// Copyright (c) 2015 GeometryFactory (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.6.2/Polygon_mesh_processing/include/CGAL/Polygon_mesh_processing/internal/Isotropic_remeshing/remesh_impl.h $
// $Id: remesh_impl.h 4dc91d18be4 2023-04-12T09:19:55+02:00 Sébastien Loriot
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s)     : Jane Tournois


#ifndef CGAL_POLYGON_MESH_PROCESSING_REMESH_IMPL_H
#define CGAL_POLYGON_MESH_PROCESSING_REMESH_IMPL_H

#include <CGAL/license/Polygon_mesh_processing/meshing_hole_filling.h>

#include <CGAL/Polygon_mesh_processing/compute_normal.h>
#include <CGAL/Polygon_mesh_processing/border.h>
#include <CGAL/Polygon_mesh_processing/repair_degeneracies.h>
#include <CGAL/Polygon_mesh_processing/measure.h>
#include <CGAL/Polygon_mesh_processing/connected_components.h>
#include <CGAL/Polygon_mesh_processing/shape_predicates.h>
#include <CGAL/Polygon_mesh_processing/tangential_relaxation.h>

#include <CGAL/AABB_tree.h>
#include <CGAL/AABB_traits.h>
#include <CGAL/AABB_triangle_primitive.h>

#include <CGAL/property_map.h>
#include <CGAL/Dynamic_property_map.h>
#include <CGAL/iterator.h>
#include <CGAL/boost/graph/Euler_operations.h>
#include <CGAL/boost/graph/properties.h>
#include <boost/graph/graph_traits.hpp>
#include <CGAL/tags.h>

#include <boost/bimap.hpp>
#include <boost/bimap/multiset_of.hpp>
#include <boost/bimap/set_of.hpp>
#include <boost/range.hpp>
#include <boost/range/join.hpp>
#include <memory>
#include <boost/container/flat_set.hpp>
#include <boost/optional.hpp>
#include <boost/property_map/function_property_map.hpp>

#include <map>
#include <list>
#include <vector>
#include <iterator>
#include <fstream>
#include <tuple>
#include <unordered_map>
#include <unordered_set>

#ifdef CGAL_PMP_REMESHING_DEBUG
#include <CGAL/Polygon_mesh_processing/self_intersections.h>
#define CGAL_DUMP_REMESHING_STEPS
#define CGAL_PMP_REMESHING_VERBOSE_PROGRESS
#endif

#ifdef CGAL_PMP_REMESHING_VERY_VERBOSE
#define CGAL_PMP_REMESHING_VERBOSE
#define CGAL_PMP_REMESHING_VERBOSE_PROGRESS
#endif

#ifdef CGAL_PMP_REMESHING_VERBOSE_PROGRESS
#define CGAL_PMP_REMESHING_VERBOSE
#define CGAL_PMP_TANGENTIAL_RELAXATION_VERBOSE
#endif

namespace CGAL {
namespace Polygon_mesh_processing {
namespace internal {

  enum Halfedge_status {
    PATCH,       //h and hopp belong to the patch to be remeshed
    PATCH_BORDER,//h belongs to the patch, hopp is MESH
    MESH,        //h and hopp belong to the mesh, not the patch
    MESH_BORDER, //h belongs to the mesh, face(hopp, pmesh) == null_face()
    ISOLATED_CONSTRAINT //h is constrained, and incident to faces that do not belong to a patch
  };

  // A property map
  template <typename PM, typename FaceIndexMap>
  struct Border_constraint_pmap
  {
    typedef typename boost::graph_traits<PM>::halfedge_descriptor halfedge_descriptor;
    typedef typename boost::graph_traits<PM>::edge_descriptor edge_descriptor;
    typedef FaceIndexMap FIMap;

    std::shared_ptr< std::set<edge_descriptor> > border_edges_ptr;
    const PM* pmesh_ptr_;

  public:
    typedef edge_descriptor                     key_type;
    typedef bool                                value_type;
    typedef value_type                          reference;
    typedef boost::read_write_property_map_tag  category;

    Border_constraint_pmap()
      : border_edges_ptr(new std::set<edge_descriptor>() )
      , pmesh_ptr_(nullptr)
    {}

    template <class FaceRange>
    Border_constraint_pmap(const PM& pmesh
                         , const FaceRange& faces
                         , const FIMap& fimap)
      : border_edges_ptr(new std::set<edge_descriptor>() )
      , pmesh_ptr_(&pmesh)
    {
      std::vector<halfedge_descriptor> border;
      border_halfedges(faces, *pmesh_ptr_, std::back_inserter(border), parameters::face_index_map(fimap));

      for(halfedge_descriptor h : border)
        border_edges_ptr->insert(edge(h, *pmesh_ptr_));
    }

    friend bool get(const Border_constraint_pmap<PM, FIMap>& map,
                    const edge_descriptor& e)
    {
      CGAL_assertion(map.pmesh_ptr_!=nullptr);
      return map.border_edges_ptr->count(e)!=0;
    }

    friend void put(Border_constraint_pmap<PM, FIMap>& map,
                    const edge_descriptor& e,
                    const bool is)
    {
      CGAL_assertion(map.pmesh_ptr_ != nullptr);
      if (is)
        map.border_edges_ptr->insert(e);
      else
        map.border_edges_ptr->erase(e);
    }
  };


  template <typename PM,
            typename FaceIndexMap>
  struct Connected_components_pmap
  {
    typedef typename boost::graph_traits<PM>::face_descriptor   face_descriptor;
    typedef std::size_t                                         Patch_id;
    typedef FaceIndexMap                                        FIMap;
    typedef Connected_components_pmap<PM, FIMap>                CCMap;

    typedef CGAL::dynamic_face_property_t<Patch_id> Face_property_tag;
    typedef typename boost::property_map<PM, Face_property_tag >::type Patch_ids_map;
    Patch_ids_map patch_ids_map;
    std::size_t nb_cc;

    template <class Range>
    bool same_range(const Range& r1, const Range& r2)
    {
      return boost::begin(r1)==boost::begin(r2) &&
             boost::end(r1)==boost::end(r2);
    }

    template <class Range1, class Range2>
    bool same_range(const Range1& r1, const Range2& r2)
    {
      return std::distance(boost::begin(r1), boost::end(r1)) ==
             std::distance(boost::begin(r2), boost::end(r2));
    }

  public:
    typedef face_descriptor                     key_type;
    typedef Patch_id                            value_type;
    typedef Patch_id                            reference;
    typedef boost::read_write_property_map_tag  category;

    //note pmesh is a non-const ref because properties are added and removed
    //modify the mesh data structure, but not the mesh itself
    template <class FaceRange, class EdgeIsConstrainedMap>
    Connected_components_pmap(const FaceRange& face_range
                            , PM& pmesh
                            , EdgeIsConstrainedMap ecmap
                            , FIMap fimap
                            , const bool do_init = true)
    {
      patch_ids_map = get(Face_property_tag(),pmesh);
      if (do_init)
      {
#ifdef CGAL_PMP_REMESHING_VERBOSE
        std::cout << "Compute connected components property map." << std::endl;
#endif
        if ( same_range(face_range, (faces(pmesh))) )
        {
          // applied on the whole mesh
          nb_cc = connected_components(pmesh, patch_ids_map,
                                       parameters::edge_is_constrained_map(ecmap)
                                                  .face_index_map(fimap));
        }
        else
        {
          // applied on a subset of the mesh
          nb_cc = connected_components(
                    pmesh, patch_ids_map,
                    parameters::edge_is_constrained_map(
                                  make_OR_property_map(ecmap,
                                                       internal::Border_constraint_pmap<PM, FIMap>(pmesh, face_range, fimap)))
                               .face_index_map(fimap));
        }
      }
      else
        nb_cc=0; // default value
    }


    friend value_type get(const CCMap& m, const key_type& f)
    {
      if (m.nb_cc == 1)
        return 0;
      return get(m.patch_ids_map, f);
    }
    friend void put(CCMap& m, const key_type& f, const value_type i)
    {
      if (m.nb_cc != 1)
        put(m.patch_ids_map, f, i);
    }
  };

  template<typename PM,
           typename EdgeConstraintMap,
           typename VertexPointMap,
           typename FacePatchMap>
  bool constraints_are_short_enough(const PM& pmesh,
                                    EdgeConstraintMap ecmap,
                                    VertexPointMap vpmap,
                                    const FacePatchMap& fpm,
                                    const double& high)
  {
    double sqh = high*high;
    typedef typename boost::graph_traits<PM>::halfedge_descriptor halfedge_descriptor;
    typedef typename boost::graph_traits<PM>::edge_descriptor     edge_descriptor;
    for(edge_descriptor e : edges(pmesh))
    {
      halfedge_descriptor h = halfedge(e, pmesh);
      if (  is_border(e, pmesh) ||
            get(ecmap, e) ||
            get(fpm, face(h,pmesh))!=get(fpm, face(opposite(h,pmesh),pmesh)) )
      {
        if (sqh < CGAL::squared_distance(get(vpmap, source(h, pmesh)),
                                         get(vpmap, target(h, pmesh))))
        {
          return false;
        }
      }
    }
    return true;
  }

  template<typename PolygonMesh
         , typename VertexPointMap
         , typename GeomTraits
         , typename EdgeIsConstrainedMap
         , typename VertexIsConstrainedMap
         , typename FacePatchMap
         , typename FaceIndexMap
  >
  class Incremental_remesher
  {
    typedef PolygonMesh PM;
    typedef typename boost::graph_traits<PM>::halfedge_descriptor halfedge_descriptor;
    typedef typename boost::graph_traits<PM>::edge_descriptor     edge_descriptor;
    typedef typename boost::graph_traits<PM>::vertex_descriptor   vertex_descriptor;
    typedef typename boost::graph_traits<PM>::face_descriptor     face_descriptor;

    typedef typename GeomTraits::Point_3    Point;
    typedef typename GeomTraits::Vector_3   Vector_3;
    typedef typename GeomTraits::Plane_3    Plane_3;
    typedef typename GeomTraits::Triangle_3 Triangle_3;

    typedef Incremental_remesher<PM, VertexPointMap
                               , GeomTraits
                               , EdgeIsConstrainedMap
                               , VertexIsConstrainedMap
                               , FacePatchMap
                               , FaceIndexMap
                               > Self;

  private:
    typedef typename boost::property_traits<FacePatchMap>::value_type Patch_id;
    typedef std::vector<Triangle_3>                      Triangle_list;
    typedef std::vector<Patch_id>                        Patch_id_list;
    typedef std::map<Patch_id,std::size_t>               Patch_id_to_index_map;

    typedef CGAL::AABB_triangle_primitive<GeomTraits,
                     typename Triangle_list::iterator>    AABB_primitive;
    typedef CGAL::AABB_traits<GeomTraits, AABB_primitive> AABB_traits;
    typedef CGAL::AABB_tree<AABB_traits>                  AABB_tree;

    typedef typename boost::property_map<
      PM, CGAL::dynamic_halfedge_property_t<Halfedge_status> >::type Halfedge_status_pmap;

  public:
    Incremental_remesher(PolygonMesh& pmesh
                       , VertexPointMap& vpmap
                       , const GeomTraits& gt
                       , const bool protect_constraints
                       , EdgeIsConstrainedMap ecmap
                       , VertexIsConstrainedMap vcmap
                       , FacePatchMap fpmap
                       , FaceIndexMap fimap
                       , const bool build_tree = true)//built by the remesher
      : mesh_(pmesh)
      , vpmap_(vpmap)
      , gt_(gt)
      , build_tree_(build_tree)
      , has_border_(false)
      , input_triangles_()
      , input_patch_ids_()
      , protect_constraints_(protect_constraints)
      , patch_ids_map_(fpmap)
      , ecmap_(ecmap)
      , vcmap_(vcmap)
      , fimap_(fimap)
    {
      halfedge_status_pmap_ = get(CGAL::dynamic_halfedge_property_t<Halfedge_status>(),
                                  pmesh);
      CGAL_warning_code(input_mesh_is_valid_ = CGAL::is_valid_polygon_mesh(pmesh));
      CGAL_warning_msg(input_mesh_is_valid_,
        "The input mesh is not a valid polygon mesh. "
        "It could lead PMP::isotropic_remeshing() to fail.");
    }

    ~Incremental_remesher()
    {
      if (build_tree_){
        for(std::size_t i=0; i < trees.size();++i){
          delete trees[i];
        }
      }
    }

    template<typename FaceRange>
    void init_remeshing(const FaceRange& face_range)
    {
      tag_halfedges_status(face_range); //called first

      for(face_descriptor f : face_range)
      {
        if(is_degenerate_triangle_face(f, mesh_, parameters::vertex_point_map(vpmap_)
                                                            .geom_traits(gt_)))
          continue;

        Patch_id pid = get_patch_id(f);
        input_triangles_.push_back(triangle(f));
        input_patch_ids_.push_back(pid);
        std::pair<typename Patch_id_to_index_map::iterator, bool>
          res = patch_id_to_index_map.insert(std::make_pair(pid,0));
        if(res.second){
          res.first->second =  patch_id_to_index_map.size()-1;
        }
      }
      CGAL_assertion(input_triangles_.size() == input_patch_ids_.size());

      if (!build_tree_)
        return;
      trees.resize(patch_id_to_index_map.size());
      for(std::size_t i=0; i < trees.size(); ++i){
        trees[i] = new AABB_tree();
      }
      typename Triangle_list::iterator it;
      typename Patch_id_list::iterator pit;
      for(it = input_triangles_.begin(), pit = input_patch_ids_.begin();
          it != input_triangles_.end();
          ++it, ++pit){
        trees[patch_id_to_index_map[*pit]]->insert(it);
      }
      for(std::size_t i=0; i < trees.size(); ++i){
        trees[i]->build();
      }
    }


    // split edges of edge_range that have their length > high
    // Note: only used to split a range of edges provided as input
    template<typename EdgeRange>
    void split_long_edges(const EdgeRange& edge_range,
                          const double& high)
    {

#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Split long edges (" << high << ")...";
      std::cout.flush();
#endif
      double sq_high = high*high;

      //collect long edges
      typedef std::pair<halfedge_descriptor, double> H_and_sql;
      std::multiset< H_and_sql, std::function<bool(H_and_sql,H_and_sql)> >
        long_edges(
          [](const H_and_sql& p1, const H_and_sql& p2)
          { return p1.second > p2.second; }
        );
      for(edge_descriptor e : edge_range)
      {
        double sqlen = sqlength(e);
        if (sqlen > sq_high)
          long_edges.emplace(halfedge(e, mesh_), sqlen);
      }

      //split long edges
#ifdef CGAL_PMP_REMESHING_VERBOSE
      unsigned int nb_splits = 0;
#endif
      while (!long_edges.empty())
      {
        //the edge with longest length
        auto eit = long_edges.begin();
        halfedge_descriptor he = eit->first;
        double sqlen = eit->second;
        long_edges.erase(eit);

        //split edge
        Point refinement_point = this->midpoint(he);
        halfedge_descriptor hnew = CGAL::Euler::split_edge(he, mesh_);
        // propagate the constrained status
        put(ecmap_, edge(hnew, mesh_), get(ecmap_, edge(he, mesh_)));
        CGAL_assertion(he == next(hnew, mesh_));
#ifdef CGAL_PMP_REMESHING_VERBOSE
        ++nb_splits;
#endif

        //move refinement point
        vertex_descriptor vnew = target(hnew, mesh_);
        put(vpmap_, vnew, refinement_point);
#ifdef CGAL_PMP_REMESHING_VERY_VERBOSE
        std::cout << "   refinement point : " << refinement_point << std::endl;
#endif

        //check sub-edges
        double sqlen_new = 0.25 * sqlen;
        if (sqlen_new > sq_high)
        {
          //if it was more than twice the "long" threshold, insert them
          long_edges.emplace(hnew, sqlen_new);
          long_edges.emplace(next(hnew, mesh_), sqlen_new);
        }

        //insert new edges to keep triangular faces, and update long_edges
        if (!is_border(hnew, mesh_))
        {
          Patch_id patch_id = get_patch_id(face(hnew, mesh_));
          halfedge_descriptor hnew2 =
            CGAL::Euler::split_face(hnew, next(next(hnew, mesh_), mesh_), mesh_);
          put(ecmap_, edge(hnew2, mesh_), false);
          set_patch_id(face(hnew2, mesh_), patch_id);
          set_patch_id(face(opposite(hnew2, mesh_), mesh_), patch_id);
        }

        //do it again on the other side if we're not on boundary
        halfedge_descriptor hnew_opp = opposite(hnew, mesh_);
        if (!is_border(hnew_opp, mesh_))
        {
          Patch_id patch_id = get_patch_id(face(hnew_opp, mesh_));
          halfedge_descriptor hnew2 =
            CGAL::Euler::split_face(prev(hnew_opp, mesh_), next(hnew_opp, mesh_), mesh_);
          put(ecmap_, edge(hnew2, mesh_), false);
          set_patch_id(face(hnew2, mesh_), patch_id);
          set_patch_id(face(opposite(hnew2, mesh_), mesh_), patch_id);
        }
      }
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << " done (" << nb_splits << " splits)." << std::endl;
#endif
#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("0-border_split.off");
#endif
    }

    // PMP book :
    // "visits all edges of the mesh
    //if an edge is longer than the given threshold `high`, the edge
    //is split at its midpoint and the two adjacent triangles are bisected (2-4 split)"
    void split_long_edges(const double& high)
    {
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Split long edges (" << high << ")..." << std::endl;
#endif
      double sq_high = high*high;

      //collect long edges
      typedef std::pair<halfedge_descriptor, double> H_and_sql;
      std::multiset< H_and_sql, std::function<bool(H_and_sql,H_and_sql)> >
      long_edges(
        [](const H_and_sql& p1, const H_and_sql& p2)
        { return p1.second > p2.second; }
      );

      for(edge_descriptor e : edges(mesh_))
      {
        if (!is_split_allowed(e))
          continue;
        double sqlen = sqlength(e);
        if(sqlen > sq_high)
          long_edges.emplace(halfedge(e, mesh_), sqlen);
      }

      //split long edges
#ifdef CGAL_PMP_REMESHING_VERBOSE
      unsigned int nb_splits = 0;
#endif
      while (!long_edges.empty())
      {
        //the edge with longest length
        auto eit = long_edges.begin();
        halfedge_descriptor he = eit->first;
        double sqlen = eit->second;
        long_edges.erase(eit);

#ifdef CGAL_PMP_REMESHING_VERBOSE_PROGRESS
        std::cout << "\r\t(" << long_edges.size() << " long edges, ";
        std::cout << nb_splits << " splits)";
        std::cout.flush();
#endif

        if (protect_constraints_ && !is_longest_on_faces(edge(he, mesh_)))
          continue;

        //collect patch_ids
        Patch_id patch_id = get_patch_id(face(he, mesh_));
        Patch_id patch_id_opp = get_patch_id(face(opposite(he, mesh_), mesh_));

        //split edge
        Point refinement_point = this->midpoint(he);
        halfedge_descriptor hnew = CGAL::Euler::split_edge(he, mesh_);
        CGAL_assertion(he == next(hnew, mesh_));
        put(ecmap_, edge(hnew, mesh_), get(ecmap_, edge(he, mesh_)) );
#ifdef CGAL_PMP_REMESHING_VERBOSE
        ++nb_splits;
#endif
        //move refinement point
        vertex_descriptor vnew = target(hnew, mesh_);
        put(vpmap_, vnew, refinement_point);
#ifdef CGAL_PMP_REMESHING_VERY_VERBOSE
        std::cout << "   Refinement point : " << refinement_point << std::endl;
#endif

        //after splitting
        halfedge_descriptor hnew_opp = opposite(hnew, mesh_);
        halfedge_added(hnew, status(he));
        halfedge_added(hnew_opp, status(opposite(he, mesh_)));

        //check sub-edges
        double sqlen_new = 0.25 * sqlen;
        if (sqlen_new > sq_high)
        {
          //if it was more than twice the "long" threshold, insert them
          long_edges.emplace(hnew, sqlen_new);
          long_edges.emplace(next(hnew, mesh_), sqlen_new);
        }

        //insert new edges to keep triangular faces, and update long_edges
        if (!is_on_border(hnew))
        {
          halfedge_descriptor hnew2 = CGAL::Euler::split_face(hnew,
                                                              next(next(hnew, mesh_), mesh_),
                                                              mesh_);
          put(ecmap_, edge(hnew2, mesh_), false);
          Halfedge_status snew = (is_on_patch(hnew) || is_on_patch_border(hnew))
            ? PATCH
            : MESH;
          halfedge_added(hnew2,                  snew);
          halfedge_added(opposite(hnew2, mesh_), snew);
          set_patch_id(face(hnew2, mesh_), patch_id);
          set_patch_id(face(opposite(hnew2, mesh_), mesh_), patch_id);

          if (snew == PATCH)
          {
            double sql = sqlength(hnew2);
            if (sql > sq_high)
              long_edges.emplace(hnew2, sql);
          }
        }

        //do it again on the other side if we're not on boundary
        if (!is_on_border(hnew_opp))
        {
          halfedge_descriptor hnew2 = CGAL::Euler::split_face(prev(hnew_opp, mesh_),
                                                              next(hnew_opp, mesh_),
                                                              mesh_);
          put(ecmap_, edge(hnew2, mesh_), false);
          Halfedge_status snew = (is_on_patch(hnew_opp) || is_on_patch_border(hnew_opp))
             ? PATCH
            : MESH;
          halfedge_added(hnew2,                  snew);
          halfedge_added(opposite(hnew2, mesh_), snew);
          set_patch_id(face(hnew2, mesh_), patch_id_opp);
          set_patch_id(face(opposite(hnew2, mesh_), mesh_), patch_id_opp);

          if (snew == PATCH)
          {
            double sql = sqlength(hnew2);
            if (sql > sq_high)
              long_edges.emplace(hnew2, sql);
          }
        }
      }
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << " done ("<< nb_splits << " splits)." << std::endl;
#endif

#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_status_map();
      debug_self_intersections();
#endif

#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("1-edge_split.off");
#endif
    }

    // PMP book :
    // "collapses and thus removes all edges that are shorter than a
    // threshold `low`. [...] testing before each collapse whether the collapse
    // would produce an edge that is longer than `high`"
    void collapse_short_edges(const double& low,
                              const double& high,
                              const bool collapse_constraints)
    {
      typedef boost::bimap<
        boost::bimaps::set_of<halfedge_descriptor>,
        boost::bimaps::multiset_of<double, std::less<double> > >  Boost_bimap;
      typedef typename Boost_bimap::value_type                    short_edge;

#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Collapse short edges (" << low << ", " << high << ")..."
                << std::endl;
#endif
#ifdef CGAL_PMP_REMESHING_VERBOSE_PROGRESS
      std::cout << "Fill bimap...";
      std::cout.flush();
#endif
      double sq_low = low*low;
      double sq_high = high*high;

      Boost_bimap short_edges;
      for(edge_descriptor e : edges(mesh_))
      {
        double sqlen = sqlength(e);
        if ((sqlen < sq_low) && is_collapse_allowed(e, collapse_constraints))
          short_edges.insert(short_edge(halfedge(e, mesh_), sqlen));
      }
#ifdef CGAL_PMP_REMESHING_VERBOSE_PROGRESS
      std::cout << "done." << std::endl;
#endif

#ifdef CGAL_PMP_REMESHING_VERBOSE
      unsigned int nb_collapses = 0;
#endif
      while (!short_edges.empty())
      {
        //the edge with shortest length
        typename Boost_bimap::right_map::iterator eit = short_edges.right.begin();
        halfedge_descriptor he = eit->second;
        short_edges.right.erase(eit);

#ifdef CGAL_PMP_REMESHING_VERBOSE_PROGRESS
        std::cout << "\r\t(" << short_edges.left.size() << " short edges, ";
        std::cout << nb_collapses << " collapses)";
        std::cout.flush();
#endif

        edge_descriptor e = edge(he, mesh_);
        if (!is_collapse_allowed(e, collapse_constraints))
          continue; //situation could have changed since it was added to the bimap

        //handle the boundary case :
        //a PATCH_BORDER edge can be collapsed,
        //and an edge incident to PATCH_BORDER can be collapsed,
        //but only if the boundary vertex is kept,
        //so re-insert opposite(he) to collapse it
        if (!is_on_patch(he))
        {
          CGAL_assertion(!protect_constraints_);//is_collapse_allowed returned false
          if (is_on_border(he) || is_on_mesh(he))
          {
            he = opposite(he, mesh_); //he now is PATCH_BORDER
            e = edge(he, mesh_);
            CGAL_assertion(is_on_patch_border(he));
          }
        }//end if(not on PATCH)

        //let's try to collapse he into vb
        vertex_descriptor va = source(he, mesh_);
        vertex_descriptor vb = target(he, mesh_);

        bool is_va_constrained = is_constrained(va) || is_corner(va);
        bool is_vb_constrained = is_constrained(vb) || is_corner(vb);

        // do not collapse edge with two constrained vertices
        if (is_va_constrained && is_vb_constrained) continue;

        bool can_swap = !is_vb_constrained;

        //do not collapse an edge connecting two different constrained polylines
        if (is_va_constrained && is_vb_constrained && !is_on_patch_border(he))
          continue;

        bool is_va_on_constrained_polyline = is_on_patch_border(va);
        bool is_vb_on_constrained_polyline = is_on_patch_border(vb);

        // swap if vb is not constrained and va is constrained or the only vertex on a constrained polyline
        if (can_swap && (is_va_constrained || (is_va_on_constrained_polyline && !is_vb_on_constrained_polyline)))
        {
          he = opposite(he, mesh_);
          e=edge(he, mesh_);
          std::swap(va, vb);
          // no need to swap is_vX_on_constrained_polyline and is_vX_constrained
          can_swap=false;
        }

        if(collapse_would_invert_face(he))
        {
          if (can_swap//if swap allowed (no constrained vertices)
              && (!is_vb_on_constrained_polyline || is_va_on_constrained_polyline)
              && !collapse_would_invert_face(opposite(he, mesh_)))
          {
            he = opposite(he, mesh_);
            e=edge(he, mesh_);
            std::swap(va, vb);
          }
          else
            continue;//both directions invert a face
        }
        CGAL_assertion(!collapse_would_invert_face(he));
        CGAL_assertion(is_collapse_allowed(e, collapse_constraints));

        if (!CGAL::Euler::does_satisfy_link_condition(e, mesh_))//necessary to collapse
          continue;

        //check that collapse would not create an edge with length > high
        //iterate on vertices va_i of the one-ring of va
        bool collapse_ok = true;
        for(halfedge_descriptor ha : halfedges_around_target(va, mesh_))
        {
          vertex_descriptor va_i = source(ha, mesh_);
          if (sqlength(vb, va_i) > sq_high)
          {
            collapse_ok = false;
            break;
          }
        }
        // before collapsing va into vb, check that it does not break a corner
        // or a constrained vertex
        if (collapse_ok)
        {
          //"collapse va into vb along e"
          // remove edges incident to va and vb, because their lengths will change
          for(halfedge_descriptor ha : halfedges_around_target(va, mesh_))
          {
            short_edges.left.erase(ha);
            short_edges.left.erase(opposite(ha, mesh_));
          }
          for(halfedge_descriptor hb : halfedges_around_target(vb, mesh_))
          {
            short_edges.left.erase(hb);
            short_edges.left.erase(opposite(hb, mesh_));
          }

          //before collapse
          halfedge_descriptor he_opp= opposite(he, mesh_);
          bool mesh_border_case     = is_on_border(he);
          bool mesh_border_case_opp = is_on_border(he_opp);
          halfedge_descriptor ep_p  = prev(he_opp, mesh_);
          halfedge_descriptor en    = next(he, mesh_);
          halfedge_descriptor ep    = prev(he, mesh_);
          halfedge_descriptor en_p  = next(he_opp, mesh_);

          // merge halfedge_status to keep the more important on both sides
          //do it before collapse is performed to be sure everything is valid
          if (!mesh_border_case)
            merge_and_update_status(en, ep);
          if (!mesh_border_case_opp)
            merge_and_update_status(en_p, ep_p);

          if (!protect_constraints_)
            put(ecmap_, e, false);
          else
            CGAL_assertion( !get(ecmap_, e) );

          //perform collapse
          CGAL_assertion(target(halfedge(e, mesh_), mesh_) == vb);
          vertex_descriptor vkept = CGAL::Euler::collapse_edge(e, mesh_, ecmap_);
          CGAL_assertion(is_valid(mesh_));
          CGAL_assertion(vkept == vb);//is the constrained point still here
#ifdef CGAL_PMP_REMESHING_VERBOSE
          ++nb_collapses;
#endif
          //fix constrained case
          CGAL_assertion((is_constrained(vkept) || is_corner(vkept) || is_on_patch_border(vkept)) ==
                         (is_va_constrained || is_vb_constrained || is_va_on_constrained_polyline || is_vb_on_constrained_polyline));
          if (fix_degenerate_faces(vkept, short_edges, sq_low, collapse_constraints))
          {
#ifdef CGAL_PMP_REMESHING_DEBUG
            debug_status_map();
            CGAL_assertion(!incident_to_degenerate(halfedge(vkept, mesh_)));
#endif

            //insert new/remaining short edges
            for (halfedge_descriptor ht : halfedges_around_target(vkept, mesh_))
            {
              double sqlen = sqlength(ht);
              if ((sqlen < sq_low) && is_collapse_allowed(edge(ht, mesh_), collapse_constraints))
                short_edges.insert(short_edge(ht, sqlen));
            }
          }
        }//end if(collapse_ok)
      }

#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << " done (" << nb_collapses << " collapses)." << std::endl;
#endif

#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("2-edge_collapse.off");
#endif

#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_status_map();
      debug_self_intersections();
      CGAL_assertion(remove_degenerate_faces(mesh_, parameters::vertex_point_map(vpmap_).geom_traits(gt_)));
#endif
    }

    // PMP book :
    // "equalizes the vertex valences by flipping edges.
    // The target valence is 6 and 4 for interior and boundary vertices, resp.
    // The algo. tentatively flips each edge `e` and checks whether the deviation
    // to the target valences decreases. If not, the edge is flipped back"
    void flip_edges_for_valence_and_shape()
    {
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Equalize valences..." << std::endl;
#endif

      typedef typename boost::property_map<PM, CGAL::dynamic_vertex_property_t<int> >::type Vertex_degree;
      Vertex_degree degree = get(CGAL::dynamic_vertex_property_t<int>(), mesh_);

      for(vertex_descriptor v : vertices(mesh_)){
        put(degree,v,0);
      }
      for(halfedge_descriptor h : halfedges(mesh_))
      {
        vertex_descriptor t = target(h, mesh_);
        put(degree, t, get(degree,t)+1);
      }

      const double cap_threshold = std::cos(160. / 180 * CGAL_PI);

#ifdef CGAL_PMP_REMESHING_VERBOSE
      unsigned int nb_flips = 0;
#endif
      for(edge_descriptor e : edges(mesh_))
      {
        //only the patch edges are allowed to be flipped
        if (!is_flip_allowed(e))
          continue;
        //add geometric test to avoid axe cuts
        if (!internal::should_flip(e, mesh_, vpmap_, gt_))
          continue;

        halfedge_descriptor he = halfedge(e, mesh_);

        std::array<halfedge_descriptor, 2> r1 = internal::is_badly_shaped(
            face(he, mesh_),
            mesh_, vpmap_, vcmap_, ecmap_, gt_,
            cap_threshold, // bound on the angle: above 160 deg => cap
            4, // bound on shortest/longest edge above 4 => needle
            0,// collapse length threshold : not needed here
            0); // flip triangle height threshold

        std::array<halfedge_descriptor, 2> r2 = internal::is_badly_shaped(
            face(opposite(he, mesh_), mesh_),
            mesh_, vpmap_, vcmap_, ecmap_, gt_, cap_threshold, 4, 0, 0);

        const bool badly_shaped = (r1[0] != boost::graph_traits<PolygonMesh>::null_halfedge()//needle
                                || r1[1] != boost::graph_traits<PolygonMesh>::null_halfedge()//cap
                                || r2[0] != boost::graph_traits<PolygonMesh>::null_halfedge()//needle
                                || r2[1] != boost::graph_traits<PolygonMesh>::null_halfedge());//cap

        vertex_descriptor va = source(he, mesh_);
        vertex_descriptor vb = target(he, mesh_);
        vertex_descriptor vc = target(next(he, mesh_), mesh_);
        vertex_descriptor vd = target(next(opposite(he, mesh_), mesh_), mesh_);

        int vva = get(degree,va), tvva = target_valence(va);
        int vvb = get(degree, vb), tvvb = target_valence(vb);
        int vvc = get(degree,vc), tvvc = target_valence(vc);
        int vvd = get(degree,vd), tvvd = target_valence(vd);

        int deviation_pre = CGAL::abs(vva - tvva)
                          + CGAL::abs(vvb - tvvb)
                          + CGAL::abs(vvc - tvvc)
                          + CGAL::abs(vvd - tvvd);

        CGAL_assertion_code(Halfedge_status s1 = status(he));
        CGAL_assertion_code(Halfedge_status s1o = status(opposite(he, mesh_)));

        CGAL_assertion( is_flip_topologically_allowed(edge(he, mesh_)) );
        CGAL_assertion( !get(ecmap_, edge(he, mesh_)) );
        CGAL::Euler::flip_edge(he, mesh_);

        if (!badly_shaped)
        {
          vva -= 1;
          vvb -= 1;
          vvc += 1;
          vvd += 1;
        }

        put(degree, va, vva);
        put(degree, vb, vvb);
        put(degree, vc, vvc);
        put(degree, vd, vvd);

#ifdef CGAL_PMP_REMESHING_VERBOSE
        ++nb_flips;
#endif
#ifdef CGAL_PMP_REMESHING_VERBOSE_PROGRESS
        std::cout << "\r\t(" << nb_flips << " flips)";
        std::cout.flush();
#endif
        CGAL_assertion_code(Halfedge_status s2 = status(he));
        CGAL_assertion_code(Halfedge_status s2o = status(opposite(he, mesh_)));
        CGAL_assertion(s1 == s2   && s1 == PATCH);
        CGAL_assertion(s1o == s2o && s1o == PATCH);
        CGAL_assertion(!is_border(he, mesh_));

        CGAL_assertion(
             (vc == target(he, mesh_) && vd == source(he, mesh_))
          || (vd == target(he, mesh_) && vc == source(he, mesh_)));

        int deviation_post;
        if(!badly_shaped)
        {
          deviation_post = CGAL::abs(vva - tvva)
                           + CGAL::abs(vvb - tvvb)
                           + CGAL::abs(vvc - tvvc)
                           + CGAL::abs(vvd - tvvd);
        }

        //check that mesh does not become non-triangle,
        //nor has inverted faces
        if ((!badly_shaped && deviation_pre <= deviation_post)
          || !check_normals(he)
          || incident_to_degenerate(he)
          || incident_to_degenerate(opposite(he, mesh_))
          || !is_on_triangle(he)
          || !is_on_triangle(opposite(he, mesh_))
          || !check_normals(target(he, mesh_))
          || !check_normals(source(he, mesh_)))
        {
          CGAL_assertion( is_flip_topologically_allowed(edge(he, mesh_)) );
          CGAL_assertion( !get(ecmap_, edge(he, mesh_)) );
          CGAL::Euler::flip_edge(he, mesh_);

          vva += 1;
          vvb += 1;
          vvc -= 1;
          vvd -= 1;

          put(degree, va, vva);
          put(degree, vb, vvb);
          put(degree, vc, vvc);
          put(degree, vd, vvd);

#ifdef CGAL_PMP_REMESHING_VERBOSE
          --nb_flips;
#endif
          CGAL_assertion_code(Halfedge_status s3 = status(he));
          CGAL_assertion(s1 == s3);
          CGAL_assertion(!is_border(he, mesh_));
          CGAL_assertion(
               (va == source(he, mesh_) && vb == target(he, mesh_))
            || (vb == source(he, mesh_) && va == target(he, mesh_)));
        }

        Patch_id pid = get_patch_id(face(he, mesh_));
        set_patch_id(face(he, mesh_), pid);
        set_patch_id(face(opposite(he, mesh_), mesh_), pid);
      }

#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "\r\tdone ("<< nb_flips << " flips)" << std::endl;
#endif

#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_status_map();
      CGAL_assertion(remove_degenerate_faces(mesh_, parameters::vertex_point_map(vpmap_).geom_traits(gt_)));
      debug_self_intersections();
#endif

#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("3-edge_flips.off");
#endif
    }

    // PMP book :
    // "applies an iterative smoothing filter to the mesh.
    // The vertex movement has to be constrained to the vertex tangent plane [...]
    // smoothing algorithm with uniform Laplacian weights"
    void tangential_relaxation_impl(const bool relax_constraints/*1d smoothing*/
                                   , const unsigned int nb_iterations)
    {
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Tangential relaxation (" << nb_iterations << " iter.)...";
      std::cout << std::endl;
#endif

      // property map of constrained edges for relaxation
      auto edge_constraint = [&](const edge_descriptor e)
      {
        return this->is_constrained(e);
      };
      auto constrained_edges_pmap
        = boost::make_function_property_map<edge_descriptor>(edge_constraint);

      // property map of constrained vertices for relaxation
      auto vertex_constraint = [&](const vertex_descriptor v)
      {
        for (halfedge_descriptor h : halfedges_around_target(v, mesh_))
        {
          Halfedge_status s = status(h);
          if ( s == PATCH
            || s == PATCH_BORDER
            || status(opposite(h, mesh_)) == PATCH_BORDER)
            return false;
        }
        return true;
      };
      auto constrained_vertices_pmap
        = boost::make_function_property_map<vertex_descriptor>(vertex_constraint);

      tangential_relaxation(
        vertices(mesh_),
        mesh_,
        CGAL::parameters::number_of_iterations(nb_iterations)
                         .vertex_point_map(vpmap_)
                         .geom_traits(gt_)
                         .edge_is_constrained_map(constrained_edges_pmap)
                         .vertex_is_constrained_map(constrained_vertices_pmap)
                         .relax_constraints(relax_constraints)
      );

      CGAL_assertion(!input_mesh_is_valid_ || is_valid_polygon_mesh(mesh_));

#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_self_intersections();
#endif
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "done." << std::endl;
#endif
#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("4-relaxation.off");
#endif
    }

    // PMP book :
    // "maps the vertices back to the surface"
    void project_to_surface(internal_np::Param_not_found)
    {
      //todo : handle the case of boundary vertices
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Project to surface...";
      std::cout.flush();
#endif

      for(vertex_descriptor v : vertices(mesh_))
      {
        if (is_constrained(v) || is_isolated(v) || !is_on_patch(v))
          continue;
        //note if v is constrained, it has not moved

        Point proj = trees[patch_id_to_index_map[get_patch_id(face(halfedge(v, mesh_), mesh_))]]->closest_point(get(vpmap_, v));
        put(vpmap_, v, proj);
      }
      CGAL_assertion(!input_mesh_is_valid_ || is_valid_polygon_mesh(mesh_));
#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_self_intersections();
#endif
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "done." << std::endl;
#endif

#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("5-project.off");
#endif
    }

    template <class ProjectionFunctor>
    void project_to_surface(const ProjectionFunctor& proj)
    {
      //todo : handle the case of boundary vertices
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "Project to surface...";
      std::cout.flush();
#endif
      for(vertex_descriptor v : vertices(mesh_))
      {
        if (is_constrained(v) || is_isolated(v) || !is_on_patch(v))
          continue;
        //note if v is constrained, it has not moved
        put(vpmap_, v,  proj(v));
      }
      CGAL_assertion(is_valid(mesh_));
#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_self_intersections();
#endif
#ifdef CGAL_PMP_REMESHING_VERBOSE
      std::cout << "done." << std::endl;
#endif

#ifdef CGAL_DUMP_REMESHING_STEPS
      dump("5-project.off");
#endif
    }

private:
  Patch_id get_patch_id(const face_descriptor& f) const
  {
    if (f == boost::graph_traits<PM>::null_face())
      return Patch_id(-1);
    return get(patch_ids_map_, f);
  }

  void set_patch_id(const face_descriptor& f, const Patch_id& i)
  {
    put(patch_ids_map_, f, i);
  }

  struct Patch_id_property_map
  {
    typedef boost::readable_property_map_tag       category;
    typedef Patch_id                               value_type;
    typedef Patch_id                               reference;
    typedef typename Triangle_list::const_iterator key_type;

    const Self* remesher_ptr_;

    Patch_id_property_map()
      : remesher_ptr_(nullptr) {}
    Patch_id_property_map(const Self& remesher)
      : remesher_ptr_(&remesher) {}

    friend value_type get(const Patch_id_property_map& m, key_type tr_it)
    {
      //tr_it is an iterator from triangles_
      std::size_t id_in_vec = std::distance(
        m.remesher_ptr_->input_triangles().begin(), tr_it);

      CGAL_assertion(id_in_vec < m.remesher_ptr_->input_patch_ids().size());
      CGAL_assertion(*tr_it == m.remesher_ptr_->input_triangles()[id_in_vec]);

      return m.remesher_ptr_->input_patch_ids()[id_in_vec];
    }
  };

  private:
    Triangle_3 triangle(face_descriptor f) const
    {
      halfedge_descriptor h = halfedge(f, mesh_);
      vertex_descriptor v1  = target(h, mesh_);
      vertex_descriptor v2  = target(next(h, mesh_), mesh_);
      vertex_descriptor v3  = target(next(next(h, mesh_), mesh_), mesh_);
      return Triangle_3(get(vpmap_, v1), get(vpmap_, v2), get(vpmap_, v3));
    }

    double sqlength(const vertex_descriptor& v1,
                    const vertex_descriptor& v2) const
    {
      return to_double(CGAL::squared_distance(get(vpmap_, v1), get(vpmap_, v2)));
    }

    double sqlength(const halfedge_descriptor& h) const
    {
      vertex_descriptor v1 = target(h, mesh_);
      vertex_descriptor v2 = source(h, mesh_);
      return sqlength(v1, v2);
    }

    double sqlength(const edge_descriptor& e) const
    {
      return sqlength(halfedge(e, mesh_));
    }

    Point midpoint(const halfedge_descriptor& he) const
    {
      Point p1 = get(vpmap_, target(he, mesh_));
      Point p2 = get(vpmap_, source(he, mesh_));
      return gt_.construct_midpoint_3_object()(p1, p2);
    }

    void dump(const char* filename) const
    {
      std::ofstream out(filename);
      out.precision(18);
      out << mesh_;
      out.close();
    }

    int target_valence(const vertex_descriptor& v) const
    {
      return (has_border_ && is_border(v, mesh_)) ? 4 : 6;
    }

    bool is_on_triangle(const halfedge_descriptor& h) const
    {
      return h == next(next(next(h, mesh_), mesh_), mesh_);
    }

    bool is_longest_on_faces(const edge_descriptor& e) const
    {
      halfedge_descriptor h = halfedge(e, mesh_);
      halfedge_descriptor hopp = opposite(h, mesh_);

      //check whether h is the longest edge in its associated face
      //otherwise refinement will go into an endless loop
      double sqh = sqlength(h);
      return sqh >= sqlength(next(h, mesh_))
          && sqh >= sqlength(next(next(h, mesh_), mesh_))
          //do the same for hopp
          && sqh >= sqlength(next(hopp, mesh_))
          && sqh >= sqlength(next(next(hopp, mesh_), mesh_));
    }

    bool is_constrained(const edge_descriptor& e) const
    {
      return is_on_border(e) || is_on_patch_border(e);
    }

    bool is_split_allowed(const edge_descriptor& e) const
    {
      halfedge_descriptor h = halfedge(e, mesh_);
      halfedge_descriptor hopp = opposite(h, mesh_);

      if (protect_constraints_ && is_constrained(e))
        return false;
      else //allow splitting constraints
      {
        if (is_on_mesh(h) && is_on_mesh(hopp))
          return false;
        else if (is_on_mesh(h) && is_on_border(hopp))
          return false;
        else if (is_on_mesh(hopp) && is_on_border(h))
          return false;
        else if (is_an_isolated_constraint(h))
          return false;
        else
          return true;
      }
    }

    bool is_collapse_allowed(const edge_descriptor& e
                           , const bool collapse_constraints) const
    {
      halfedge_descriptor he = halfedge(e, mesh_);
      halfedge_descriptor hopp = opposite(he, mesh_);

      if (is_on_mesh(he) && is_on_mesh(hopp))
        return false;

      if (is_an_isolated_constraint(he) || is_an_isolated_constraint(hopp))
        return false;

      if ( (protect_constraints_ || !collapse_constraints) && is_constrained(e))
        return false;
      if (is_on_patch(he)) //hopp is also on patch
      {
        CGAL_assertion(is_on_patch(hopp));
        if (is_on_patch_border(target(he, mesh_)) && is_on_patch_border(source(he, mesh_)))
          return false;//collapse would induce pinching the selection
        else
          return (is_collapse_allowed_on_patch(he)
               && is_collapse_allowed_on_patch(hopp));
      }
      else if (is_on_patch_border(he))
        return is_collapse_allowed_on_patch_border(he);
      else if (is_on_patch_border(hopp))
        return is_collapse_allowed_on_patch_border(hopp);
      return false;
    }

    bool is_collapse_allowed_on_patch(const halfedge_descriptor& he) const
    {
      halfedge_descriptor hopp = opposite(he, mesh_);

      if (is_on_patch_border(next(he, mesh_)) && is_on_patch_border(prev(he, mesh_)))
        return false;//too many cases to be handled
      if (is_on_patch_border(next(hopp, mesh_)) && is_on_patch_border(prev(hopp, mesh_)))
        return false;//too many cases to be handled
      else if (is_on_patch_border(next(he, mesh_)))
      {
        //avoid generation of degenerate faces, and self-intersections
        if (source(he, mesh_) ==
          target(next(next_on_patch_border(next(he, mesh_)), mesh_), mesh_))
          return false;
      }
      else if (is_on_patch_border(prev(hopp, mesh_)))
      {
        //avoid generation of degenerate faces, and self-intersections
        if (target(hopp, mesh_) ==
          source(prev(prev_on_patch_border(prev(hopp, mesh_)), mesh_), mesh_))
          return false;
      }
      return true;
    }

    bool is_collapse_allowed_on_patch_border(const halfedge_descriptor& h) const
    {
      CGAL_precondition(is_on_patch_border(h));
      halfedge_descriptor hopp = opposite(h, mesh_);

      if (is_on_patch_border(next(h, mesh_)) && is_on_patch_border(prev(h, mesh_)))
        return false;

      if (is_on_patch_border(hopp))
      {
        if (is_on_patch_border(next(hopp, mesh_)) && is_on_patch_border(prev(hopp, mesh_)))
          return false;
        else if (next_on_patch_border(h) == hopp && prev_on_patch_border(h) == hopp)
          return false; //isolated patch border
        else
          return true;
      }
      CGAL_assertion(is_on_mesh(hopp) || is_on_border(hopp));
      return true;//we already checked we're not pinching a hole in the patch
    }

    bool is_flip_topologically_allowed(const edge_descriptor& e) const
    {
      halfedge_descriptor h=halfedge(e, mesh_);
      return !halfedge(target(next(h, mesh_), mesh_),
               target(next(opposite(h, mesh_), mesh_), mesh_),
               mesh_).second;
    }

    bool is_flip_allowed(const edge_descriptor& e) const
    {
      bool flip_possible = is_flip_allowed(halfedge(e, mesh_))
                        && is_flip_allowed(opposite(halfedge(e, mesh_), mesh_));

      if (!flip_possible) return false;

      // the flip is not possible if the edge already exists
      return is_flip_topologically_allowed(e);
    }

    bool is_flip_allowed(const halfedge_descriptor& h) const
    {
      if (!is_on_patch(h))
        return false;
      if (!is_on_patch_border(target(h, mesh_)))
        return true;
      if ( is_on_patch_border(next(h, mesh_))
        && is_on_patch_border(prev(opposite(h, mesh_), mesh_)))
        return false;
      return true;
    }

    halfedge_descriptor next_on_patch_border(const halfedge_descriptor& h) const
    {
      CGAL_precondition(is_on_patch_border(h));
      CGAL_assertion_code(const Patch_id& pid = get_patch_id(face(h, mesh_)));

      halfedge_descriptor end = opposite(h, mesh_);
      halfedge_descriptor nxt = next(h, mesh_);
      do
      {
        if (is_on_patch_border(nxt))
        {
          CGAL_assertion(get_patch_id(face(nxt, mesh_)) == pid);
          return nxt;
        }
        nxt = next(opposite(nxt, mesh_), mesh_);
      }
      while (end != nxt);

      CGAL_assertion(get_patch_id(face(nxt, mesh_)) == pid);
      CGAL_assertion(is_on_patch_border(end));
      return end;
    }

    halfedge_descriptor prev_on_patch_border(const halfedge_descriptor& h) const
    {
      CGAL_precondition(is_on_patch_border(h));
      CGAL_assertion_code(const Patch_id& pid = get_patch_id(face(h, mesh_)));

      halfedge_descriptor end = opposite(h, mesh_);
      halfedge_descriptor prv = prev(h, mesh_);
      do
      {
        if (is_on_patch_border(prv))
        {
          CGAL_assertion(get_patch_id(face(prv, mesh_)) == pid);
          return prv;
        }
        prv = prev(opposite(prv, mesh_), mesh_);
      }
      while (end != prv);

      CGAL_assertion(is_on_patch_border(end));
      CGAL_assertion(get_patch_id(face(prv, mesh_)) == pid);
      return end;
    }

    bool collapse_would_invert_face(const halfedge_descriptor& h) const
    {
      vertex_descriptor tv = target(h, mesh_);
      typename boost::property_traits<VertexPointMap>::reference
        s = get(vpmap_, source(h, mesh_)); //s for source
      typename boost::property_traits<VertexPointMap>::reference
        t = get(vpmap_, target(h, mesh_)); //t for target

      //check if collapsing the edge [src; tgt] towards tgt
      //would inverse the normal to the considered face
      //src and tgt are the endpoints of the edge to be collapsed
      //p and q are the vertices that form the face to be tested
      //along with src before collapse, and with tgt after collapse
      for(halfedge_descriptor hd :
          halfedges_around_target(opposite(h, mesh_), mesh_))
      {
        if (face(hd, mesh_) == boost::graph_traits<PM>::null_face())
          continue;

        vertex_descriptor tnhd = target(next(hd, mesh_), mesh_);
        vertex_descriptor tnnhd = target(next(next(hd, mesh_), mesh_), mesh_);
        typename boost::property_traits<VertexPointMap>::reference
          p = get(vpmap_, tnhd);
        typename boost::property_traits<VertexPointMap>::reference
          q = get(vpmap_, tnnhd);

#ifdef CGAL_PMP_REMESHING_DEBUG
        CGAL_assertion((Triangle_3(t, p, q).is_degenerate())
                     == GeomTraits().collinear_3_object()(t, p, q));
#endif

        if((tv == tnnhd) || (tv == tnhd))
          continue;

        if ( GeomTraits().collinear_3_object()(s, p, q)
          || GeomTraits().collinear_3_object()(t, p, q))
          continue;

        typename GeomTraits::Construct_cross_product_vector_3 cross_product
          = GeomTraits().construct_cross_product_vector_3_object();
#ifdef CGAL_PMP_REMESHING_DEBUG
        typename GeomTraits::Construct_normal_3 normal
          = GeomTraits().construct_normal_3_object();
        Vector_3 normal_before_collapse = normal(s, p, q);
        Vector_3 normal_after_collapse  = normal(t, p, q);

        CGAL::Sign s1 = CGAL::sign(normal_before_collapse * normal_after_collapse);
        CGAL::Sign s2 = CGAL::sign(cross_product(Vector_3(s, p), Vector_3(s, q))
                                 * cross_product(Vector_3(t, p), Vector_3(t, q)));
        CGAL_assertion(s1 == s2);
#endif

        if(CGAL::sign(cross_product(Vector_3(s, p), Vector_3(s, q))
                    * cross_product(Vector_3(t, p), Vector_3(t, q)))
          != CGAL::POSITIVE)
          return true;
      }
      return false;
    }

    bool is_constrained(const vertex_descriptor& v) const
    {
      return get(vcmap_, v);
    }
    bool is_isolated(const vertex_descriptor& v) const
    {
      return halfedges_around_target(v, mesh_).empty();
    }

    bool is_corner(const vertex_descriptor& v) const
    {
      if(! has_border_){
        return false;
      }
      unsigned int nb_incident_features = 0;
      for(halfedge_descriptor h : halfedges_around_target(v, mesh_))
      {
        halfedge_descriptor hopp = opposite(h, mesh_);
        if ( is_on_border(h) || is_on_patch_border(h)
          || is_on_border(hopp) || is_on_patch_border(hopp)
          || is_an_isolated_constraint(h))
          ++nb_incident_features;
        if (nb_incident_features > 2)
          return true;
      }
      return (nb_incident_features == 1);
    }

    Vector_3 compute_normal(const face_descriptor& f) const
    {
      if (f == boost::graph_traits<PM>::null_face())
        return CGAL::NULL_VECTOR;

      return compute_face_normal(f, mesh_, parameters::vertex_point_map(vpmap_).geom_traits(gt_));
    }

    template<typename FaceRange>
    void tag_halfedges_status(const FaceRange& face_range)
    {
      //init halfedges as:
      //  - MESH,        //h and hopp belong to the mesh, not the patch
      //  - MESH_BORDER  //h belongs to the mesh, face(hopp, pmesh) == null_face()
      for(halfedge_descriptor h : halfedges(mesh_))
      {
        //being part of the border of the mesh is predominant
        if (is_border(h, mesh_)){
          set_status(h, MESH_BORDER); //erase previous value if exists
          has_border_ = true;
        } else {
          set_status(h, MESH);
        }
      }

      //tag PATCH,       //h and hopp belong to the patch to be remeshed
      std::vector<halfedge_descriptor> patch_halfedges;
      for(face_descriptor f : face_range)
      {
        for(halfedge_descriptor h :
            halfedges_around_face(halfedge(f, mesh_), mesh_))
        {
          set_status(h, PATCH);
          patch_halfedges.push_back(h);
        }
      }

      // tag patch border halfedges
      for(halfedge_descriptor h : patch_halfedges)
      {
        CGAL_assertion(status(h) == PATCH);
        if( status(opposite(h, mesh_)) != PATCH
         || get_patch_id(face(h, mesh_)) != get_patch_id(face(opposite(h, mesh_), mesh_)))
        {
          set_status(h, PATCH_BORDER);
          has_border_ = true;
        }
      }

      // update status using constrained edge map
      if (!std::is_same<EdgeIsConstrainedMap,
                          Static_boolean_property_map<edge_descriptor, false> >::value)
      {
        for(edge_descriptor e : edges(mesh_))
        {
          if (get(ecmap_, e))
          {
            //deal with h and hopp for borders that are sharp edges to be preserved
            halfedge_descriptor h = halfedge(e, mesh_);
            Halfedge_status hs = status(h);
            if (hs == PATCH) {
              set_status(h, PATCH_BORDER);
              hs = PATCH_BORDER;
              has_border_ = true;
            }

            halfedge_descriptor hopp = opposite(h, mesh_);
            Halfedge_status hsopp = status(hopp);
            if (hsopp == PATCH) {
              set_status(hopp, PATCH_BORDER);
              hsopp = PATCH_BORDER;
              has_border_ = true;
            }

            if (hs != PATCH_BORDER && hsopp != PATCH_BORDER)
            {
              if(hs != MESH_BORDER)
                set_status(h, ISOLATED_CONSTRAINT);
              if(hsopp != MESH_BORDER)
                set_status(hopp, ISOLATED_CONSTRAINT);
            }
          }
        }
      }

#ifdef CGAL_PMP_REMESHING_DEBUG
      std::ofstream ofs("dump_isolated.polylines.txt");
      for (edge_descriptor e : edges(mesh_))
      {
        halfedge_descriptor h = halfedge(e, mesh_);
        Halfedge_status so = status(opposite(h, mesh_));
        bool isolated = (status(h) == ISOLATED_CONSTRAINT || so == ISOLATED_CONSTRAINT);
        CGAL_assertion(!isolated
                    || so == ISOLATED_CONSTRAINT
                    || so == MESH_BORDER);
        if(isolated)
          ofs << "2 " << get(vpmap_, target(h, mesh_))
              << " " << get(vpmap_, source(h, mesh_)) << std::endl;
      }
      ofs.close();
#endif
    }

    Halfedge_status status(const halfedge_descriptor& h) const
    {
      return get(halfedge_status_pmap_,h);
    }

    void set_status(const halfedge_descriptor& h,
                    const Halfedge_status& s)
    {
      put(halfedge_status_pmap_,h,s);
    }

    void merge_and_update_status(halfedge_descriptor en,
                                 halfedge_descriptor ep)
    {

      halfedge_descriptor eno = opposite(en, mesh_);
      halfedge_descriptor epo = opposite(ep, mesh_);
      Halfedge_status s_eno = status(eno);
      Halfedge_status s_epo = status(epo);

      Halfedge_status s_ep = status(ep);
      if(s_epo == MESH_BORDER
        || s_ep == MESH_BORDER
        || s_epo == PATCH_BORDER
        || s_ep == PATCH_BORDER)
      {
        set_status(en, s_epo);
        set_status(eno, s_ep);
      }
      else
      {
        Halfedge_status s_en = status(en);
        if(s_eno == MESH_BORDER
          || s_en == MESH_BORDER
          || s_eno == PATCH_BORDER
          || s_en == PATCH_BORDER)
        {
          set_status(ep, s_epo);
          set_status(epo, s_ep);
        }
      }
      // else keep current status for en and eno
    }

    template<typename Bimap>
    bool fix_degenerate_faces(const vertex_descriptor& v,
                              Bimap& short_edges,
                              const double& sq_low,
                              const bool collapse_constraints)
    {
      std::unordered_set<halfedge_descriptor> degenerate_faces;
      for(halfedge_descriptor h :
          halfedges_around_target(halfedge(v, mesh_), mesh_))
      {
        if(!is_border(h, mesh_) &&
           is_degenerate_triangle_face(face(h, mesh_), mesh_,
                                       parameters::vertex_point_map(vpmap_)
                                                   .geom_traits(gt_)))
          degenerate_faces.insert(h);
      }

      if(degenerate_faces.empty())
        return true;

      bool done = false;
      while(!degenerate_faces.empty())
      {
        halfedge_descriptor h = *(degenerate_faces.begin());
        degenerate_faces.erase(degenerate_faces.begin());

        if (!is_degenerate_triangle_face(face(h, mesh_), mesh_,
                                         parameters::vertex_point_map(vpmap_)
                                                    .geom_traits(gt_)))
          //this can happen when flipping h has consequences further in the mesh
          continue;

        //check that opposite is not also degenerate
        degenerate_faces.erase(opposite(h, mesh_));

        if(is_border(h, mesh_))
          continue;

        for(halfedge_descriptor hf :
            halfedges_around_face(h, mesh_))
        {
          if(face(opposite(hf, mesh_), mesh_) == boost::graph_traits<PM>::null_face())
            continue;

          vertex_descriptor vc = target(hf, mesh_);
          vertex_descriptor va = target(next(hf, mesh_), mesh_);
          vertex_descriptor vb = target(next(next(hf, mesh_), mesh_), mesh_);
          Vector_3 ab(get(vpmap_,va), get(vpmap_,vb));
          Vector_3 ac(get(vpmap_,va), get(vpmap_,vc));
          if (ab * ac < 0)
          {
            halfedge_descriptor hfo = opposite(hf, mesh_);
            halfedge_descriptor h_ab = prev(hf, mesh_);
            halfedge_descriptor h_ca = next(hf, mesh_);

            short_edges.left.erase(hf);
            short_edges.left.erase(hfo);

            CGAL_assertion( !get(ecmap_, edge(hf, mesh_)) );

            if (!is_flip_topologically_allowed(edge(hf, mesh_)))
              continue;

            CGAL::Euler::flip_edge(hf, mesh_);
            done = true;

            //update status
            set_status(h_ab, merge_status(h_ab, hf, hfo));
            set_status(h_ca, merge_status(h_ca, hf, hfo));
            if (is_on_patch(h_ca) || is_on_patch_border(h_ca))
            {
              set_status(hf, PATCH);
              set_status(hfo, PATCH);
            }
#ifdef CGAL_PMP_REMESHING_DEBUG
            debug_status_map();
#endif

            //insert new edges in 'short_edges'
            if (is_collapse_allowed(edge(hf, mesh_), collapse_constraints))
            {
              double sqlen = sqlength(hf);
              if (sqlen < sq_low)
                short_edges.insert(typename Bimap::value_type(hf, sqlen));
            }

            if(!is_border(hf, mesh_) &&
               is_degenerate_triangle_face(face(hf, mesh_), mesh_,
                                           parameters::vertex_point_map(vpmap_)
                                                      .geom_traits(gt_)))
              degenerate_faces.insert(hf);
            if(!is_border(hfo, mesh_) &&
               is_degenerate_triangle_face(face(hfo, mesh_), mesh_,
                                           parameters::vertex_point_map(vpmap_)
                                                      .geom_traits(gt_)))
              degenerate_faces.insert(hfo);

            break;
          }
        }
      }
#ifdef CGAL_PMP_REMESHING_DEBUG
      debug_status_map();
#endif
      return done;
    }

    bool incident_to_degenerate(const halfedge_descriptor& he)
    {
      for(halfedge_descriptor h :
          halfedges_around_target(he, mesh_))
      {
        if(!is_border(h, mesh_) &&
           is_degenerate_triangle_face(face(h, mesh_), mesh_,
                                       parameters::vertex_point_map(vpmap_)
                                                  .geom_traits(gt_)))
          return true;
      }
      return false;
    }

    Halfedge_status merge_status(const halfedge_descriptor& h1,
      const halfedge_descriptor& h2,
      const halfedge_descriptor& h3)
    {
      Halfedge_status s1 = status(h1);
      if (s1 == MESH_BORDER) return s1;
      Halfedge_status s2 = status(h2);
      if (s2 == MESH_BORDER) return s2;
      Halfedge_status s3 = status(h3);
      if (s3 == MESH_BORDER) return s3;
      else if (s1 == PATCH_BORDER) return s1;
      else if (s2 == PATCH_BORDER) return s2;
      else if (s3 == PATCH_BORDER) return s3;

      CGAL_assertion(s1 == s2 && s1 == s3);
      return s1;
    }

    bool is_on_patch(const halfedge_descriptor& h) const
    {
      bool res =(status(h) == PATCH);
      CGAL_assertion(res == (status(opposite(h, mesh_)) == PATCH));
      return res;
    }

    bool is_on_patch(const face_descriptor& f) const
    {
      for(halfedge_descriptor h :
          halfedges_around_face(halfedge(f, mesh_), mesh_))
      {
        if (is_on_patch(h) || is_on_patch_border(h))
          return true;
      }
      return false;
    }

    bool is_on_patch(const vertex_descriptor& v) const
    {
      if(! has_border_){
        return true;
      }
      for(halfedge_descriptor h :
          halfedges_around_target(v, mesh_))
      {
        if (!is_on_patch(h))
          return false;
      }
      return true;
    }

public:
    bool is_on_patch_border(const halfedge_descriptor& h) const
    {
      bool res = (status(h) == PATCH_BORDER);
      if (res)
      {
        CGAL_assertion_code(Halfedge_status hs = status(opposite(h, mesh_)));
        CGAL_assertion(hs == MESH_BORDER
                    || hs == MESH
                    || hs == PATCH_BORDER);//when 2 incident patches are remeshed
      }
      return res;
    }
    bool is_on_patch_border(const edge_descriptor& e) const
    {
      return is_on_patch_border(halfedge(e,mesh_))
          || is_on_patch_border(opposite(halfedge(e, mesh_), mesh_));
    }
    bool is_on_patch_border(const vertex_descriptor& v) const
    {
      if(! has_border_){
        return false;
      }
      for(halfedge_descriptor h : halfedges_around_target(v, mesh_))
      {
        if (is_on_patch_border(h) || is_on_patch_border(opposite(h, mesh_)))
          return true;
      }
      return false;
    }

    bool is_on_border(const halfedge_descriptor& h) const
    {
      bool res = (status(h) == MESH_BORDER);
      CGAL_assertion(res == is_border(h, mesh_));
      CGAL_assertion(res == is_border(next(h, mesh_), mesh_));
      return res;
    }

    bool is_on_border(const edge_descriptor& e) const
    {
      return is_on_border(halfedge(e, mesh_))
          || is_on_border(opposite(halfedge(e, mesh_), mesh_));
    }

    bool is_on_mesh(const halfedge_descriptor& h) const
    {
      return status(h) == MESH;
    }

    bool is_an_isolated_constraint(const halfedge_descriptor& h) const
    {
      bool res = (status(h) == ISOLATED_CONSTRAINT);
      CGAL_assertion_code(Halfedge_status so = status(opposite(h, mesh_)));
      CGAL_assertion(!res || so == ISOLATED_CONSTRAINT || so == MESH_BORDER);
      return res;
    }

private:
    void halfedge_added(const halfedge_descriptor& h,
                        const Halfedge_status& s)
    {
        set_status(h, s);
    }

    std::size_t nb_valid_halfedges() const
    {
      return static_cast<std::size_t>(
        std::distance(halfedges(mesh_).first, halfedges(mesh_).second));
    }

    void debug_status_map() const
    {
      unsigned int nb_border = 0;
      unsigned int nb_mesh = 0;
      unsigned int nb_patch = 0;
      unsigned int nb_patch_border = 0;
      unsigned int nb_isolated = 0;

      for(halfedge_descriptor h : halfedges(mesh_))
      {
        if(is_on_patch(h))              nb_patch++;
        else if(is_on_patch_border(h))  nb_patch_border++;
        else if(is_on_mesh(h))          nb_mesh++;
        else if(is_on_border(h))        nb_border++;
        else if(is_an_isolated_constraint(h)) nb_isolated++;
        else CGAL_assertion(false);
      }
      CGAL_USE(nb_border);
      CGAL_USE(nb_mesh);
      CGAL_USE(nb_patch);
      CGAL_USE(nb_patch_border);
      CGAL_USE(nb_isolated);
    }

#ifdef CGAL_PMP_REMESHING_DEBUG
    void debug_self_intersections() const
    {
      std::cout << "Test self intersections...";
      std::vector<std::pair<face_descriptor, face_descriptor> > facets;
      self_intersections(mesh_, std::back_inserter(facets),
                         parameters::vertex_point_map(vpmap_).geom_traits(gt_));
      //CGAL_assertion(facets.empty());
      std::cout << "done ("<< facets.size() <<" facets)." << std::endl;
    }

    void debug_self_intersections(const vertex_descriptor& v) const
    {
      std::cout << "Test self intersections...";
      std::vector<std::pair<face_descriptor, face_descriptor> > facets;
      self_intersections(faces_around_target(halfedge(v, mesh_), mesh_), mesh_, std::back_inserter(facets),
                         parameters::vertex_point_map(vpmap_).geom_traits(gt_));
      //CGAL_assertion(facets.empty());
      std::cout << "done ("<< facets.size() <<" facets)." << std::endl;
    }
#endif

    //check whether the normals to faces incident to v
    //have all their 2 by 2 dot products > 0
    bool check_normals(const vertex_descriptor& v) const
    {
      return check_normals(halfedges_around_target(halfedge(v, mesh_), mesh_));
    }

    template <typename HalfedgeRange>
    bool check_normals(const HalfedgeRange& hedges) const
    {
      std::size_t nb_patches = patch_id_to_index_map.size();
      //std::vector<boost::optional<Vector_3> > normal_per_patch(nb_patches,boost::none);
      std::vector<bool> initialized(nb_patches,false);
      std::vector<Vector_3> normal_per_patch(nb_patches);

      for(halfedge_descriptor hd : hedges)
      {
        Halfedge_status s = status(hd);

        if(s != PATCH && s != PATCH_BORDER)
          continue;

        Vector_3 n = compute_normal(face(hd, mesh_));
        if (n == CGAL::NULL_VECTOR) //for degenerate faces
          continue;
        Patch_id pid = get_patch_id(face(hd, mesh_));
        std::size_t index = patch_id_to_index_map.at(pid);
        //if(normal_per_patch[index]){
        if(initialized[index]){
          const Vector_3& vec = normal_per_patch[index];
          double dot = to_double(n * vec);
          if (dot <= 0.){
            return false;
          }
        }
        //normal_per_patch[index] = boost::make_optional(n);
        normal_per_patch[index] = n;
        initialized[index] = true;
      }
      return true;
    }

    bool check_normals(const halfedge_descriptor& h) const
    {
      if (!is_on_patch(h))
        return true;//nothing to say
      Vector_3 n = compute_normal(face(h, mesh_));
      Vector_3 no = compute_normal(face(opposite(h, mesh_), mesh_));
      return n * no > 0.;
    }

  public:
    const Triangle_list& input_triangles() const {
      return input_triangles_;
    }

    const Patch_id_list& input_patch_ids() const {
      return input_patch_ids_;
    }

  private:
    PolygonMesh& mesh_;
    VertexPointMap& vpmap_;
    const GeomTraits& gt_;
    bool build_tree_;
    bool has_border_;
    std::vector<AABB_tree*> trees;
    Patch_id_to_index_map patch_id_to_index_map;
    Triangle_list input_triangles_;
    Patch_id_list input_patch_ids_;
    Halfedge_status_pmap halfedge_status_pmap_;
    bool protect_constraints_;
    FacePatchMap patch_ids_map_;
    EdgeIsConstrainedMap ecmap_;
    VertexIsConstrainedMap vcmap_;
    FaceIndexMap fimap_;
    CGAL_assertion_code(bool input_mesh_is_valid_;)

  };//end class Incremental_remesher
}//end namespace internal
}//end namespace Polygon_mesh_processing
}//end namespace CGAL

#endif //CGAL_POLYGON_MESH_PROCESSING_REMESH_IMPL_H
