dust3d/thirdparty/cgal/CGAL-5.1/include/CGAL/Delaunay_triangulation.h

// Copyright (c) 2009-2014 INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.1/Triangulation/include/CGAL/Delaunay_triangulation.h $
// $Id: Delaunay_triangulation.h 0779373 2020-03-26T13:31:46+01:00 Sébastien Loriot
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)    : Samuel Hornus

#ifndef CGAL_DELAUNAY_COMPLEX_H
#define CGAL_DELAUNAY_COMPLEX_H

#include <CGAL/license/Triangulation.h>

#include <CGAL/disable_warnings.h>

#include <CGAL/tss.h>
#include <CGAL/Triangulation.h>
#include <CGAL/Dimension.h>
#include <CGAL/Default.h>

#include <CGAL/boost/iterator/transform_iterator.hpp>

#include <algorithm>

namespace CGAL {

template< typename DCTraits, typename _TDS = Default >
class Delaunay_triangulation
: public Triangulation<DCTraits,
            typename Default::Get<_TDS, Triangulation_data_structure<
                             typename DCTraits::Dimension,
                             Triangulation_vertex<DCTraits>,
                             Triangulation_full_cell<DCTraits> >
                    >::type >
{
    typedef typename DCTraits::Dimension            Maximal_dimension_;
    typedef typename Default::Get<_TDS, Triangulation_data_structure<
                         Maximal_dimension_,
                         Triangulation_vertex<DCTraits>,
                         Triangulation_full_cell<DCTraits> >
                >::type                             TDS;
    typedef Triangulation<DCTraits, TDS>            Base;
    typedef Delaunay_triangulation<DCTraits, _TDS>  Self;

    typedef typename DCTraits::Side_of_oriented_sphere_d
                                                    Side_of_oriented_sphere_d;
    typedef typename DCTraits::Orientation_d        Orientation_d;

public: // PUBLIC NESTED TYPES

    typedef DCTraits                                Geom_traits;
    typedef typename Base::Triangulation_ds         Triangulation_ds;

    typedef typename Base::Vertex                   Vertex;
    typedef typename Base::Full_cell                Full_cell;
    typedef typename Base::Facet                    Facet;
    typedef typename Base::Face                     Face;

    typedef typename Base::Maximal_dimension        Maximal_dimension;
    typedef typename DCTraits::Point_d              Point;
    typedef typename DCTraits::Point_d              Point_d;

    typedef typename Base::Vertex_handle            Vertex_handle;
    typedef typename Base::Vertex_iterator          Vertex_iterator;
    typedef typename Base::Vertex_const_handle      Vertex_const_handle;
    typedef typename Base::Vertex_const_iterator    Vertex_const_iterator;

    typedef typename Base::Full_cell_handle         Full_cell_handle;
    typedef typename Base::Full_cell_iterator       Full_cell_iterator;
    typedef typename Base::Full_cell_const_handle   Full_cell_const_handle;
    typedef typename Base::Full_cell_const_iterator Full_cell_const_iterator;
    typedef typename Base::Finite_full_cell_const_iterator
                                                    Finite_full_cell_const_iterator;

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

    typedef typename Base::Locate_type              Locate_type;

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

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

public:
    typedef typename Base::Rotor Rotor;
    using Base::maximal_dimension;
    using Base::are_incident_full_cells_valid;
    using Base::coaffine_orientation_predicate;
    using Base::reset_flat_orientation;
    using Base::current_dimension;
    //using Base::star;
    //using Base::incident_full_cells;
    using Base::geom_traits;
    using Base::index_of_covertex;
    //using Base::index_of_second_covertex;
    using Base::infinite_vertex;
    using Base::rotate_rotor;
    using Base::insert_in_hole;
    using Base::insert_outside_convex_hull_1;
    using Base::is_infinite;
    using Base::locate;
    using Base::points_begin;
    using Base::points_end;
    using Base::set_neighbors;
    using Base::new_full_cell;
    using Base::number_of_vertices;
    using Base::orientation;
    using Base::tds;
    using Base::reorient_full_cells;
    using Base::full_cell;
    using Base::full_cells_begin;
    using Base::full_cells_end;
    using Base::finite_full_cells_begin;
    using Base::finite_full_cells_end;
    using Base::vertices_begin;
    using Base::vertices_end;
    // using Base::

private:
    //*** Side_of_oriented_subsphere_d ***
    typedef typename Base::Flat_orientation_d Flat_orientation_d;
    typedef typename Base::Construct_flat_orientation_d Construct_flat_orientation_d;
    typedef typename DCTraits::In_flat_side_of_oriented_sphere_d In_flat_side_of_oriented_sphere_d;
    // Wrapper
    struct Side_of_oriented_subsphere_d
    {
      boost::optional<Flat_orientation_d>* fop;
      Construct_flat_orientation_d cfo;
      In_flat_side_of_oriented_sphere_d ifsoos;

      Side_of_oriented_subsphere_d(
        boost::optional<Flat_orientation_d>& x,
        Construct_flat_orientation_d const&y,
        In_flat_side_of_oriented_sphere_d const&z)
      : fop(&x), cfo(y), ifsoos(z) {}

      template<class Iter>
      CGAL::Orientation operator()(Iter a, Iter b, const Point & p)const
      {
        if(!*fop)
          *fop=cfo(a,b);
        return ifsoos(fop->get(),a,b,p);
      }
    };
public:

// - - - - - - - - - - - - - - - - - - - - - - - - - - CREATION / CONSTRUCTORS

    Delaunay_triangulation(int dim, const Geom_traits &k = Geom_traits())
    : Base(dim, k)
    {
    }

    // With this constructor,
    // the user can specify a Flat_orientation_d object to be used for
    // orienting simplices of a specific dimension
    // (= preset_flat_orientation_.first)
    // It it used by the dark triangulations created by DT::remove
    Delaunay_triangulation(
      int dim,
      const std::pair<int, const Flat_orientation_d *> &preset_flat_orientation,
      const Geom_traits &k = Geom_traits())
    : Base(dim, preset_flat_orientation, k)
    {
    }

    ~Delaunay_triangulation() {}

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ACCESS

    // Not Documented
    Side_of_oriented_subsphere_d side_of_oriented_subsphere_predicate() const
    {
      return Side_of_oriented_subsphere_d (
        flat_orientation_,
        geom_traits().construct_flat_orientation_d_object(),
        geom_traits().in_flat_side_of_oriented_sphere_d_object()
      );
    }


    // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - REMOVALS

    Full_cell_handle remove(Vertex_handle);
    Full_cell_handle remove(const Point & p, Full_cell_handle hint = Full_cell_handle())
    {
        Locate_type lt;
        Face f(maximal_dimension());
        Facet ft;
        Full_cell_handle s = locate(p, lt, f, ft, hint);
        if( Base::ON_VERTEX == lt )
        {
            return remove(s->vertex(f.index(0)));
        }
        return Full_cell_handle();
    }

    template< typename ForwardIterator >
    void remove(ForwardIterator start, ForwardIterator end)
    {
        while( start != end )
            remove(*start++);
    }

    // Not documented
    void remove_decrease_dimension(Vertex_handle);

    // - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - INSERTIONS

    template< typename ForwardIterator >
    size_type insert(ForwardIterator start, ForwardIterator end)
    {
        size_type n = number_of_vertices();
        std::vector<Point> points(start, end);
        spatial_sort(points.begin(), points.end(), geom_traits());
        Full_cell_handle hint;
        for( typename std::vector<Point>::const_iterator p = points.begin(); p != points.end(); ++p )
        {
            hint = insert(*p, hint)->full_cell();
        }
        return number_of_vertices() - n;
    }
    Vertex_handle insert(const Point &, Locate_type, const Face &, const Facet &, Full_cell_handle);
    Vertex_handle insert(const Point & p, Full_cell_handle start = Full_cell_handle())
    {
        Locate_type lt;
        Face f(maximal_dimension());
        Facet ft;
        Full_cell_handle s = locate(p, lt, f, ft, start);
        return insert(p, lt, f, ft, s);
    }
    Vertex_handle insert(const Point & p, Vertex_handle hint)
    {
        CGAL_assertion( Vertex_handle() != hint );
        return insert(p, hint->full_cell());
    }
    Vertex_handle insert_outside_affine_hull(const Point &);
    Vertex_handle insert_in_conflicting_cell(const Point &, Full_cell_handle);

// - - - - - - - - - - - - - - - - - - - - - - - - - GATHERING CONFLICTING SIMPLICES

    bool is_in_conflict(const Point &, Full_cell_const_handle) const;
    template< class OrientationPredicate >
    Oriented_side perturbed_side_of_positive_sphere(const Point &,
            Full_cell_const_handle, const OrientationPredicate &) const;

    template< typename OutputIterator >
    Facet compute_conflict_zone(const Point &, Full_cell_handle, OutputIterator) const;

    template < typename OrientationPredicate, typename SideOfOrientedSpherePredicate >
    class Conflict_predicate
    {
        const Self & dc_;
        const Point & p_;
        OrientationPredicate ori_;
        SideOfOrientedSpherePredicate side_of_s_;
        int cur_dim_;
    public:
        Conflict_predicate(
                const Self & dc,
                const Point & p,
                const OrientationPredicate & ori,
                const SideOfOrientedSpherePredicate & side)
        : dc_(dc), p_(p), ori_(ori), side_of_s_(side), cur_dim_(dc.current_dimension()) {}

        inline
        bool operator()(Full_cell_const_handle s) const
        {
            bool ok;
            if( ! dc_.is_infinite(s) )
            {
                Oriented_side side = side_of_s_(dc_.points_begin(s), dc_.points_begin(s) + cur_dim_ + 1, p_);
                if( ON_POSITIVE_SIDE == side )
                    ok = true;
                else if( ON_NEGATIVE_SIDE == side )
                    ok = false;
                else
                    ok = ON_POSITIVE_SIDE == dc_.perturbed_side_of_positive_sphere<OrientationPredicate>(p_, s, ori_);
            }
            else
            {
                typedef typename Full_cell::Vertex_handle_const_iterator VHCI;
                typedef Substitute_point_in_vertex_iterator<VHCI> F;
                F spivi(dc_.infinite_vertex(), &p_);

                Orientation o =  ori_(
                  boost::make_transform_iterator(s->vertices_begin(), spivi),
                  boost::make_transform_iterator(s->vertices_begin() + cur_dim_ + 1,
                                                 spivi));

                if( POSITIVE == o )
                    ok = true;
                else if( o == NEGATIVE )
                    ok = false;
                else
                    ok = (*this)(s->neighbor( s->index( dc_.infinite_vertex() ) ));
            }
            return ok;
        }
    };

    template < typename ConflictPredicate >
    class Conflict_traversal_predicate
    {
        const Self & dc_;
        const ConflictPredicate & pred_;
    public:
        Conflict_traversal_predicate(const Self & dc, const ConflictPredicate & pred)
        : dc_(dc), pred_(pred)
        {}
        inline
        bool operator()(const Facet & f) const
        {
            return pred_(dc_.full_cell(f)->neighbor(dc_.index_of_covertex(f)));
        }
    };

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  VALIDITY

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

private:
    // Some internal types to shorten notation
    typedef typename Base::Coaffine_orientation_d Coaffine_orientation_d;
    using Base::flat_orientation_;
    typedef Conflict_predicate<Coaffine_orientation_d, Side_of_oriented_subsphere_d>
            Conflict_pred_in_subspace;
    typedef Conflict_predicate<Orientation_d, Side_of_oriented_sphere_d>
            Conflict_pred_in_fullspace;
    typedef Conflict_traversal_predicate<Conflict_pred_in_subspace>
            Conflict_traversal_pred_in_subspace;
    typedef Conflict_traversal_predicate<Conflict_pred_in_fullspace>
            Conflict_traversal_pred_in_fullspace;
};

// = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
// FUNCTIONS THAT ARE MEMBER METHODS:

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - REMOVALS

template< typename DCTraits, typename TDS >
typename Delaunay_triangulation<DCTraits, TDS>::Full_cell_handle
Delaunay_triangulation<DCTraits, TDS>
::remove( Vertex_handle v )
{
    CGAL_precondition( ! is_infinite(v) );
    CGAL_expensive_precondition( is_vertex(v) );

    // THE CASE cur_dim == 0
    if( 0 == current_dimension() )
    {
        remove_decrease_dimension(v);
        return Full_cell_handle();
    }
    else if( 1 == current_dimension() )
    {   // THE CASE cur_dim == 1
        if( 2 == number_of_vertices() )
        {
            remove_decrease_dimension(v);
            return Full_cell_handle();
        }
        Full_cell_handle left = v->full_cell();
        if( 0 == left->index(v) )
            left = left->neighbor(1);
        CGAL_assertion( 1 == left->index(v) );
        Full_cell_handle right = left->neighbor(0);

            tds().associate_vertex_with_full_cell(left, 1, right->vertex(1));
            set_neighbors(left, 0, right->neighbor(0), right->mirror_index(0));

        tds().delete_vertex(v);
        tds().delete_full_cell(right);
        return left;
    }

    // THE CASE cur_dim >= 2
    // Gather the finite vertices sharing an edge with |v|
    typedef typename Base::template Full_cell_set<Full_cell_handle> Simplices;
    Simplices simps;
    std::back_insert_iterator<Simplices> out(simps);
    tds().incident_full_cells(v, out);
    typedef std::set<Vertex_handle> Vertex_set;
    Vertex_set verts;
    Vertex_handle vh;
    for( typename Simplices::iterator it = simps.begin(); it != simps.end(); ++it )
        for( int i = 0; i <= current_dimension(); ++i )
        {
            vh = (*it)->vertex(i);
            if( is_infinite(vh) )
                continue;
            if( vh == v )
                continue;
            verts.insert(vh);
        }

    // After gathering finite neighboring vertices, create their Dark Delaunay triangulation
    typedef Triangulation_vertex<Geom_traits, Vertex_handle> Dark_vertex_base;
    typedef Triangulation_full_cell<Geom_traits,
        internal::Triangulation::Dark_full_cell_data<Self> > Dark_full_cell_base;
    typedef Triangulation_data_structure<Maximal_dimension, Dark_vertex_base, Dark_full_cell_base> Dark_tds;
    typedef Delaunay_triangulation<DCTraits, Dark_tds>   Dark_triangulation;
    typedef typename Dark_triangulation::Face             Dark_face;
    typedef typename Dark_triangulation::Facet            Dark_facet;
    typedef typename Dark_triangulation::Vertex_handle    Dark_v_handle;
    typedef typename Dark_triangulation::Full_cell_handle   Dark_s_handle;

    // If flat_orientation_ is defined, we give it the Dark triangulation
    // so that the orientation it uses for "current_dimension()"-simplices is
    // coherent with the global triangulation
    Dark_triangulation dark_side(
      maximal_dimension(),
      flat_orientation_ ?
      std::pair<int, const Flat_orientation_d *>(current_dimension(), flat_orientation_.get_ptr())
      : std::pair<int, const Flat_orientation_d *>((std::numeric_limits<int>::max)(), (Flat_orientation_d*) nullptr) );

    Dark_s_handle dark_s;
    Dark_v_handle dark_v;
    typedef std::map<Vertex_handle, Dark_v_handle> Vertex_map;
    Vertex_map light_to_dark;
    typename Vertex_set::iterator vit = verts.begin();
    while( vit != verts.end() )
    {
        dark_v = dark_side.insert((*vit)->point(), dark_s);
        dark_s = dark_v->full_cell();
        dark_v->data() = *vit;
        light_to_dark[*vit] = dark_v;
        ++vit;
    }

    if( dark_side.current_dimension() != current_dimension() )
    {
        CGAL_assertion( dark_side.current_dimension() + 1 == current_dimension() );
        // Here, the finite neighbors of |v| span a affine subspace of
        // dimension one less than the current dimension. Two cases are possible:
        if( (size_type)(verts.size() + 1) == number_of_vertices() )
        {
            remove_decrease_dimension(v);
            return Full_cell_handle();
        }
        else
        {   // |v| is strictly outside the convex hull of the rest of the points. This is an
            // easy case: first, modify the finite full_cells, then, delete the infinite ones.
            // We don't even need the Dark triangulation.
            Simplices infinite_simps;
            {
                Simplices finite_simps;
                for( typename Simplices::iterator it = simps.begin(); it != simps.end(); ++it )
                    if( is_infinite(*it) )
                        infinite_simps.push_back(*it);
                    else
                        finite_simps.push_back(*it);
                simps.swap(finite_simps);
            } // now, simps only contains finite simplices
            // First, modify the finite full_cells:
            for( typename Simplices::iterator it = simps.begin(); it != simps.end(); ++it )
            {
                int v_idx = (*it)->index(v);
                tds().associate_vertex_with_full_cell(*it, v_idx, infinite_vertex());
                }
            // Make the handles to infinite full cells searchable
            infinite_simps.make_searchable();
            // Then, modify the neighboring relation
            for( typename Simplices::iterator it = simps.begin(); it != simps.end(); ++it )
            {
                for( int i = 0; i <= current_dimension(); ++i )
                {
                    if (is_infinite((*it)->vertex(i)))
                        continue;
                    (*it)->vertex(i)->set_full_cell(*it);
                    Full_cell_handle n = (*it)->neighbor(i);
                    // Was |n| a finite full cell prior to removing |v| ?
                    if( ! infinite_simps.contains(n) )
                        continue;
                    int n_idx = n->index(v);
                    set_neighbors(*it, i, n->neighbor(n_idx), n->neighbor(n_idx)->index(n));
                }
            }
            Full_cell_handle ret_s;
            // Then, we delete the infinite full_cells
            for( typename Simplices::iterator it = infinite_simps.begin(); it != infinite_simps.end(); ++it )
                tds().delete_full_cell(*it);
            tds().delete_vertex(v);
            return simps.front();
        }
    }
    else //  From here on, dark_side.current_dimension() == current_dimension()
    {
        dark_side.infinite_vertex()->data() = infinite_vertex();
        light_to_dark[infinite_vertex()] = dark_side.infinite_vertex();
    }

    // Now, compute the conflict zone of v->point() in
    // the dark side. This is precisely the set of full_cells
    // that we have to glue back into the light side.
    Dark_face       dark_f(dark_side.maximal_dimension());
    Dark_facet      dark_ft;
    typename Dark_triangulation::Locate_type     lt;
    dark_s = dark_side.locate(v->point(), lt, dark_f, dark_ft);
    CGAL_assertion( lt != Dark_triangulation::ON_VERTEX
        && lt != Dark_triangulation::OUTSIDE_AFFINE_HULL );

    // |ret_s| is the full_cell that we return
    Dark_s_handle dark_ret_s = dark_s;
    Full_cell_handle ret_s;

    typedef typename Base::template Full_cell_set<Dark_s_handle> Dark_full_cells;
    Dark_full_cells conflict_zone;
    std::back_insert_iterator<Dark_full_cells> dark_out(conflict_zone);

    dark_ft = dark_side.compute_conflict_zone(v->point(), dark_s, dark_out);
    // Make the dark simplices in the conflict zone searchable
    conflict_zone.make_searchable();

    // THE FOLLOWING SHOULD MAYBE GO IN TDS.
    // Here is the plan:
    // 1. Pick any Facet from boundary of the light zone
    // 2. Find corresponding Facet on boundary of dark zone
    // 3. stitch.

    // 1. Build a facet on the boudary of the light zone:
    Full_cell_handle light_s = *simps.begin();
    Facet light_ft(light_s, light_s->index(v));

    // 2. Find corresponding Dark_facet on boundary of the dark zone
    Dark_full_cells dark_incident_s;
    for( int i = 0; i <= current_dimension(); ++i )
    {
        if( index_of_covertex(light_ft) == i )
            continue;
        Dark_v_handle dark_v = light_to_dark[full_cell(light_ft)->vertex(i)];
        dark_incident_s.clear();
        dark_out = std::back_inserter(dark_incident_s);
        dark_side.tds().incident_full_cells(dark_v, dark_out);
        for( typename Dark_full_cells::iterator it = dark_incident_s.begin(); it != dark_incident_s.end(); ++it )
        {
            (*it)->data().count_ += 1;
        }
    }

    for( typename Dark_full_cells::iterator it = dark_incident_s.begin(); it != dark_incident_s.end(); ++it )
    {
        if( current_dimension() != (*it)->data().count_ )
            continue;
        if( ! conflict_zone.contains(*it) )
            continue;
        // We found a full_cell incident to the dark facet corresponding to the light facet |light_ft|
        int ft_idx = 0;
        while( light_s->has_vertex( (*it)->vertex(ft_idx)->data() ) )
            ++ft_idx;
        dark_ft = Dark_facet(*it, ft_idx);
        break;
    }
    // Pre-3. Now, we are ready to traverse both boundary and do the stiching.

    // But first, we create the new full_cells in the light triangulation,
    // with as much adjacency information as possible.

    // Create new full_cells with vertices
    for( typename Dark_full_cells::iterator it = conflict_zone.begin(); it != conflict_zone.end(); ++it )
    {
        Full_cell_handle new_s = new_full_cell();
        (*it)->data().light_copy_ = new_s;
        for( int i = 0; i <= current_dimension(); ++i )
            tds().associate_vertex_with_full_cell(new_s, i, (*it)->vertex(i)->data());
        if( dark_ret_s == *it )
            ret_s = new_s;
    }

    // Setup adjacencies inside the hole
    for( typename Dark_full_cells::iterator it = conflict_zone.begin(); it != conflict_zone.end(); ++it )
    {
        Full_cell_handle new_s = (*it)->data().light_copy_;
        for( int i = 0; i <= current_dimension(); ++i )
            if( conflict_zone.contains((*it)->neighbor(i)) )
                tds().set_neighbors(new_s, i, (*it)->neighbor(i)->data().light_copy_, (*it)->mirror_index(i));
    }

    // 3. Stitch
    simps.make_searchable();
    typedef std::queue<std::pair<Facet, Dark_facet> > Queue;
    Queue q;
    q.push(std::make_pair(light_ft, dark_ft));
    dark_s = dark_side.full_cell(dark_ft);
    int dark_i = dark_side.index_of_covertex(dark_ft);
    // mark dark_ft as visited:
    // TODO try by marking with Dark_v_handle (vertex)
    dark_s->neighbor(dark_i)->set_neighbor(dark_s->mirror_index(dark_i), Dark_s_handle());
    while( ! q.empty() )
    {
        std::pair<Facet, Dark_facet> p = q.front();
        q.pop();
        light_ft = p.first;
        dark_ft = p.second;
        light_s = full_cell(light_ft);
        int light_i = index_of_covertex(light_ft);
        dark_s = dark_side.full_cell(dark_ft);
        int dark_i = dark_side.index_of_covertex(dark_ft);
        Full_cell_handle light_n = light_s->neighbor(light_i);
        set_neighbors(dark_s->data().light_copy_, dark_i, light_n, light_s->mirror_index(light_i));
        for( int di = 0; di <= current_dimension(); ++di )
        {
            if( di == dark_i )
                continue;
            int li = light_s->index(dark_s->vertex(di)->data());
            Rotor light_r(light_s, li, light_i);
            typename Dark_triangulation::Rotor dark_r(dark_s, di, dark_i);

            while (simps.contains(std::get<0>(light_r)->neighbor(std::get<1>(light_r))))
                light_r = rotate_rotor(light_r);

            while (conflict_zone.contains(std::get<0>(dark_r)->neighbor(std::get<1>(dark_r))))
                dark_r = dark_side.rotate_rotor(dark_r);

            Dark_s_handle dark_ns = std::get<0>(dark_r);
            int dark_ni = std::get<1>(dark_r);
            Full_cell_handle light_ns = std::get<0>(light_r);
            int light_ni = std::get<1>(light_r);
            // mark dark_r as visited:
            // TODO try by marking with Dark_v_handle (vertex)
            Dark_s_handle outside = dark_ns->neighbor(dark_ni);
            Dark_v_handle mirror = dark_ns->mirror_vertex(dark_ni, current_dimension());
            int dn = outside->index(mirror);
            if( Dark_s_handle() == outside->neighbor(dn) )
                continue;
            outside->set_neighbor(dn, Dark_s_handle());
            q.push(std::make_pair(Facet(light_ns, light_ni), Dark_facet(dark_ns, dark_ni)));
        }
    }
    tds().delete_full_cells(simps.begin(), simps.end());
    tds().delete_vertex(v);
    return ret_s;
}

template< typename DCTraits, typename TDS >
void
Delaunay_triangulation<DCTraits, TDS>
::remove_decrease_dimension(Vertex_handle v)
{
    CGAL_precondition( current_dimension() >= 0 );
    tds().remove_decrease_dimension(v, infinite_vertex());
    // reset the predicates:
    reset_flat_orientation();
    if( 1 <= current_dimension() )
    {
        Full_cell_handle inf_v_cell = infinite_vertex()->full_cell();
        int inf_v_index = inf_v_cell->index(infinite_vertex());
        Full_cell_handle s = inf_v_cell->neighbor(inf_v_index);
        Orientation o = orientation(s);
        CGAL_assertion( ZERO != o );
        if( NEGATIVE == o )
            reorient_full_cells();
    }
}

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - INSERTIONS

template< typename DCTraits, typename TDS >
typename Delaunay_triangulation<DCTraits, TDS>::Vertex_handle
Delaunay_triangulation<DCTraits, TDS>
::insert(const Point & p, Locate_type lt, const Face & f, const Facet &, Full_cell_handle s)
{
    switch( lt )
    {
        case Base::OUTSIDE_AFFINE_HULL:
            return insert_outside_affine_hull(p);
            break;
        case Base::ON_VERTEX:
        {
            Vertex_handle v = s->vertex(f.index(0));
            v->set_point(p);
            return v;
            break;
        }
        default:
            if( 1 == current_dimension() )
            {
                if( Base::OUTSIDE_CONVEX_HULL == lt )
                {
                    return insert_outside_convex_hull_1(p, s);
                }
                Vertex_handle v = tds().insert_in_full_cell(s);
                v->set_point(p);
                return v;
            }
            else
                return insert_in_conflicting_cell(p, s);
            break;
    }
}

/*
[Undocumented function]

Inserts the point `p` in the Delaunay triangulation. Returns a handle to the
(possibly newly created) vertex at that position.
\pre The point `p`
must lie outside the affine hull of the Delaunay triangulation. This implies that
`dt`.`current_dimension()` must be less than `dt`.`maximal_dimension()`.
*/
template< typename DCTraits, typename TDS >
typename Delaunay_triangulation<DCTraits, TDS>::Vertex_handle
Delaunay_triangulation<DCTraits, TDS>
::insert_outside_affine_hull(const Point & p)
{
    // we don't use Base::insert_outside_affine_hull(...) because here, we
    // also need to reset the side_of_oriented_subsphere functor.
    CGAL_precondition( current_dimension() < maximal_dimension() );
    Vertex_handle v = tds().insert_increase_dimension(infinite_vertex());
    // reset the predicates:
    reset_flat_orientation();
    v->set_point(p);
    if( current_dimension() >= 1 )
    {
        Full_cell_handle inf_v_cell = infinite_vertex()->full_cell();
        int inf_v_index = inf_v_cell->index(infinite_vertex());
        Full_cell_handle s = inf_v_cell->neighbor(inf_v_index);
        Orientation o = orientation(s);
        CGAL_assertion( ZERO != o );
            if( NEGATIVE == o )
                reorient_full_cells();

        // We just inserted the second finite point and the right infinite
        // cell is like : (inf_v, v), but we want it to be (v, inf_v) to be
        // consistent with the rest of the cells
        if (current_dimension() == 1)
        {
            // Is "inf_v_cell" the right infinite cell?
            // Then inf_v_index should be 1
            if (inf_v_cell->neighbor(inf_v_index)->index(inf_v_cell) == 0
                && inf_v_index == 0)
            {
                inf_v_cell->swap_vertices(
                    current_dimension() - 1, current_dimension());
            }
            // Otherwise, let's find the right infinite cell
            else
            {
                inf_v_cell = inf_v_cell->neighbor((inf_v_index + 1) % 2);
                inf_v_index = inf_v_cell->index(infinite_vertex());
                // Is "inf_v_cell" the right infinite cell?
                // Then inf_v_index should be 1
                if (inf_v_cell->neighbor(inf_v_index)->index(inf_v_cell) == 0
                    && inf_v_index == 0)
                {
                    inf_v_cell->swap_vertices(
                        current_dimension() - 1, current_dimension());
                }
            }
        }
    }
    return v;
}

/*!
[Undocumented function]

Inserts the point `p` in the Delaunay triangulation. Returns a handle to the
(possibly newly created) vertex at that position.
\pre The point `p` must be in conflict with the full cell `c`.
*/
template< typename DCTraits, typename TDS >
typename Delaunay_triangulation<DCTraits, TDS>::Vertex_handle
Delaunay_triangulation<DCTraits, TDS>
::insert_in_conflicting_cell(const Point & p, Full_cell_handle s)
{
    CGAL_precondition(is_in_conflict(p, s));

    // for storing conflicting full_cells.
    typedef std::vector<Full_cell_handle> Full_cell_h_vector;
    CGAL_STATIC_THREAD_LOCAL_VARIABLE(Full_cell_h_vector,cs,0);
    cs.clear();

    std::back_insert_iterator<Full_cell_h_vector> out(cs);
    Facet ft = compute_conflict_zone(p, s, out);
    return insert_in_hole(p, cs.begin(), cs.end(), ft);
}

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - GATHERING CONFLICTING SIMPLICES

// NOT DOCUMENTED
template< typename DCTraits, typename TDS >
template< typename OrientationPred >
Oriented_side
Delaunay_triangulation<DCTraits, TDS>
::perturbed_side_of_positive_sphere(const Point & p, Full_cell_const_handle s,
        const OrientationPred & ori) const
{
    CGAL_precondition_msg( ! is_infinite(s), "full cell must be finite");
    CGAL_expensive_precondition( POSITIVE == orientation(s) );
    typedef std::vector<const Point *> Points;
    Points points(current_dimension() + 2);
    int i(0);
    for( ; i <= current_dimension(); ++i )
        points[i] = &(s->vertex(i)->point());
    points[i] = &p;
    std::sort(points.begin(), points.end(),
            internal::Triangulation::Compare_points_for_perturbation<Self>(*this));
    typename Points::const_reverse_iterator cut_pt = points.rbegin();
    Points test_points;
    while( cut_pt != points.rend() )
    {
        if( &p == *cut_pt )
            // because the full_cell "s" is assumed to be positively oriented
            return ON_NEGATIVE_SIDE; // we consider |p| to lie outside the sphere
        test_points.clear();
        typename Base::Point_const_iterator spit = points_begin(s);
        int adjust_sign = -1;
        for( i = 0; i < current_dimension(); ++i )
        {
            if( &(*spit) == *cut_pt )
            {
                ++spit;
                adjust_sign = (((current_dimension() + i) % 2) == 0) ? -1 : +1;
            }
            test_points.push_back(&(*spit));
            ++spit;
        }
        test_points.push_back(&p);

        typedef typename CGAL::Iterator_project<typename Points::iterator,
                        internal::Triangulation::Point_from_pointer<Self>,
                const Point &, const Point *> Point_pointer_iterator;

        Orientation ori_value = ori(
                Point_pointer_iterator(test_points.begin()),
                Point_pointer_iterator(test_points.end()));

        if( ZERO != ori_value )
            return Oriented_side( - adjust_sign * ori_value );

        ++cut_pt;
    }
    CGAL_assertion(false); // we should never reach here
    return ON_NEGATIVE_SIDE;
}

template< typename DCTraits, typename TDS >
bool
Delaunay_triangulation<DCTraits, TDS>
::is_in_conflict(const Point & p, Full_cell_const_handle s) const
{
    CGAL_precondition( 2 <= current_dimension() );
    if( current_dimension() < maximal_dimension() )
    {
        Conflict_pred_in_subspace c(*this, p, coaffine_orientation_predicate(), side_of_oriented_subsphere_predicate());
        return c(s);
    }
    else
    {
        Orientation_d ori = geom_traits().orientation_d_object();
        Side_of_oriented_sphere_d side = geom_traits().side_of_oriented_sphere_d_object();
        Conflict_pred_in_fullspace c(*this, p, ori, side);
        return c(s);
    }
}

template< typename DCTraits, typename TDS >
template< typename OutputIterator >
typename Delaunay_triangulation<DCTraits, TDS>::Facet
Delaunay_triangulation<DCTraits, TDS>
::compute_conflict_zone(const Point & p, Full_cell_handle s, OutputIterator out) const
{
    CGAL_precondition( 2 <= current_dimension() );
    if( current_dimension() < maximal_dimension() )
    {
        Conflict_pred_in_subspace c(*this, p, coaffine_orientation_predicate(), side_of_oriented_subsphere_predicate());
        Conflict_traversal_pred_in_subspace tp(*this, c);
        return tds().gather_full_cells(s, tp, out);
    }
    else
    {
        Orientation_d ori = geom_traits().orientation_d_object();
        Side_of_oriented_sphere_d side = geom_traits().side_of_oriented_sphere_d_object();
        Conflict_pred_in_fullspace c(*this, p, ori, side);
        Conflict_traversal_pred_in_fullspace tp(*this, c);
        return tds().gather_full_cells(s, tp, out);
    }
}

// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - VALIDITY

template< typename DCTraits, typename TDS >
bool
Delaunay_triangulation<DCTraits, TDS>
::is_valid(bool verbose, int level) const
{
  if (!Base::is_valid(verbose, level))
    return false;

  int dim = current_dimension();
  if (dim == maximal_dimension())
  {
    for (Finite_full_cell_const_iterator cit = this->finite_full_cells_begin() ;
         cit != this->finite_full_cells_end() ; ++cit )
    {
      Full_cell_const_handle ch = cit.base();
      for(int i = 0; i < dim+1 ; ++i )
      {
        // If the i-th neighbor is not an infinite cell
        Vertex_handle opposite_vh =
          ch->neighbor(i)->vertex(ch->neighbor(i)->index(ch));
        if (!is_infinite(opposite_vh))
        {
          Side_of_oriented_sphere_d side =
            geom_traits().side_of_oriented_sphere_d_object();
          if (side(points_begin(ch),
                   points_end(ch),
                   opposite_vh->point()) == ON_POSITIVE_SIDE)
          {
            if (verbose)
              CGAL_warning_msg(false, "Non-empty sphere");
            return false;
          }
        }
      }
    }
  }
  return true;
}


} //namespace CGAL

#include <CGAL/enable_warnings.h>

#endif // CGAL_DELAUNAY_COMPLEX_H