,
                 Vertex_handle v2, Vertex_handle v3,
                 const Point& p, bool perturb) const;
public:
  // Queries
  Bounded_side side_of_sphere(Cell_handle c, const Point& p, bool perturb = false) const
  {
    return side_of_sphere(c->vertex(0), c->vertex(1),
                          c->vertex(2), c->vertex(3), p, perturb);
  }

  Bounded_side side_of_circle(const Facet& f, const Point& p, bool perturb = false) const
  {
    return side_of_circle(f.first, f.second, p, perturb);
  }

  Bounded_side side_of_circle(Cell_handle c, int i, const Point& p, bool perturb = false) const;

  Vertex_handle nearest_vertex_in_cell(const Point& p, Cell_handle c) const;
  Vertex_handle nearest_vertex(const Point& p, Cell_handle c = Cell_handle()) const;

  bool is_Gabriel(Cell_handle c, int i) const;
  bool is_Gabriel(Cell_handle c, int i, int j) const;
  bool is_Gabriel(const Facet& f)const ;
  bool is_Gabriel(const Edge& e) const;

  bool is_delaunay_after_displacement(Vertex_handle v, const Point& p) const;

// Dual functions
  Point dual(Cell_handle c) const;
  Object dual(const Facet& f) const { return dual(f.first, f.second); }
  Object dual(Cell_handle c, int i) const;

  Line dual_support(Cell_handle c, int i) const;

  bool is_valid(bool verbose = false, int level = 0) const;
  bool is_valid(Cell_handle c, bool verbose = false, int level = 0) const;

  template < class Stream>
  Stream& draw_dual(Stream& os)
  {
    for(Finite_facets_iterator fit = finite_facets_begin(),
        end = finite_facets_end();
        fit != end; ++fit) {
      Object o = dual(*fit);
      if(const Segment* s = object_cast<Segment>(&o))
        os << *s;
      else if(const Ray* r = object_cast<Ray>(&o))
        os << *r;
      else if(const Point* p = object_cast<Point>(&o))
        os << *p;
    }
    return os;
  }

protected:

  Vertex_handle
  nearest_vertex(const Point& p, Vertex_handle v, Vertex_handle w) const
  {
    // In case of equality, v is returned.
    CGAL_precondition(v != w);

    if(is_infinite(v))
      return w;
    if(is_infinite(w))
      return v;
    return less_distance(p, w->point(), v->point()) ? w : v;
  }

  class Conflict_tester_3
  {
      const Point& p;
      const Self *t;

  public:
    Conflict_tester_3(const Point& pt, const Self *tr)
      : p(pt), t(tr) {}

    bool operator()(const Cell_handle c) const
    {
      return t->side_of_sphere(c, p, true) == ON_BOUNDED_SIDE;
    }

    Oriented_side compare_weight(const Point& , const Point& ) const
    {
      return ZERO;
    }

    bool test_initial_cell(Cell_handle) const
    {
      return true;
    }
  };

  class Conflict_tester_2
  {
      const Point& p;
      const Self *t;

  public:
    Conflict_tester_2(const Point& pt, const Self *tr)
      : p(pt), t(tr) {}

    bool operator()(const Cell_handle c) const
    {
      return t->side_of_circle(c, 3, p, true) == ON_BOUNDED_SIDE;
    }

    Oriented_side compare_weight(const Point& , const Point& ) const
    {
      return ZERO;
    }

    bool test_initial_cell(Cell_handle) const
    {
      return true;
    }
  };
  class Hidden_point_visitor
  {
  public:

    Hidden_point_visitor() {}

    template <class InputIterator>
    void process_cells_in_conflict(InputIterator, InputIterator) const {}
    void reinsert_vertices(Vertex_handle) {}
    Vertex_handle replace_vertex(Cell_handle c, int index, const Point& )
    {
      return c->vertex(index);
    }
    void hide_point(Cell_handle, const Point& ) {}
  };

#ifdef CGAL_LINKED_WITH_TBB
  // Functor for parallel insert(begin, end) function
  template <typename DT>
  class Insert_point
  {
    typedef typename DT::Point                          Point;
    typedef typename DT::Vertex_handle                  Vertex_handle;

    DT& m_dt;
    const std::vector<Point>& m_points;
    tbb::enumerable_thread_specific<Vertex_handle>& m_tls_hint;

  public:
    // Constructor
    Insert_point(DT& dt,
                 const std::vector<Point>& points,
                 tbb::enumerable_thread_specific<Vertex_handle>& tls_hint)
    : m_dt(dt), m_points(points), m_tls_hint(tls_hint)
    {}

    // operator()
    void operator()(const tbb::blocked_range<size_t>& r) const
    {
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
      static Profile_branch_counter_3 bcounter(
        "early withdrawals / late withdrawals / successes [Delaunay_tri_3::insert]");
#endif

      Vertex_handle& hint = m_tls_hint.local();
      for(std::size_t i_point = r.begin() ; i_point != r.end() ; ++i_point)
      {
        bool success = false;
        while(!success)
        {
          if(m_dt.try_lock_vertex(hint) && m_dt.try_lock_point(m_points[i_point]))
          {
            bool could_lock_zone;
            Vertex_handle new_hint = m_dt.insert(m_points[i_point], hint, &could_lock_zone);

            m_dt.unlock_all_elements();

            if(could_lock_zone)
            {
              hint = new_hint;
              success = true;
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
              ++bcounter;
#endif
            }
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
            else
            {
              bcounter.increment_branch_1(); // THIS is a late withdrawal!
            }
#endif
          }
          else
          {
            m_dt.unlock_all_elements();

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
            bcounter.increment_branch_2(); // THIS is an early withdrawal!
#endif
          }
        }

      }
    }
  };

  // Functor for parallel insert_with_info(begin, end) function
  template <typename DT>
  class Insert_point_with_info
  {
    typedef typename DT::Point                                  Point;
    typedef typename DT::Vertex_handle                          Vertex_handle;
    typedef typename DT::Triangulation_data_structure::Vertex::Info Info;

    DT& m_dt;
    const std::vector<Point>& m_points;
    const std::vector<Info>& m_infos;
    const std::vector<std::size_t>& m_indices;
    tbb::enumerable_thread_specific<Vertex_handle>& m_tls_hint;

  public:
    // Constructor
    Insert_point_with_info(DT& dt,
                 const std::vector<Point>& points,
                 const std::vector<Info>& infos,
                 const std::vector<std::size_t>& indices,
                 tbb::enumerable_thread_specific<Vertex_handle>& tls_hint)
    : m_dt(dt), m_points(points), m_infos(infos), m_indices(indices),
      m_tls_hint(tls_hint)
    {}

    // operator()
    void operator()(const tbb::blocked_range<size_t>& r) const
    {
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
      static Profile_branch_counter_3 bcounter(
        "early withdrawals / late withdrawals / successes [Delaunay_tri_3::insert_with_info]");
#endif

      Vertex_handle& hint = m_tls_hint.local();
      for(std::size_t i_idx = r.begin() ; i_idx != r.end() ; ++i_idx)
      {
        bool success = false;
        std::ptrdiff_t i_point = m_indices[i_idx];
        const Point& p = m_points[i_point];
        while(!success)
        {
          if(m_dt.try_lock_vertex(hint) && m_dt.try_lock_point(p))
          {
            bool could_lock_zone;
            Vertex_handle new_hint = m_dt.insert(p, hint, &could_lock_zone);

            m_dt.unlock_all_elements();

            if(could_lock_zone)
            {
              hint = new_hint;
              if(hint!=Vertex_handle()) hint->info()=m_infos[i_point];
              success = true;
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
              ++bcounter;
#endif
            }
#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
            else
            {
              bcounter.increment_branch_1(); // THIS is a late withdrawal!
            }
#endif
          }
          else
          {
            m_dt.unlock_all_elements();

#ifdef CGAL_CONCURRENT_TRIANGULATION_3_PROFILING
            bcounter.increment_branch_2(); // THIS is an early withdrawal!
#endif
          }
        }

      }
    }
  };

  // Functor for parallel remove(begin, end) function
  template <typename DT>
  class Remove_point
  {
    typedef typename DT::Point                          Point;
    typedef typename DT::Vertex_handle                  Vertex_handle;

    DT& m_dt;
    const std::vector<Vertex_handle>& m_vertices;
    tbb::concurrent_vector<Vertex_handle>& m_vertices_to_remove_sequentially;

  public:
    // Constructor
    Remove_point(DT& dt,
                 const std::vector<Vertex_handle>& vertices,
                 tbb::concurrent_vector<Vertex_handle>& vertices_to_remove_sequentially)
    : m_dt(dt), m_vertices(vertices),
      m_vertices_to_remove_sequentially(vertices_to_remove_sequentially)
    {}

    // operator()
    void operator()(const tbb::blocked_range<size_t>& r) const
    {
      for(size_t i_vertex = r.begin() ; i_vertex != r.end() ; ++i_vertex)
      {
        Vertex_handle v = m_vertices[i_vertex];
        bool could_lock_zone, needs_to_be_done_sequentially;
        do
        {
          needs_to_be_done_sequentially =
            !m_dt.remove(v, &could_lock_zone);
          m_dt.unlock_all_elements();
        }
        while(!could_lock_zone);

        if(needs_to_be_done_sequentially)
          m_vertices_to_remove_sequentially.push_back(v);
      }
    }
  };
#endif // CGAL_LINKED_WITH_TBB

  template < class DelaunayTriangulation_3 >
  class Vertex_remover;

  template < class DelaunayTriangulation_3 >
  class Vertex_inserter;

  friend class Conflict_tester_3;
  friend class Conflict_tester_2;

  Hidden_point_visitor hidden_point_visitor;
};

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
insert(const Point& p, Cell_handle start, bool *could_lock_zone)
{
  Locate_type lt;
  int li = -1, lj = -1;

  // Parallel
  if(could_lock_zone)
  {
    Cell_handle c = locate(p, lt, li, lj, start, could_lock_zone);
    if(*could_lock_zone)
      return insert(p, lt, c, li, lj, could_lock_zone);
    else
      return Vertex_handle();
  }
  // Sequential
  else
  {
    Cell_handle c = locate(p, lt, li, lj, start);
    return insert(p, lt, c, li, lj);
  }
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
insert(const Point& p, Locate_type lt, Cell_handle c, int li, int lj, bool *could_lock_zone)
{
  switch(dimension())
  {
    case 3:
    {
      Conflict_tester_3 tester(p, this);
      Vertex_handle v = insert_in_conflict(p, lt, c, li, lj,
                                           tester, hidden_point_visitor, could_lock_zone);
      return v;
    }// dim 3
    case 2:
    {
      Conflict_tester_2 tester(p, this);
      return insert_in_conflict(p, lt, c, li, lj,
                                tester, hidden_point_visitor, could_lock_zone);
    }//dim 2
    default :
      // dimension <= 1
      // Do not use the generic insert.
      return Tr_Base::insert(p, c);
  }
}

template < class Gt, class Tds, class Lds >
template <class OutputItCells>
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,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 // dimension = 0
  {
    *fit++ = v->cell();
  }
  return v;
}

template < class Gt, class Tds, class Lds >
template <class OutputItCells>
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
insert_and_give_new_cells(const Point& p, OutputItCells fit, Vertex_handle hint)
{
  Vertex_handle v = insert(p, hint);
  int dimension = this->dimension();
  if(dimension == 3)
  {
    this->incident_cells(v, fit);
  }
  else if(dimension == 2)
  {
    Cell_handle c = v->cell(), end = c;
    do
    {
      *fit++ = c;
      int i = c->index(v);
      c = c->neighbor((i+1)%3);
    }
    while(c != end);
  }
  else if(dimension == 1)
  {
    Cell_handle c = v->cell();
    *fit++ = c;
    *fit++ = c->neighbor((~(c->index(v)))&1);
  }
  else // dimension = 0
  {
    *fit++ = v->cell();
  }
  return v;
}

template < class Gt, class Tds, class Lds >
template <class OutputItCells>
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,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 // dimension = 0
  {
    *fit++ = v->cell();
  }
  return v;
}


template <class Gt, class Tds, class Lds >
template <class DelaunayTriangulation_3>
class Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_remover
{
  typedef DelaunayTriangulation_3 Delaunay;

public:
  typedef std::nullptr_t Hidden_points_iterator;

  Vertex_remover(Delaunay &tmp_) : tmp(tmp_) {}

  Delaunay &tmp;

  void add_hidden_points(Cell_handle) {}
  Hidden_points_iterator hidden_points_begin() { return nullptr; }
  Hidden_points_iterator hidden_points_end() { return nullptr; }

  Bounded_side side_of_bounded_circle(const Point& p, const Point& q,
                                      const Point& r, const Point& s,
                                      bool perturb = false) const
  {
    return tmp.coplanar_side_of_bounded_circle(p,q,r,s,perturb);
  }
};

template <class Gt, class Tds, class Lds >
template <class DelaunayTriangulation_3>
class Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_inserter
{
  typedef DelaunayTriangulation_3 Delaunay;

public:
  typedef std::nullptr_t Hidden_points_iterator;

  Vertex_inserter(Delaunay &tmp_) : tmp(tmp_) {}

  Delaunay &tmp;

  void add_hidden_points(Cell_handle) {}
  Hidden_points_iterator hidden_points_begin() { return nullptr; }
  Hidden_points_iterator hidden_points_end() { return nullptr; }

  Vertex_handle insert(const Point& p, Locate_type lt, Cell_handle c, int li, int lj)
  {
    return tmp.insert(p, lt, c, li, lj);
  }

  Vertex_handle insert(const Point& p, Cell_handle c)
  {
    return tmp.insert(p, c);
  }

  Vertex_handle insert(const Point& p)
  {
    return tmp.insert(p);
  }
};

template < class Gt, class Tds, class Lds >
void
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
remove(Vertex_handle v)
{
  Self tmp;
  Vertex_remover<Self> remover(tmp);
  Tr_Base::remove(v, remover);

  CGAL_expensive_postcondition(is_valid());
}

template < class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
remove(Vertex_handle v, bool *could_lock_zone)
{
  Self tmp;
  Vertex_remover<Self> remover(tmp);
  bool ret = Tr_Base::remove(v, remover, could_lock_zone);

  CGAL_expensive_postcondition(is_valid());
  return ret;
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
move_if_no_collision(Vertex_handle v, const Point& p)
{
  Self tmp;
  Vertex_remover<Self> remover(tmp);
  Vertex_inserter<Self> inserter(*this);
  Vertex_handle res = Tr_Base::move_if_no_collision(v,p,remover,inserter);

  CGAL_expensive_postcondition(is_valid());
        return res;
}

template <class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
move(Vertex_handle v, const Point& p)
{
  CGAL_precondition(!is_infinite(v));
  if(v->point() == p)
    return v;

  Self tmp;
  Vertex_remover<Self> remover(tmp);
  Vertex_inserter<Self> inserter(*this);
        return Tr_Base::move(v,p,remover,inserter);
}

template < class Gt, class Tds, class Lds >
template <class OutputItCells>
void
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
remove_and_give_new_cells(Vertex_handle v, OutputItCells fit)
{
  Self tmp;
  Vertex_remover<Self> remover(tmp);
  Tr_Base::remove_and_give_new_cells(v,remover,fit);

  CGAL_expensive_postcondition(is_valid());
}

template < class Gt, class Tds, class Lds >
template <class OutputItCells>
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
move_if_no_collision_and_give_new_cells(Vertex_handle v, const Point& p,
                                        OutputItCells fit)
{
  Self tmp;
  Vertex_remover<Self> remover(tmp);
  Vertex_inserter<Self> inserter(*this);
  Vertex_handle res =
    Tr_Base::move_if_no_collision_and_give_new_cells(v,p,
      remover,inserter,fit);

  CGAL_expensive_postcondition(is_valid());
        return res;
}

template < class Gt, class Tds, class Lds >
Oriented_side
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
side_of_oriented_sphere(const Point& p0, const Point& p1, const Point& p2,
                        const Point& p3, const Point& p, bool perturb) const
{
  CGAL_precondition(orientation(p0, p1, p2, p3) == POSITIVE);

  Oriented_side os =
      geom_traits().side_of_oriented_sphere_3_object()(p0, p1, p2, p3, p);

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

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

  // We sort the points lexicographically.
  const Point * points[5] = {&p0, &p1, &p2, &p3, &p};
  std::sort(points, points + 5, typename Tr_Base::Perturbation_order(this));

  // We successively look whether the leading monomial, then 2nd monomial
  // of the determinant has non null coefficient.
  // 2 iterations are enough (cf paper)
  for(int i=4; i>2; --i)
  {
    if(points[i] == &p)
      return ON_NEGATIVE_SIDE; // since p0 p1 p2 p3 are non coplanar
    // and positively oriented
    Orientation o;
    if(points[i] == &p3 && (o = orientation(p0,p1,p2,p)) != COPLANAR)
      return o;
    if(points[i] == &p2 && (o = orientation(p0,p1,p,p3)) != COPLANAR)
      return o;
    if(points[i] == &p1 && (o = orientation(p0,p,p2,p3)) != COPLANAR)
      return o;
    if(points[i] == &p0 && (o = orientation(p,p1,p2,p3)) != COPLANAR)
      return o;
  }

  CGAL_assertion(false);
  return ON_NEGATIVE_SIDE;
}

template < class Gt, class Tds, class Lds >
Bounded_side
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
coplanar_side_of_bounded_circle(const Point& p0, const Point& p1,
                                const Point& p2, const Point& p, bool perturb) const
{
  // In dim==2, we should even be able to assert orient == POSITIVE.
  CGAL_precondition(coplanar_orientation(p0, p1, p2)
                    != COLLINEAR);

  Bounded_side bs =
      geom_traits().coplanar_side_of_bounded_circle_3_object()(p0, p1, p2, p);

  if(bs != ON_BOUNDARY || !perturb)
    return bs;

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

  // We sort the points lexicographically.
  const Point * points[4] = {&p0, &p1, &p2, &p};
  std::sort(points, points + 4, typename Tr_Base::Perturbation_order(this));

  Orientation local = coplanar_orientation(p0, p1, p2);

  // we successively look whether the leading monomial, then 2nd monimial,
  // then 3rd monomial, of the determinant which has non null coefficient
  // [syl] : TODO : Probably it can be stopped earlier like the 3D version
  for(int i=3; i>0; --i)
  {
    if(points[i] == &p)
      return Bounded_side(NEGATIVE); // since p0 p1 p2 are non collinear
                                     // but not necessarily positively oriented
    Orientation o;
    if(points[i] == &p2 && (o = coplanar_orientation(p0,p1,p)) != COLLINEAR)
      // [syl] : TODO : I'm not sure of the signs here (nor the rest :)
      return Bounded_side(o*local);
    if(points[i] == &p1 && (o = coplanar_orientation(p0,p,p2)) != COLLINEAR)
      return Bounded_side(o*local);
    if(points[i] == &p0 && (o = coplanar_orientation(p,p1,p2)) != COLLINEAR)
      return Bounded_side(o*local);
  }

  // case when the first non null coefficient is the coefficient of
  // the 4th monomial
  // moreover, the tests (points[] == &p) were false up to here, so the
  // monomial corresponding to p is the only monomial with non-zero
  // coefficient, it is equal to coplanar_orient(p0,p1,p2) == positive
  // so, no further test is required
  return Bounded_side(-local); //ON_UNBOUNDED_SIDE;
}

template < class Gt, class Tds, class Lds >
Bounded_side
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
side_of_sphere(Vertex_handle v0, Vertex_handle v1,
               Vertex_handle v2, Vertex_handle v3,
               const Point& p, bool perturb) const
{
  CGAL_precondition(dimension() == 3);

  if(is_infinite(v0))
  {
    Orientation o = orientation(v2->point(), v1->point(), v3->point(), p);
    if(o != COPLANAR)
      return Bounded_side(o);

    return coplanar_side_of_bounded_circle(v2->point(), v1->point(), v3->point(), p, perturb);
  }

  if(is_infinite(v1))
  {
    Orientation o = orientation(v2->point(), v3->point(), v0->point(), p);
    if(o != COPLANAR)
      return Bounded_side(o);

    return coplanar_side_of_bounded_circle(v2->point(), v3->point(), v0->point(), p, perturb);
  }

  if(is_infinite(v2))
  {
    Orientation o = orientation(v1->point(), v0->point(), v3->point(), p);
    if(o != COPLANAR)
      return Bounded_side(o);

    return coplanar_side_of_bounded_circle(v1->point(), v0->point(), v3->point(), p, perturb);
  }

  if(is_infinite(v3))
  {
    Orientation o = orientation(v0->point(), v1->point(), v2->point(), p);
    if(o != COPLANAR)
      return Bounded_side(o);

    return coplanar_side_of_bounded_circle(v0->point(), v1->point(), v2->point(), p, perturb);
  }

  return (Bounded_side) side_of_oriented_sphere(v0->point(), v1->point(), v2->point(), v3->point(), p, perturb);
}

template < class Gt, class Tds, class Lds >
Bounded_side
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
side_of_circle(Cell_handle c, int i, const Point& p, bool perturb) const
{
  // precondition : dimension >=2
  // in dimension 3, - for a finite facet
  // returns ON_BOUNDARY if the point lies on the circle,
  // ON_UNBOUNDED_SIDE when exterior, ON_BOUNDED_SIDE
  // interior
  // for an infinite facet, considers the plane defined by the
  // adjacent finite facet of the same cell, and does the same as in
  // dimension 2 in this plane
  // in dimension 2, for an infinite facet
  // in this case, returns ON_BOUNDARY if the point lies on the
  // finite edge (endpoints included)
  // ON_BOUNDED_SIDE for a point in the open half-plane
  // ON_UNBOUNDED_SIDE elsewhere

  CGAL_precondition(dimension() >= 2);
  int i3 = 5;

  if(dimension() == 2)
  {
    CGAL_precondition(i == 3);
    // the triangulation is supposed to be valid, ie the facet
    // with vertices 0 1 2 in this order is positively oriented
    if(! c->has_vertex(infinite_vertex(), i3))
      return coplanar_side_of_bounded_circle(c->vertex(0)->point(),
                                              c->vertex(1)->point(),
                                              c->vertex(2)->point(),
                                              p, perturb);
    // else infinite facet
    // v1, v2 finite vertices of the facet such that v1,v2,infinite
    // is positively oriented
    Vertex_handle v1 = c->vertex(ccw(i3)),
                  v2 = c->vertex(cw(i3));
    CGAL_assertion(coplanar_orientation(v1->point(), v2->point(),
                                        mirror_vertex(c, i3)->point()) == NEGATIVE);
    Orientation o = coplanar_orientation(v1->point(), v2->point(), p);
    if(o != COLLINEAR)
        return Bounded_side(o);
    // because p is in f iff
    // it does not lie on the same side of v1v2 as vn
    int i_e;
    Locate_type lt;
    // case when p collinear with v1v2
    return side_of_segment(p, v1->point(), v2->point(), lt, i_e);
  }

  // else dimension == 3
  CGAL_precondition(i >= 0 && i < 4);
  if((! c->has_vertex(infinite_vertex(),i3)) || (i3 != i))
  {
    // finite facet
    // initialization of i0 i1 i2, vertices of the facet positively
    // oriented (if the triangulation is valid)
    int i0 = (i>0) ? 0 : 1;
    int i1 = (i>1) ? 1 : 2;
    int i2 = (i>2) ? 2 : 3;
    CGAL_precondition(coplanar(c->vertex(i0)->point(),
                               c->vertex(i1)->point(),
                               c->vertex(i2)->point(),
                               p));
    return coplanar_side_of_bounded_circle(c->vertex(i0)->point(),
                                            c->vertex(i1)->point(),
                                            c->vertex(i2)->point(),
                                            p, perturb);
  }

  //else infinite facet
  // v1, v2 finite vertices of the facet such that v1,v2,infinite
  // is positively oriented
  Vertex_handle v1 = c->vertex(next_around_edge(i3,i)),
                v2 = c->vertex(next_around_edge(i,i3));
  Orientation o = (Orientation)
                  (coplanar_orientation(v1->point(), v2->point(),
                                         c->vertex(i)->point()) *
                   coplanar_orientation(v1->point(), v2->point(), p));
  // then the code is duplicated from 2d case
  if(o != COLLINEAR)
      return Bounded_side(-o);
  // because p is in f iff
  // it is not on the same side of v1v2 as c->vertex(i)
  int i_e;
  Locate_type lt;
  // case when p collinear with v1v2
  return side_of_segment(p, v1->point(), v2->point(), lt, i_e);
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
nearest_vertex_in_cell(const Point& p, Cell_handle c) const
{
// Returns the finite vertex of the cell c which is the closest to p.
  CGAL_precondition(dimension() >= 0);

  Vertex_handle nearest = nearest_vertex(p, c->vertex(0), c->vertex(1));
  if(dimension() >= 2)
  {
    nearest = nearest_vertex(p, nearest, c->vertex(2));
    if(dimension() == 3)
      nearest = nearest_vertex(p, nearest, c->vertex(3));
  }
  return nearest;
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Vertex_handle
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
nearest_vertex(const Point& p, Cell_handle start) const
{
  if(number_of_vertices() == 0)
    return Vertex_handle();

  // Use a brute-force algorithm if dimension < 3.
  if(dimension() < 3)
  {
    Finite_vertices_iterator vit = finite_vertices_begin();
    Vertex_handle res = vit;
    ++vit;
    for(Finite_vertices_iterator end = finite_vertices_end(); vit != end; ++vit)
      res = nearest_vertex(p, res, vit);
    return res;
  }

  Locate_type lt;
  int li, lj;
  Cell_handle c = locate(p, lt, li, lj, start);
  if(lt == Tr_Base::VERTEX)
    return c->vertex(li);

  // - start with the closest vertex from the located cell.
  // - repeatedly take the nearest of its incident vertices if any
  // - if not, we're done.
  Vertex_handle nearest = nearest_vertex_in_cell(p, c);
  std::vector<Vertex_handle> vs;
  vs.reserve(32);
  while(true)
  {
    Vertex_handle tmp = nearest;
    adjacent_vertices(nearest, std::back_inserter(vs));
    for(typename std::vector<Vertex_handle>::const_iterator
        vsit = vs.begin(); vsit != vs.end(); ++vsit)
    {
      tmp = nearest_vertex(p, tmp, *vsit);
    }

    if(tmp == nearest)
      break;

    vs.clear();
    nearest = tmp;
  }

  return nearest;
}

// This is not a fast version.
// The optimized version needs an int for book-keeping in
// tds() so as to avoiding the need to clear
// the tds marker in each cell (which is an unsigned char)
// Also the visitor in TDS could be more clever.
// The Delaunay triangulation which filters displacements
// will do these optimizations.
template <class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_delaunay_after_displacement(Vertex_handle v, const Point& p) const
{
  CGAL_precondition(!this->is_infinite(v));
  CGAL_precondition(this->dimension() == 2);
  CGAL_precondition(!this->test_dim_down(v));
  if(v->point() == p)
    return true;

  Point ant = v->point();
  v->set_point(p);

  std::size_t size;

  // are incident cells well-oriented
  std::vector<Cell_handle> cells;
  cells.reserve(64);
  this->incident_cells(v, std::back_inserter(cells));
  size = cells.size();
  for(std::size_t i=0; i<size; i++)
  {
    Cell_handle c = cells[i];
    if(this->is_infinite(c)) continue;
    if(this->orientation(c->vertex(0)->point(), c->vertex(1)->point(),
                         c->vertex(2)->point(), c->vertex(3)->point()) != POSITIVE)
    {
      v->set_point(ant);
      return false;
    }
  }

  // are incident bi-cells Delaunay?
  std::vector<Facet> facets;
  facets.reserve(128);
  this->incident_facets(v, std::back_inserter(facets));
  size = facets.size();
  for(std::size_t i=0; i<size; i++)
  {
    const Facet& f = facets[i];
    Cell_handle c = f.first;
    int j = f.second;
    Cell_handle cj = c->neighbor(j);
    int mj = this->mirror_index(c, j);
    Vertex_handle h1 = c->vertex(j);
    if(this->is_infinite(h1))
    {
      if(this->side_of_sphere(c, cj->vertex(mj)->point(), true) != ON_UNBOUNDED_SIDE)
      {
        v->set_point(ant);
        return false;
      }
    }
    else
    {
      if(this->side_of_sphere(cj, h1->point(), true) != ON_UNBOUNDED_SIDE)
      {
        v->set_point(ant);
        return false;
      }
    }
  }

  v->set_point(ant);
  return true;
}

template < class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_Gabriel(const Facet& f) const
{
  return is_Gabriel(f.first, f.second);
}

template < class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_Gabriel(Cell_handle c, int i) const
{
  CGAL_precondition(dimension() == 3 && !is_infinite(c,i));
  typename Geom_traits::Side_of_bounded_sphere_3 side_of_bounded_sphere =
    geom_traits().side_of_bounded_sphere_3_object();

  if((!is_infinite(c->vertex(i))) &&
     side_of_bounded_sphere (c->vertex(vertex_triple_index(i,0))->point(),
                             c->vertex(vertex_triple_index(i,1))->point(),
                             c->vertex(vertex_triple_index(i,2))->point(),
                             c->vertex(i)->point()) == ON_BOUNDED_SIDE)
    return false;

  Cell_handle neighbor = c->neighbor(i);
  int in = neighbor->index(c);

  if((!is_infinite(neighbor->vertex(in))) &&
     side_of_bounded_sphere(c->vertex(vertex_triple_index(i,0))->point(),
                            c->vertex(vertex_triple_index(i,1))->point(),
                            c->vertex(vertex_triple_index(i,2))->point(),
                            neighbor->vertex(in)->point()) == ON_BOUNDED_SIDE)
    return false;

  return true;
}

template < class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_Gabriel(const Edge& e) const
{
  return is_Gabriel(e.first, e.second, e.third);
}

template < class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_Gabriel(Cell_handle c, int i, int j) const
{
  CGAL_precondition(dimension() == 3 && !is_infinite(c,i,j));
  typename Geom_traits::Side_of_bounded_sphere_3 side_of_bounded_sphere =
    geom_traits().side_of_bounded_sphere_3_object();

  Facet_circulator fcirc = incident_facets(c,i,j), fdone(fcirc);
  Vertex_handle v1 = c->vertex(i);
  Vertex_handle v2 = c->vertex(j);
  do
  {
      // test whether the vertex of cc opposite to *fcirc
      // is inside the sphere defined by the edge e = (s, i,j)
      Cell_handle cc = (*fcirc).first;
      int ii = (*fcirc).second;
      if(!is_infinite(cc->vertex(ii)) &&
           side_of_bounded_sphere(v1->point(), v2->point(), cc->vertex(ii)->point())
          == ON_BOUNDED_SIDE) return false;
  }
  while(++fcirc != fdone);

  return true;
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Point
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
dual(Cell_handle c) const
{
  CGAL_precondition(dimension()==3);
  CGAL_precondition(! is_infinite(c));
  return c->circumcenter(geom_traits());
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Object
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
dual(Cell_handle c, int i) const
{
  CGAL_precondition(dimension()>=2);
  CGAL_precondition(! is_infinite(c,i));

  if(dimension() == 2)
  {
    CGAL_precondition(i == 3);
    return construct_object(construct_circumcenter(c->vertex(0)->point(),
                                                   c->vertex(1)->point(),
                                                   c->vertex(2)->point()));
  }

  // dimension() == 3
  Cell_handle n = c->neighbor(i);
  if(! is_infinite(c) && ! is_infinite(n))
    return construct_object(construct_segment(dual(c), dual(n)));

  // either n or c is infinite
  int in;
  if(is_infinite(c))
  {
    in = n->index(c);
  }
  else
  {
    n = c;
    in = i;
  }

  // n now denotes a finite cell, either c or c->neighbor(i)
  int ind[3] = {(in+1)&3,(in+2)&3,(in+3)&3};
  if((in&1) == 1)
      std::swap(ind[0], ind[1]);

  // in=0: 1 2 3
  // in=1: 3 2 0
  // in=2: 3 0 1
  // in=3: 1 0 2
  const Point& p = n->vertex(ind[0])->point();
  const Point& q = n->vertex(ind[1])->point();
  const Point& r = n->vertex(ind[2])->point();

  Line l = construct_equidistant_line(p, q, r);
  return construct_object(construct_ray(dual(n), l));
}

template < class Gt, class Tds, class Lds >
typename Delaunay_triangulation_3<Gt,Tds,Default,Lds>::Line
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
dual_support(Cell_handle c, int i) const
{
  CGAL_precondition(dimension()>=2);
  CGAL_precondition(! is_infinite(c,i));

  if(dimension() == 2)
  {
    CGAL_precondition(i == 3);
    return construct_equidistant_line(c->vertex(0)->point(),
                                       c->vertex(1)->point(),
                                       c->vertex(2)->point());
  }

  return construct_equidistant_line(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 >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_valid(bool verbose, int level) const
{
  if(! tds().is_valid(verbose,level))
  {
    if(verbose)
      Rcpp::Rcerr << "invalid data structure" << std::endl;

    CGAL_assertion(false);
    return false;
  }

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

    CGAL_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);
        for(int i=0; i<4; i++)
        {
          if(!is_infinite(it->neighbor(i)->vertex(it->neighbor(i)->index(it))))
          {
            if(side_of_sphere(it, it->neighbor(i)->vertex(
                                it->neighbor(i)->index(it))->point()) == ON_BOUNDED_SIDE)
            {
              if(verbose)
                Rcpp::Rcerr << "non-empty sphere " << std::endl;

              CGAL_assertion(false);
              return false;
            }
          }
        }
      }
      break;
    }
    case 2:
    {
      for(Finite_facets_iterator it = finite_facets_begin(), end = finite_facets_end(); it != end; ++it)
      {
        is_valid_finite((*it).first);
        for(int i=0; i<3; i++)
        {
          if(!is_infinite((*it).first->neighbor(i)->vertex(
                            (((*it).first)->neighbor(i))->index(
                              (*it).first))))
          {
            if(side_of_circle((*it).first, 3, (*it).first->neighbor(i)->
                              vertex((((*it).first)->neighbor(i))->index(
                                       (*it).first))->point()) == ON_BOUNDED_SIDE)
            {
              if(verbose)
                Rcpp::Rcerr << "non-empty circle " << std::endl;

              CGAL_assertion(false);
              return false;
            }
          }
        }
      }
      break;
    }
    case 1:
    {
      for(Finite_edges_iterator it = finite_edges_begin(), end = finite_edges_end(); it != end; ++it)
      {
        is_valid_finite((*it).first);
      }
      break;
    }
  }
  if(verbose)
      Rcpp::Rcerr << "Delaunay valid triangulation" << std::endl;

  return true;
}

template < class Gt, class Tds, class Lds >
bool
Delaunay_triangulation_3<Gt,Tds,Default,Lds>::
is_valid(Cell_handle c, bool verbose, int level) const
{
  if(! Tr_Base::is_valid(c, verbose, level))
  {
    if(verbose)
    {
      Rcpp::Rcerr << "combinatorically invalid cell" ;
      for(int i=0; i <= dimension(); i++)
        Rcpp::Rcerr << c->vertex(i)->point() << ", " ;
      Rcpp::Rcerr << std::endl;
    }
    CGAL_assertion(false);
    return false;
  }
  switch(dimension())
  {
    case 3:
    {
      if(! is_infinite(c))
      {
        is_valid_finite(c, verbose, level);
        for(int i=0; i<4; i++) {
          if(side_of_sphere(c, c->vertex((c->neighbor(i))->index(c))->point()) == ON_BOUNDED_SIDE)
          {
            if(verbose)
              Rcpp::Rcerr << "non-empty sphere " << std::endl;

            CGAL_assertion(false);
            return false;
          }
        }
      }
      break;
    }
    case 2:
    {
      if(! is_infinite(c,3))
      {
        for(int i=0; i<2; i++)
        {
          if(side_of_circle(c, 3, c->vertex(c->neighbor(i)->index(c))->point()) == ON_BOUNDED_SIDE)
          {
            if(verbose)
              Rcpp::Rcerr << "non-empty circle " << std::endl;

            CGAL_assertion(false);
            return false;
          }
        }
      }
      break;
    }
  }
  if(verbose)
    Rcpp::Rcerr << "Delaunay valid cell" << std::endl;

  return true;
}

} //namespace CGAL

#include <CGAL/Triangulation_3/internal/Delaunay_triangulation_hierarchy_3.h>

#include <CGAL/enable_warnings.h>

#endif // CGAL_DELAUNAY_TRIANGULATION_3_H

