#include <Rcpp.h>
// Copyright (c) 2005,2006,2007,2008,2009,2010,2011,2012,2013 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL$
// $Id$
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s): Ron Wein          <wein@post.tau.ac.il>
//            Efi Fogel         <efif@post.tau.ac.il>
//            Eric Berberich    <eric.berberich@cgal.org>
//            (based on old version by: Iddo Hanniel,
//                                      Eyal Flato,
//                                      Oren Nechushtan,
//                                      Ester Ezra,
//                                      Shai Hirsch,
//                                      and Eugene Lipovetsky)

#ifndef CGAL_ARRANGEMENT_ON_SURFACE_2_IMPL_H
#define CGAL_ARRANGEMENT_ON_SURFACE_2_IMPL_H

#include <CGAL/license/Arrangement_on_surface_2.h>

#ifndef CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
#define CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE 0
#endif

/*! \file
 * Member-function definitions for the Arrangement_2<GeomTraits, TopTraits>
 * class-template.
 */

#include <variant>

#include <CGAL/function_objects.h>
#include <CGAL/use.h>

namespace CGAL {

//-----------------------------------------------------------------------------
// Default constructor.
//
template <typename GeomTraits, typename TopTraits>
Arrangement_on_surface_2<GeomTraits, TopTraits>::Arrangement_on_surface_2() :
  m_topol_traits()
{
  typedef has_Left_side_category<GeomTraits> Cond_left;
  typedef internal::Validate_left_side_category<GeomTraits, Cond_left::value>
    Validate_left_side_category;
  void (Validate_left_side_category::*pleft)(void) =
    &Validate_left_side_category::template missing__Left_side_category<int>;
  (void)pleft;

  typedef has_Bottom_side_category<GeomTraits> Cond_bottom;
  typedef internal::Validate_bottom_side_category<GeomTraits,
                                                  Cond_bottom::value>
    Validate_bottom_side_category;
  void (Validate_bottom_side_category::*pbottom)(void) =
    &Validate_bottom_side_category::template missing__Bottom_side_category<int>;
  (void)pbottom;

  typedef has_Top_side_category<GeomTraits> Cond_top;
  typedef internal::Validate_top_side_category<GeomTraits, Cond_top::value>
    Validate_top_side_category;
  void (Validate_top_side_category::*ptop)(void) =
    &Validate_top_side_category::template missing__Top_side_category<int>;
  (void)ptop;

  typedef has_Right_side_category<GeomTraits> Cond_right;
  typedef internal::Validate_right_side_category<GeomTraits, Cond_right::value>
    Validate_right_side_category;
  void (Validate_right_side_category::*pright)(void) =
    &Validate_right_side_category::template missing__Right_side_category<int>;
  (void)pright;

  // Initialize the DCEL structure to represent an empty arrangement.
  m_topol_traits.init_dcel();

  // Allocate the traits.
  m_geom_traits = new Traits_adaptor_2;
  m_own_traits = true;
}

//-----------------------------------------------------------------------------
// Copy constructor.
//
template <typename GeomTraits, typename TopTraits>
Arrangement_on_surface_2<GeomTraits, TopTraits>::
Arrangement_on_surface_2(const Self& arr) :
  m_geom_traits(nullptr),
  m_own_traits(false)
{ assign(arr); }

//-----------------------------------------------------------------------------
// Constructor given a traits object.
//
template <typename GeomTraits, typename TopTraits>
Arrangement_on_surface_2<GeomTraits, TopTraits>::
Arrangement_on_surface_2(const Geometry_traits_2* geom_traits) :
  m_topol_traits(geom_traits)
{
  typedef has_Left_side_category<GeomTraits> Cond_left;
  typedef internal::Validate_left_side_category<GeomTraits, Cond_left::value>
    Validate_left_side_category;
  void (Validate_left_side_category::*pleft)(void) =
    &Validate_left_side_category::template missing__Left_side_category<int>;
  (void)pleft;

  typedef has_Bottom_side_category<GeomTraits> Cond_bottom;
  typedef internal::Validate_bottom_side_category<GeomTraits,
                                                  Cond_bottom::value>
    Validate_bottom_side_category;
  void (Validate_bottom_side_category::*pbottom)(void) =
    &Validate_bottom_side_category::template missing__Bottom_side_category<int>;
  (void)pbottom;

  typedef has_Top_side_category<GeomTraits> Cond_top;
  typedef internal::Validate_top_side_category<GeomTraits, Cond_top::value>
    Validate_top_side_category;
  void (Validate_top_side_category::*ptop)(void) =
    &Validate_top_side_category::template missing__Top_side_category<int>;
  (void)ptop;

  typedef has_Right_side_category<GeomTraits> Cond_right;
  typedef internal::Validate_right_side_category<GeomTraits, Cond_right::value>
    Validate_right_side_category;
  void (Validate_right_side_category::*pright)(void) =
    &Validate_right_side_category::template missing__Right_side_category<int>;
  (void)pright;

  // Initialize the DCEL structure to represent an empty arrangement.
  m_topol_traits.init_dcel();

  // Set the traits.
  m_geom_traits = static_cast<const Traits_adaptor_2*>(geom_traits);
  m_own_traits = false;
}

//-----------------------------------------------------------------------------
// Assignment operator.
//
template <typename GeomTraits, typename TopTraits>
Arrangement_on_surface_2<GeomTraits, TopTraits>&
Arrangement_on_surface_2<GeomTraits, TopTraits>::operator=(const Self& arr)
{
  if (this == &arr) return (*this);     // handle self-assignment
  assign(arr);
  return (*this);
}

//-----------------------------------------------------------------------------
// Assign an arrangement.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::assign(const Self& arr)
{
  // Clear the current contents of the arrangement.
  clear();

  // Notify the observers that an assignment is to take place.
  _notify_before_assign(arr);

  // Assign the topology-traits object.
  m_topol_traits.assign(arr.m_topol_traits);

  // Go over the vertices and create duplicates of the stored points.
  typename Dcel::Vertex_iterator vit;
  for (vit = _dcel().vertices_begin(); vit != _dcel().vertices_end(); ++vit) {
    DVertex* p_v = &(*vit);

    if (! p_v->has_null_point()) {
      // Create the duplicate point and store it in the points container.
      Point_2* dup_p = _new_point(p_v->point());

      // Associate the vertex with the duplicated point.
      p_v->set_point(dup_p);
    }
  }

  // Go over the edge and create duplicates of the stored curves.
  typename Dcel::Edge_iterator eit;
  for (eit = _dcel().edges_begin(); eit != _dcel().edges_end(); ++eit) {
    DHalfedge* p_e = &(*eit);

    if (! p_e->has_null_curve()) {
      // Create the duplicate curve and store it in the curves container.
      X_monotone_curve_2* dup_cv = _new_curve(p_e->curve());

      // Associate the halfedge (and its twin) with the duplicated curve.
      p_e->set_curve(dup_cv);
    }
  }

  // Take care of the traits object.
  if (m_own_traits && (m_geom_traits != nullptr)) {
    delete m_geom_traits;
    m_geom_traits = nullptr;
  }

  m_geom_traits = (arr.m_own_traits) ? new Traits_adaptor_2 : arr.m_geom_traits;
  m_own_traits = arr.m_own_traits;

  // Notify the observers that the assignment has been performed.
  _notify_after_assign();
}

//-----------------------------------------------------------------------------
// Destructor.
//
template <typename GeomTraits, typename TopTraits>
Arrangement_on_surface_2<GeomTraits, TopTraits>::~Arrangement_on_surface_2()
{
  // Free all stored points.
  typename Dcel::Vertex_iterator vit;
  for (vit = _dcel().vertices_begin(); vit != _dcel().vertices_end(); ++vit)
    if (! vit->has_null_point())
      _delete_point(vit->point());

  // Free all stores curves.
  typename Dcel::Edge_iterator eit;
  for (eit = _dcel().edges_begin(); eit != _dcel().edges_end(); ++eit)
    if (! eit->has_null_curve())
      _delete_curve(eit->curve());

  // Free the traits object, if necessary.
  if (m_own_traits && (m_geom_traits != nullptr)) {
    delete m_geom_traits;
    m_geom_traits = nullptr;
  }

  // Detach all observers still attached to the arrangement.
  Observers_iterator  iter = m_observers.begin();
  Observers_iterator  next;
  Observers_iterator  end = m_observers.end();

  while (iter != end) {
    next = iter;
    ++next;
    (*iter)->detach();
    iter = next;
  }
}

//-----------------------------------------------------------------------------
// Clear the arrangement.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::clear()
{
  // Notify the observers that we are about to clear the arrangement.
  _notify_before_clear();

  // Free all stored points.
  typename Dcel::Vertex_iterator vit;
  for (vit = _dcel().vertices_begin(); vit != _dcel().vertices_end(); ++vit)
    if (! vit->has_null_point()) _delete_point(vit->point());

  // Free all stores curves.
  typename Dcel::Edge_iterator eit;
  for (eit = _dcel().edges_begin(); eit != _dcel().edges_end(); ++eit)
    if (! eit->has_null_curve()) _delete_curve(eit->curve());

  // Clear the DCEL and construct an empty arrangement.
  _dcel().delete_all();
  m_topol_traits.init_dcel();

  // Notify the observers that we have just cleared the arrangement.
  _notify_after_clear();
}

//-----------------------------------------------------------------------------
// Insert a point as an isolated vertex in the interior of a given face.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Vertex_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_in_face_interior(const Point_2& p, Face_handle f)
{
  DFace* p_f = _face(f);

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: insert_in_face_interior (interface)" << std::endl;
  Rcpp::Rcout << "pt   : " << p << std::endl;
  Rcpp::Rcout << "face : " << &(*f) << std::endl;
#endif

  // Obtain the boundary conditions:
  auto ps_x = m_geom_traits->parameter_space_in_x_2_object()(p);
  auto ps_y = m_geom_traits->parameter_space_in_y_2_object()(p);

  // Create a new vertex associated with the given point.
  // The point is either the interior of the parameter space or on the boundary.
  DVertex* v(nullptr);
  if ((ps_x == ARR_INTERIOR) && (ps_y == ARR_INTERIOR)) v = _create_vertex(p);
  else {
    v = _create_boundary_vertex(p, ps_x, ps_y);

    // Notify the topology traits on the creation of the boundary vertex.
    m_topol_traits.notify_on_boundary_vertex_creation(v, p, ps_x, ps_y);
  }

  // Insert v as an isolated vertex inside the given face.
  _insert_isolated_vertex(p_f, v);

  // Return a handle to the new isolated vertex.
  Vertex_handle vh(v);
  return vh;
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement as a new hole (inner
// component) inside the given face.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_in_face_interior(const X_monotone_curve_2& cv, Face_handle f)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: insert_in_face_interior (interface)" << std::endl;
  Rcpp::Rcout << "cv   : " << cv << std::endl;
  Rcpp::Rcout << "face : " << &(*f) << std::endl;
#endif

  DFace* p_f = _face(f);

  // Check if cv's left end has boundary conditions, and obtain a vertex v1
  // that corresponds to this end.
  const Arr_parameter_space  ps_x1 =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
  const Arr_parameter_space  ps_y1 =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);
  DHalfedge* fict_prev1 = nullptr;

  DVertex* v1 = ((ps_x1 == ARR_INTERIOR) && (ps_y1 == ARR_INTERIOR)) ?
    // The curve has a valid left endpoint: Create a new vertex associated
    // with the curve's left endpoint.
    _create_vertex(m_geom_traits->construct_min_vertex_2_object()(cv)) :
    // Locate the DCEL features that will be used for inserting the curve's
    // left end.
    _place_and_set_curve_end(p_f, cv, ARR_MIN_END, ps_x1, ps_y1, &fict_prev1);

  // Check if cv's right end has boundary conditions, and obtain a vertex v2
  // that corresponds to this end.
  const Arr_parameter_space  ps_x2 =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END);
  const Arr_parameter_space  ps_y2 =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END);
  DHalfedge* fict_prev2 = nullptr;

  DVertex* v2 = ((ps_x2 == ARR_INTERIOR) && (ps_y2 == ARR_INTERIOR)) ?
    // The curve has a valid right endpoint: Create a new vertex associated
    // with the curve's right endpoint.
    _create_vertex(m_geom_traits->construct_max_vertex_2_object()(cv)) :
    // Locate the DCEL features that will be used for inserting the curve's
    // right end.
    _place_and_set_curve_end(p_f, cv, ARR_MAX_END, ps_x2, ps_y2, &fict_prev2);

  // Create the edge connecting the two vertices (note we know v1 is
  // lexicographically smaller than v2).
  DHalfedge* new_he;

  if ((fict_prev1 == nullptr) && (fict_prev2 == nullptr))
    // Both vertices represent valid points.
    new_he = _insert_in_face_interior(p_f, cv, ARR_LEFT_TO_RIGHT, v1, v2);
  else if ((fict_prev1 == nullptr) && (fict_prev2 != nullptr)) {
    // v1 represents a valid point and v2 is inserted using its predecessor.
    new_he = _insert_from_vertex(fict_prev2, cv, ARR_RIGHT_TO_LEFT, v1);
    new_he = new_he->opposite();
  }
  else if ((fict_prev1 != nullptr) && (fict_prev2 == nullptr))
    // v1 is inserted using its predecessor and v2 represents a valid point.
    new_he = _insert_from_vertex(fict_prev1, cv, ARR_LEFT_TO_RIGHT, v2);
  else {
    // Both vertices are inserted using their predecessor halfedges.

    // Comment:
    // In case the inserted curve has two vertical asymptotes at the top
    // it happens that fict_prev1 is split by the max end and becomes the
    // prev edge, which is fict_prev2. Since both pointers are equal they
    // both point to the max end. Thus, we advance fict_prev1 by one
    // such that it points to the min end again.
    // Note that this only happens at the top. At the bottom everything
    // goes fine since the insertion order is reverted with respect to the
    // orientation of the edges.
    //
    // In the other function such a fix is not needed, as at most one
    // curve-end reaches the boundary. It is also not possible to delay
    // it to _insert_at_vertices, as that expects the two predecessor
    // halfedges as input. An early detecting is also not possible
    // (i.e.~in _place_and_set_curve_end), as that needs to know to be
    // called from here!
    if (fict_prev1 == fict_prev2) fict_prev1 = fict_prev1->next();

    // Note that in this case we may create a new face.
    bool new_face_created = false;
    bool check_swapped_predecessors = false;
    new_he = _insert_at_vertices(fict_prev1, cv, ARR_LEFT_TO_RIGHT,
                                 fict_prev2->next(), new_face_created,
                                 check_swapped_predecessors);
    // Comment EBEB 2012-10-21: Swapping does not take place as there is no local minimum so far
    CGAL_assertion(!check_swapped_predecessors);
    // usually one would expect to have an new_he (and its twin) lying on the
    // same _inner_ CCB ...

    if (new_face_created) {
      // ... but in case that a new face got created new_he should lie on an
      // outer CCB
      CGAL_assertion(new_he->is_on_outer_ccb());
      // Note! new_he is not needed to define the new outer CCB of the new face
      // Here, it can be the outer ccb of the old face, as there is also not
      // swapping taking place!

      // In case a new face has been created (pointed by the new halfedge we
      // obtained), we have to examine the holes and isolated vertices in the
      // existing face (pointed by the twin halfedge) and move the relevant
      // holes and isolated vertices into the new face.
      _relocate_in_new_face(new_he);
    }
  }

  // Return a handle to the new halfedge directed from left to right.
  CGAL_postcondition(new_he->direction() == ARR_LEFT_TO_RIGHT);
  return (Halfedge_handle(new_he));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that its left
// endpoint corresponds to a given arrangement vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_from_left_vertex(const X_monotone_curve_2& cv,
                        Vertex_handle v,
                        Face_handle f)
{
  CGAL_precondition_code
    (const bool at_obnd1 =
     !m_geom_traits->is_closed_2_object()(cv, ARR_MIN_END));
  CGAL_precondition_msg
    ((! at_obnd1 &&
      m_geom_traits->equal_2_object()
      (v->point(),
       m_geom_traits->construct_min_vertex_2_object()(cv))) ||
     (at_obnd1 && v->is_at_open_boundary()),
     "The input vertex should be the left curve end.");

  // Check if cv's right end has boundary conditions. If not, create a vertex
  // that corresponds to the right endpoint.
  auto ps_x2 = m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END);
  auto ps_y2 = m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END);
  DVertex* v2 = nullptr;
  DHalfedge* fict_prev2 = nullptr;

  if ((ps_x2 == ARR_INTERIOR) && (ps_y2 == ARR_INTERIOR))
    // The curve has a valid right endpoint: Create a new vertex associated
    // with the curve's right endpoint.
    v2 = _create_vertex(m_geom_traits->construct_max_vertex_2_object()(cv));

  // Check if the given vertex, corresponding to the left curve end, has no
  // incident edges.
  if (v->degree() == 0) {
    // The given vertex is an isolated one: We should in fact insert the curve
    // in the interior of the face containing this vertex.
    DVertex* v1 = _vertex(v);
    DIso_vertex* iv = nullptr;
    DFace* p_f = nullptr;

    if (v->is_isolated()) {
      // Obtain the face from the isolated vertex.
      iv = v1->isolated_vertex();
      p_f = iv->face();
    }
    else {
      // In this case the face that contains v should be provided by the user.
      CGAL_precondition(f != Face_handle());
      p_f = _face(f);
    }

    // If the vertex that corresponds to cv's right end has boundary
    // conditions, create it now.
    if (v2 == nullptr)
      // Locate the DCEL features that will be used for inserting the curve's
      // right end.
      v2 = _place_and_set_curve_end(p_f, cv, ARR_MAX_END, ps_x2, ps_y2,
                                    &fict_prev2);

    if (iv != nullptr) {
      // Remove the isolated vertex v1, as it will not be isolated any more.
      p_f->erase_isolated_vertex(iv);
      _dcel().delete_isolated_vertex(iv);
    }

    // Create the edge connecting the two vertices (note that we know that
    // v1 is smaller than v2).
    DHalfedge* new_he;
    if (fict_prev2 == nullptr)
      new_he = _insert_in_face_interior(p_f, cv, ARR_LEFT_TO_RIGHT, v1, v2);
    else {
      new_he = _insert_from_vertex(fict_prev2, cv, ARR_RIGHT_TO_LEFT, v1);
      new_he = new_he->opposite();
    }

    // Return a handle to the new halfedge directed from v1 to v2.
    CGAL_postcondition(new_he->direction() == ARR_LEFT_TO_RIGHT);
    return (Halfedge_handle(new_he));
  }

  // Go over the incident halfedges around v and find the halfedge after
  // which the new curve should be inserted.
  DHalfedge* prev1 = _locate_around_vertex(_vertex(v), cv, ARR_MIN_END);
  CGAL_assertion_msg
    (prev1 != nullptr,
     "The inserted curve cannot be located in the arrangement.");

  DFace* f1 = prev1->is_on_inner_ccb() ? prev1->inner_ccb()->face() :
    prev1->outer_ccb()->face();

  // If the vertex that corresponds to cv's right end has boundary conditions,
  // create it now.
  if (v2 == nullptr)
    // Locate the DCEL features that will be used for inserting the curve's
    // right end.
    v2 =
      _place_and_set_curve_end(f1, cv, ARR_MAX_END, ps_x2, ps_y2, &fict_prev2);

  // Perform the insertion (note that we know that prev1->vertex is smaller
  // than v2).
  DHalfedge* new_he;

  if (fict_prev2 == nullptr)
    // Insert the halfedge given the predecessor halfedge of v1.
    new_he = _insert_from_vertex(prev1, cv, ARR_LEFT_TO_RIGHT, v2);
  else {
    // Insert the halfedge given the two predecessor halfedges.
    // Note that in this case we may create a new face.
    bool new_face_created = false;
    bool check_swapped_predecessors = false;
    new_he = _insert_at_vertices(prev1, cv, ARR_LEFT_TO_RIGHT,
                                 fict_prev2->next(),
                                 new_face_created, check_swapped_predecessors);
    // Comment EBEB 2012-10-21: Swapping does not take place as the insertion
    // merges the CCB as an "interior" extension into an outer CCB of a face
    // incident the parameter space's boundary.
    CGAL_assertion(!check_swapped_predecessors);

    if (new_face_created) {
      CGAL_assertion(new_he->is_on_outer_ccb());
      // Note! new_he is not needed to define the new outer CCB of the new face
      // Here, it can be the outer ccb of the old face, as there is also not
      // swapping taking place!

      // In case a new face has been created (pointed by the new halfedge we
      // obtained), we have to examine the holes and isolated vertices in the
      // existing face (pointed by the twin halfedge) and move the relevant
      // holes and isolated vertices into the new face.
      _relocate_in_new_face(new_he);
    }
  }

  // Return a handle to the halfedge directed toward the new vertex v2.
  CGAL_postcondition(new_he->direction() == ARR_LEFT_TO_RIGHT);
  return (Halfedge_handle(new_he));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that one its left
// endpoint corresponds to a given arrangement vertex, given the exact place
// for the curve in the circular list around this vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_from_left_vertex(const X_monotone_curve_2& cv, Halfedge_handle prev)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: insert_from_left_vertex (interface)" << std::endl;
  Rcpp::Rcout << "cv   : " << cv << std::endl;
  if (!prev->is_fictitious()) {
    Rcpp::Rcout << "prev : " << prev ->curve() << std::endl;
  } else {
    Rcpp::Rcout << "prev : fictitious" << std::endl;
  }
  Rcpp::Rcout << "dir  : " << prev->direction() << std::endl;
#endif

  CGAL_precondition_code
    (const bool at_obnd1 =
     !m_geom_traits->is_closed_2_object()(cv, ARR_MIN_END));
  CGAL_precondition_msg
    ((! at_obnd1 &&
      m_geom_traits->equal_2_object()
      (prev->target()->point(),
       m_geom_traits->construct_min_vertex_2_object()(cv))) ||
     (at_obnd1 && prev->target()->is_at_open_boundary()),
     "The target of the input halfedge should be the left curve end.");

  CGAL_precondition_msg
    (at_obnd1 || _locate_around_vertex(_vertex(prev->target()),
                                       cv, ARR_MIN_END) == _halfedge(prev),
     "In the clockwise order of curves around the vertex, "
     " cv must succeed the curve of prev.");

  // Get the predecessor halfedge for the insertion of the left curve end.
  DHalfedge* prev1 = _halfedge(prev);
  DFace* f1 = _face(prev->face());

  // Check if cv's right end has boundary conditions, and obtain a vertex
  // that corresponds to this end.
  const Arr_parameter_space  ps_x2 =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END);
  const Arr_parameter_space  ps_y2 =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END);
  DHalfedge* fict_prev2 = nullptr;

  DVertex* v2 = ((ps_x2 == ARR_INTERIOR) && (ps_y2 == ARR_INTERIOR)) ?
    // The curve has a valid right endpoint: Create a new vertex associated
    // with the curve's right endpoint.
    _create_vertex(m_geom_traits->construct_max_vertex_2_object()(cv)) :
    // Locate the DCEL features that will be used for inserting the curve's
    // right end.
    _place_and_set_curve_end(f1, cv, ARR_MAX_END, ps_x2, ps_y2, &fict_prev2);

  // Perform the insertion (note that we know that prev1->vertex is smaller
  // than v2).
  DHalfedge* new_he;

  if (fict_prev2 == nullptr)
    // Insert the halfedge given the predecessor halfedge of the left vertex.
    new_he = _insert_from_vertex(prev1, cv, ARR_LEFT_TO_RIGHT, v2);
  else {
    // Insert the halfedge given the two predecessor halfedges.
    // Note that in this case we may create a new face.
    bool new_face_created = false;
    bool check_swapped_predecessors = false;
    new_he = _insert_at_vertices(prev1, cv, ARR_LEFT_TO_RIGHT,
                                 fict_prev2->next(), new_face_created,
                                 check_swapped_predecessors);
    // Comment EBEB 2012-10-21: Swapping does not take place as the insertion
    // merges the CCB as an "interior" extension into an outer CCB of a face
    // incident the parameter space's boundary.
    CGAL_assertion(!check_swapped_predecessors);

    if (new_face_created) {
      CGAL_assertion(new_he->is_on_outer_ccb());
      // Note! new_he is not needed to define the new outer CCB of the new face
      // Here, it can be the outer ccb of the old face, as there is also not
      // swapping taking place!

      // In case a new face has been created (pointed by the new halfedge we
      // obtained), we have to examine the holes and isolated vertices in the
      // existing face (pointed by the twin halfedge) and move the relevant
      // holes and isolated vertices into the new face.
      _relocate_in_new_face(new_he);
    }
  }

  // Return a handle to the halfedge directed toward the new vertex v2.
  CGAL_postcondition(new_he->direction() == ARR_LEFT_TO_RIGHT);
  return (Halfedge_handle(new_he));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that its right
// endpoint corresponds to a given arrangement vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_from_right_vertex(const X_monotone_curve_2& cv,
                         Vertex_handle v, Face_handle f)
{
  CGAL_precondition_code
    (const bool at_obnd2 =
     !m_geom_traits->is_closed_2_object()(cv, ARR_MAX_END));
  CGAL_precondition_msg
    ((! at_obnd2 &&
      m_geom_traits->equal_2_object()
      (v->point(),
       m_geom_traits->construct_max_vertex_2_object()(cv))) ||
     (at_obnd2 && v->is_at_open_boundary()),
     "The input vertex should be the right curve end.");

  // Check if cv's left end has boundary conditions. If not, create a vertex
  // that corresponds to the left endpoint.
  const Arr_parameter_space  ps_x1 =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
  const Arr_parameter_space  ps_y1 =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);
  DVertex* v1 = nullptr;
  DHalfedge* fict_prev1 = nullptr;

  if ((ps_x1 == ARR_INTERIOR) && (ps_y1 == ARR_INTERIOR))
    // The curve has a valid left endpoint: Create a new vertex associated
    // with the curve's left endpoint.
    v1 = _create_vertex(m_geom_traits->construct_min_vertex_2_object()(cv));

  // Check if the given vertex, corresponding to the right curve end, has no
  // incident edges.
  if (v->degree() == 0) {
    // The given vertex is an isolated one: We should in fact insert the curve
    // in the interior of the face containing this vertex.
    DVertex* v2 = _vertex(v);
    DIso_vertex* iv = nullptr;
    DFace* p_f = nullptr;

    if (v->is_isolated()) {
      // Obtain the face from the isolated vertex.
      iv = v2->isolated_vertex();
      p_f = iv->face();
    }
    else {
      // In this case the face that contains v should be provided by the user.
      CGAL_precondition(f != Face_handle());
      p_f = _face(f);
    }

    // If the vertex that corresponds to cv's left end has boundary
    // conditions, create it now.
    if (v1 == nullptr)
      // Locate the DCEL features that will be used for inserting the curve's
      // left end.
      v1 = _place_and_set_curve_end(p_f, cv, ARR_MIN_END, ps_x1, ps_y1,
                                    &fict_prev1);

    if (iv != nullptr) {
      // Remove the isolated vertex v2, as it will not be isolated any more.
      p_f->erase_isolated_vertex(iv);
      _dcel().delete_isolated_vertex(iv);
    }

    // Create the edge connecting the two vertices (note that we know that
    // v1 is smaller than v2).
    DHalfedge* new_he = (fict_prev1 == nullptr) ?
      _insert_in_face_interior(p_f, cv, ARR_LEFT_TO_RIGHT, v1, v2) :
      _insert_from_vertex(fict_prev1, cv, ARR_LEFT_TO_RIGHT, v2);

    // Return a handle to the new halfedge whose target is the new vertex v1.
    CGAL_postcondition(new_he->opposite()->direction() == ARR_RIGHT_TO_LEFT);
    return (Halfedge_handle(new_he->opposite()));
  }

  // Go over the incident halfedges around v and find the halfedge after
  // which the new curve should be inserted.
  DHalfedge* prev2 = _locate_around_vertex(_vertex(v), cv, ARR_MAX_END);
  CGAL_assertion_msg
    (prev2 != nullptr, "The inserted curve cannot be located in the arrangement.");

  DFace* f2 = prev2->is_on_inner_ccb() ? prev2->inner_ccb()->face() :
    prev2->outer_ccb()->face();

  // If the vertex that corresponds to cv's left end has boundary conditions,
  // create it now.
  if (v1 == nullptr)
    // Locate the DCEL features that will be used for inserting the curve's
    // left end.
    v1 =
      _place_and_set_curve_end(f2, cv, ARR_MIN_END, ps_x1, ps_y1, &fict_prev1);

  // Perform the insertion (note that we know that prev2->vertex is larger
  // than v1).
  DHalfedge* new_he;

  if (fict_prev1 == nullptr)
    // Insert the halfedge given the predecessor halfedge of v2.
    new_he = _insert_from_vertex(prev2, cv, ARR_RIGHT_TO_LEFT, v1);
  else {
    // Insert the halfedge given the two predecessor halfedges.
    // Note that in this case we may create a new face.
    bool new_face_created = false;
    bool check_swapped_predecessors = false;
    new_he = _insert_at_vertices(prev2, cv, ARR_RIGHT_TO_LEFT,
                                 fict_prev1->next(), new_face_created,
                                 check_swapped_predecessors);
    // Comment EBEB 2012-10-21: Swapping does not take place as the insertion
    // merges the CCB as an "interior" extension into an outer CCB of a face
    // incident the parameter space's boundary.
    CGAL_assertion(!check_swapped_predecessors);

    if (new_face_created) {
      CGAL_assertion(new_he->is_on_outer_ccb());
      // Note! new_he is not needed to define the new outer CCB of the new face
      // Here, it can be the outer ccb of the old face, as there is also not
      // swapping taking place!

      // In case a new face has been created (pointed by the new halfedge we
      // obtained), we have to examine the holes and isolated vertices in the
      // existing face (pointed by the twin halfedge) and move the relevant
      // holes and isolated vertices into the new face.
      _relocate_in_new_face(new_he);
    }

  }

  // Return a handle to the halfedge directed toward the new vertex v1.
  CGAL_postcondition(new_he->direction() == ARR_RIGHT_TO_LEFT);
  return (Halfedge_handle(new_he));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that its right
// endpoint corresponds to a given arrangement vertex, given the exact place
// for the curve in the circular list around this vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_from_right_vertex(const X_monotone_curve_2& cv,
                         Halfedge_handle prev)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: insert_from_right_vertex (interface)" << std::endl;
  Rcpp::Rcout << "cv   : " << cv << std::endl;
  if (!prev->is_fictitious())
    Rcpp::Rcout << "prev : " << prev ->curve() << std::endl;
  else
    Rcpp::Rcout << "prev : fictitious" << std::endl;
  Rcpp::Rcout << "dir  : " << prev->direction() << std::endl;
#endif

  CGAL_precondition_code
    (const bool at_obnd2 =
     !m_geom_traits->is_closed_2_object()(cv, ARR_MAX_END));
  CGAL_precondition_msg
    ((! at_obnd2 &&
      m_geom_traits->equal_2_object()
      (prev->target()->point(),
       m_geom_traits->construct_max_vertex_2_object()(cv))) ||
     (at_obnd2 && prev->target()->is_at_open_boundary()),
     "The input vertex should be the right curve end.");

  CGAL_precondition_msg
    (at_obnd2 ||
     (_locate_around_vertex(_vertex(prev->target()), cv, ARR_MAX_END) ==
      _halfedge(prev)),
     "In the clockwise order of curves around the vertex, "
     " cv must succeed the curve of prev.");

  // Get the predecessor halfedge for the insertion of the right curve end.
  DHalfedge* prev2 = _halfedge(prev);
  DFace* f2 = _face(prev->face());

  // Check if cv's left end has boundary conditions, and obtain a vertex v1
  // that corresponds to this end.
  const Arr_parameter_space  ps_x1 =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
  const Arr_parameter_space  ps_y1 =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);
  DHalfedge* fict_prev1 = nullptr;

  DVertex* v1 = ((ps_x1 == ARR_INTERIOR) && (ps_y1 == ARR_INTERIOR)) ?
    // The curve has a valid left endpoint: Create a new vertex associated
    // with the curve's left endpoint.
    _create_vertex(m_geom_traits->construct_min_vertex_2_object()(cv)) :
    // Locate the DCEL features that will be used for inserting the curve's
    // left end.
    _place_and_set_curve_end(f2, cv, ARR_MIN_END, ps_x1, ps_y1, &fict_prev1);

  // Perform the insertion (note that we know that prev2->vertex is larger
  // than v1).
  DHalfedge* new_he;

  if (fict_prev1 == nullptr)
    // Insert the halfedge given the predecessor halfedge of the right vertex.
    new_he = _insert_from_vertex(prev2, cv, ARR_RIGHT_TO_LEFT, v1);
  else {
    // Insert the halfedge given the two predecessor halfedges.
    // Note that in this case we may create a new face.
    bool new_face_created = false;
    bool check_swapped_predecessors = false;
    new_he = _insert_at_vertices(prev2, cv, ARR_RIGHT_TO_LEFT,
                                 fict_prev1->next(), new_face_created,
                                 check_swapped_predecessors);
    // Comment EBEB 2012-10-21: Swapping does not take place as the insertion
    // merges the CCB as an "interior" extension into an outer CCB of a face
    // incident the parameter space's boundary.
    CGAL_assertion(!check_swapped_predecessors);

    if (new_face_created) {
      CGAL_assertion(new_he->is_on_outer_ccb());
      // Note! new_he is not needed to define the new outer CCB of the new face
      // Here, it can be the outer ccb of the old face, as there is also not
      // swapping taking place!

      // In case a new face has been created (pointed by the new halfedge we
      // obtained), we have to examine the holes and isolated vertices in the
      // existing face (pointed by the twin halfedge) and move the relevant
      // holes and isolated vertices into the new face.
      _relocate_in_new_face(new_he);
    }
  }

  // Return a handle to the halfedge directed toward the new vertex v1.
  CGAL_postcondition(new_he->direction() == ARR_RIGHT_TO_LEFT);
  return (Halfedge_handle(new_he));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that both its
// endpoints corresponds to a given arrangement vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_at_vertices(const X_monotone_curve_2& cv,
                   Vertex_handle v1, Vertex_handle v2,
                   Face_handle f)
{
  CGAL_USE(f);

  // Determine which one of the given vertices matches the left end of the
  // given curve.
  const bool at_obnd1 = !m_geom_traits->is_closed_2_object()(cv, ARR_MIN_END);
  const bool at_obnd2 = !m_geom_traits->is_closed_2_object()(cv, ARR_MAX_END);

  Arr_curve_end ind1;
  Arr_curve_end ind2;

  if (! at_obnd1) {
    CGAL_precondition_code(Vertex_handle v_right);

    if (! v1->is_at_open_boundary() &&
        m_geom_traits->equal_2_object()
        (v1->point(),
         m_geom_traits->construct_min_vertex_2_object()(cv)))
    {
      ind1 = ARR_MIN_END;
      ind2 = ARR_MAX_END;
      CGAL_precondition_code(v_right = v2);
    }
    else {
      CGAL_precondition_msg
        (! v2->is_at_open_boundary() &&
         m_geom_traits->equal_2_object()
         (v2->point(),
          m_geom_traits->construct_min_vertex_2_object()(cv)),
         "One of the input vertices should be the left curve end.");

      ind1 = ARR_MAX_END;
      ind2 = ARR_MIN_END;
      CGAL_precondition_code(v_right = v1);
    }

    CGAL_precondition_msg
      ((! at_obnd2 &&
        m_geom_traits->equal_2_object()
        (v_right->point(),
         m_geom_traits->construct_max_vertex_2_object()(cv))) ||
       (at_obnd2 && v_right->is_at_open_boundary()),
       "One of the input vertices should be the right curve end.");
  }
  else {
    if (! at_obnd2) {
      CGAL_precondition_code(Vertex_handle v_left);

      if (! v1->is_at_open_boundary() &&
          m_geom_traits->equal_2_object()
          (v1->point(),
           m_geom_traits->construct_max_vertex_2_object()(cv)))
      {
        ind1 = ARR_MAX_END;
        ind2 = ARR_MIN_END;
        CGAL_precondition_code(v_left = v2);
      }
      else {
        CGAL_precondition_msg
          (! v2->is_at_open_boundary() &&
           m_geom_traits->equal_2_object()
           (v2->point(),
            m_geom_traits->construct_max_vertex_2_object()(cv)),
           "One of the input vertices should be the right curve end.");

        ind1 = ARR_MIN_END;
        ind2 = ARR_MAX_END;
        CGAL_precondition_code(v_left = v1);
      }

      CGAL_precondition_msg
        (at_obnd1 && v_left->is_at_open_boundary(),
         "One of the input vertices should be the left curve end.");
    }
    else {
      Arr_parameter_space ps_x1 =
        m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
      Arr_parameter_space ps_y1 =
        m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);

      // Check which vertex should be associated with the minimal curve-end
      // (so the other is associated with the maximal curve-end).
      if (m_topol_traits.are_equal(_vertex(v1), cv, ARR_MIN_END, ps_x1, ps_y1))
      {
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(v2), cv, ARR_MAX_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END)));

        ind1 = ARR_MIN_END;
        ind2 = ARR_MAX_END;
      }
      else {
        CGAL_assertion(m_topol_traits.are_equal
                       (_vertex(v2), cv, ARR_MIN_END, ps_x1, ps_y1));
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(v1), cv, ARR_MAX_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END)));

        ind1 = ARR_MAX_END;
        ind2 = ARR_MIN_END;
      }
    }
  }

  // Check whether one of the vertices has no incident halfedges.
  if (v1->degree() == 0) {
    // Get the face containing the isolated vertex v1.
    DVertex* p_v1 = _vertex(v1);
    DIso_vertex* iv1 = nullptr;
    DFace* f1 = nullptr;

    if (p_v1->is_isolated()) {
      // Obtain the containing face from the isolated vertex record.
      iv1 = p_v1->isolated_vertex();
      f1 = iv1->face();

      // Remove the isolated vertex v1, as it will not be isolated any more.
      f1->erase_isolated_vertex(iv1);
      _dcel().delete_isolated_vertex(iv1);
    }

    // Check whether the other vertex also has no incident halfedges.
    if (v2->degree() == 0) {
      // Both end-vertices are isolated. Make sure they are contained inside
      // the same face.
      DVertex* p_v2 = _vertex(v2);
      DIso_vertex* iv2 = nullptr;
      DFace* f2 = nullptr;

      if (p_v2->is_isolated()) {
        // Obtain the containing face from the isolated vertex record.
        iv2 = p_v2->isolated_vertex();
        f2 = iv2->face();

        CGAL_assertion_msg
          ((f1 == nullptr) || (f1 == f2),
           "The two isolated vertices must be located inside the same face.");

        // Remove the isolated vertex v2, as it will not be isolated any more.
        f2->erase_isolated_vertex(iv2);
        _dcel().delete_isolated_vertex(iv2);
      }
      else if (f1 == nullptr)
        // In this case the containing face must be given by the user.
        CGAL_precondition(f != Face_handle());

      // Perform the insertion.
      Arr_halfedge_direction cv_dir =
        (ind1 == ARR_MIN_END) ? ARR_LEFT_TO_RIGHT : ARR_RIGHT_TO_LEFT;
      DHalfedge* new_he = _insert_in_face_interior(f1, cv, cv_dir, p_v1, p_v2);

      return (Halfedge_handle(new_he));
    }

    // Go over the incident halfedges around v2 and find the halfedge after
    // which the new curve should be inserted.
    DHalfedge* prev2 = _locate_around_vertex(_vertex(v2), cv, ind2);
    CGAL_assertion_msg
      (prev2 != nullptr,
       "The inserted curve cannot be located in the arrangement.");

    CGAL_assertion_code
      (DFace* f2 = prev2->is_on_inner_ccb() ? prev2->inner_ccb()->face() :
       prev2->outer_ccb()->face());

    CGAL_assertion_msg
      ((f1 == nullptr) || (f1 == f2),
       "The inserted curve should not intersect the existing arrangement.");

    // Perform the insertion. Note that the returned halfedge is directed
    // toward v1 (and not toward v2), so we return its twin.
    Arr_halfedge_direction cv_dir =
      (ind2 == ARR_MIN_END) ? ARR_LEFT_TO_RIGHT : ARR_RIGHT_TO_LEFT;
    DHalfedge* new_he = _insert_from_vertex(prev2, cv, cv_dir, p_v1);

    return (Halfedge_handle(new_he->opposite()));
  }
  else if (v2->degree() == 0) {
    // Get the face containing the isolated vertex v2.
    DVertex* p_v2 = _vertex(v2);
    DIso_vertex* iv2 = nullptr;
    DFace* f2 = nullptr;

    if (v2->is_isolated()) {
      // Obtain the containing face from the isolated vertex record.
      iv2 = p_v2->isolated_vertex();
      f2 = iv2->face();

      // Remove the isolated vertex v2, as it will not be isolated any more.
      f2->erase_isolated_vertex(iv2);
      _dcel().delete_isolated_vertex(iv2);
    }

    // Go over the incident halfedges around v1 and find the halfedge after
    // which the new curve should be inserted.
    DHalfedge* prev1 = _locate_around_vertex(_vertex(v1), cv, ind1);
    CGAL_assertion_msg
      (prev1 != nullptr,
       "The inserted curve cannot be located in the arrangement.");

    CGAL_assertion_code
      (DFace* f1 = prev1->is_on_inner_ccb() ? prev1->inner_ccb()->face() :
       prev1->outer_ccb()->face());

    CGAL_assertion_msg
      ((f2 == nullptr) || (f2 == f1),
       "The inserted curve should not intersect the existing arrangement.");

    // Perform the insertion.
    Arr_halfedge_direction cv_dir =
      (ind1 == ARR_MIN_END) ? ARR_LEFT_TO_RIGHT : ARR_RIGHT_TO_LEFT;
    DHalfedge* new_he = _insert_from_vertex(prev1, cv, cv_dir, p_v2);

    return (Halfedge_handle(new_he));
  }

  // Go over the incident halfedges around v1 and v2 and find the two
  // halfedges after which the new curve should be inserted, respectively.
  DHalfedge* prev1 = _locate_around_vertex(_vertex(v1), cv, ind1);
  DHalfedge* prev2 = _locate_around_vertex(_vertex(v2), cv, ind2);

  CGAL_assertion_msg
    (((prev1 != nullptr) && (prev2 != nullptr)),
     "The inserted curve cannot be located in the arrangement.");

  // Perform the insertion.
  return insert_at_vertices(cv, Halfedge_handle(prev1), Halfedge_handle(prev2));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that both its
// endpoints correspond to given arrangement vertices, given the exact
// place for the curve in one of the circular lists around a vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_at_vertices(const X_monotone_curve_2& cv,
                   Halfedge_handle prev1,
                   Vertex_handle v2)
{
  // Determine which one of the given vertices matches the left end of the
  // given curve.
  const bool at_obnd1 = !m_geom_traits->is_closed_2_object()(cv, ARR_MIN_END);
  const bool at_obnd2 = !m_geom_traits->is_closed_2_object()(cv, ARR_MAX_END);

  Arr_curve_end      ind2;

  if (! at_obnd1) {
    CGAL_precondition_code(Vertex_handle  v_right);

    if (! prev1->target()->is_at_open_boundary() &&
        m_geom_traits->equal_2_object()
        (prev1->target()->point(),
         m_geom_traits->construct_min_vertex_2_object()(cv)))
    {
      ind2 = ARR_MAX_END;
      CGAL_precondition_code(v_right = v2);
    }
    else {
      CGAL_precondition_msg
        (! v2->is_at_open_boundary() &&
         m_geom_traits->equal_2_object()
         (v2->point(),
          m_geom_traits->construct_min_vertex_2_object()(cv)),
         "One of the input vertices should be the left curve end.");

      ind2 = ARR_MIN_END;
      CGAL_precondition_code(v_right = prev1->target());
    }

    CGAL_precondition_msg
      ((! at_obnd2 &&
        m_geom_traits->equal_2_object()
        (v_right->point(),
         m_geom_traits->construct_max_vertex_2_object()(cv))) ||
       (at_obnd2 && v_right->is_at_open_boundary()),
       "One of the input vertices should be the right curve end.");
  }
  else {
    if (! at_obnd2) {
      CGAL_precondition_code(Vertex_handle v_left);

      if (! prev1->target()->is_at_open_boundary() &&
          m_geom_traits->equal_2_object()
          (prev1->target()->point(),
           m_geom_traits->construct_max_vertex_2_object()(cv)))
      {
        ind2 = ARR_MIN_END;
        CGAL_precondition_code(v_left = v2);
      }
      else {
        CGAL_precondition_msg
          (! v2->is_at_open_boundary() &&
           m_geom_traits->equal_2_object()
           (v2->point(),
            m_geom_traits->construct_max_vertex_2_object()(cv)),
           "One of the input vertices should be the right curve end.");

        ind2 = ARR_MAX_END;
        CGAL_precondition_code(v_left = prev1->target());
      }

      CGAL_precondition_msg
        (at_obnd1 && v_left->is_at_open_boundary(),
         "One of the input vertices should be the left curve end.");
    }
    else {
      Arr_parameter_space  ps_x1 =
        m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
      Arr_parameter_space  ps_y1 =
        m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);

      // Check which vertex should be associated with the minimal curve-end
      // (so the other is associated with the maximal curve-end).
      if (m_topol_traits.are_equal(_vertex(prev1->target()),
                                   cv, ARR_MIN_END, ps_x1, ps_y1))
      {
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(v2), cv, ARR_MAX_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END)));

        ind2 = ARR_MAX_END;
      }
      else {
        CGAL_assertion(m_topol_traits.are_equal
                       (_vertex(v2), cv, ARR_MIN_END, ps_x1, ps_y1));
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(prev1->target()), cv, ARR_MAX_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END)));

        ind2 = ARR_MIN_END;
      }
    }
  }

  // Check whether v2 is has no incident edges.
  if (v2->degree() == 0) {
    // Get the face containing the isolated vertex v2.
    DVertex* p_v2 = _vertex(v2);
    DIso_vertex* iv2 = nullptr;
    DFace* f2 = nullptr;

    if (v2->is_isolated()) {
      iv2 = p_v2->isolated_vertex();
      f2 = iv2->face();

      CGAL_assertion_msg
        (f2 == _face(prev1->face()),
         "The inserted curve should not intersect the existing arrangement.");

      // Remove the isolated vertex v2, as it will not be isolated any more.
      f2->erase_isolated_vertex(iv2);
      _dcel().delete_isolated_vertex(iv2);
    }

    // Perform the insertion.
    Arr_halfedge_direction cv_dir =
      (ind2 == ARR_MAX_END) ? ARR_LEFT_TO_RIGHT : ARR_RIGHT_TO_LEFT;
    DHalfedge* new_he = _insert_from_vertex(_halfedge(prev1), cv, cv_dir, p_v2);

    return (Halfedge_handle(new_he));
  }

  // Go over the incident halfedges around v2 and find the halfedge after
  // which the new curve should be inserted.
  DHalfedge* prev2 = _locate_around_vertex(_vertex(v2), cv, ind2);
  CGAL_assertion_msg
    (prev2 != nullptr, "The inserted curve cannot be located in the arrangement.");

  // Perform the insertion.
  return (insert_at_vertices(cv, prev1, Halfedge_handle(prev2)));
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that both its
// endpoints correspond to given arrangement vertices, given the exact
// place for the curve in both circular lists around these two vertices.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
insert_at_vertices(const X_monotone_curve_2& cv,
                   Halfedge_handle prev1, Halfedge_handle prev2)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: insert_at_vertices (interface)" << std::endl;
  Rcpp::Rcout << "cv   : " << cv << std::endl;
  if (!prev1->is_fictitious())
    Rcpp::Rcout << "prev1: " << prev1->curve() << std::endl;
  else
    Rcpp::Rcout << "prev1: fictitious" << std::endl;
  Rcpp::Rcout << "dir1 : " << prev1->direction() << std::endl;
  if (!prev2->is_fictitious())
    Rcpp::Rcout << "prev2: " << prev2->curve() << std::endl;
  else
    Rcpp::Rcout << "prev2: fictitious" << std::endl;
  Rcpp::Rcout << "dir2 : " << prev2->direction() << std::endl;
#endif

  // Determine which one of the given vertices (the target vertices of the
  // given halfedges) matches the left end of the given curve.
  // Thus, we can determine the comparison result between prev1->target()
  // and prev2->target().
  const bool at_obnd1 = !m_geom_traits->is_closed_2_object()(cv, ARR_MIN_END);
  const bool at_obnd2 = !m_geom_traits->is_closed_2_object()(cv, ARR_MAX_END);
  Comparison_result  res;

  if (! at_obnd1) {
    CGAL_precondition_code(Vertex_handle v_right);

    if (! prev1->target()->is_at_open_boundary() &&
        m_geom_traits->equal_2_object()
        (prev1->target()->point(),
         m_geom_traits->construct_min_vertex_2_object()(cv)))
    {
      res = SMALLER;
      CGAL_precondition_code(v_right = prev2->target());
    }
    else {
      CGAL_precondition_msg
        (! prev2->target()->is_at_open_boundary() &&
         m_geom_traits->equal_2_object()
         (prev2->target()->point(),
          m_geom_traits->construct_min_vertex_2_object()(cv)),
         "One of the input vertices should be the left curve end.");

      res = LARGER;
      CGAL_precondition_code(v_right = prev1->target());
    }

    CGAL_precondition_msg
      ((! at_obnd2 &&
        m_geom_traits->equal_2_object()
        (v_right->point(),
         m_geom_traits->construct_max_vertex_2_object()(cv))) ||
       (at_obnd2 && v_right->is_at_open_boundary()),
       "One of the input vertices should be the right curve end.");
  }
  else {
    if (! at_obnd2) {
      CGAL_precondition_code(Vertex_handle  v_left);

      if (! prev1->target()->is_at_open_boundary() &&
          m_geom_traits->equal_2_object()
          (prev1->target()->point(),
           m_geom_traits->construct_max_vertex_2_object()(cv)))
      {
        res = LARGER;
        CGAL_precondition_code(v_left = prev2->target());
      }
      else {
        CGAL_precondition_msg
          (! prev2->target()->is_at_open_boundary() &&
           m_geom_traits->equal_2_object()
           (prev2->target()->point(),
            m_geom_traits->construct_max_vertex_2_object()(cv)),
           "One of the input vertices should be the right curve end.");

        res = SMALLER;
        CGAL_precondition_code(v_left = prev1->target());
      }

      CGAL_precondition_msg
        (at_obnd1 && v_left->is_at_open_boundary(),
         "One of the input vertices should be the left curve end.");
    }
    else {
      Arr_parameter_space ps_x1 =
        m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
      Arr_parameter_space ps_y1 =
        m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);
      // Check which vertex should be associated with the minimal curve-end
      // (so the other is associated with the maximal curve-end), and
      // determine the comparison result of the two vertices accordingly.
      if (m_topol_traits.are_equal(_vertex(prev1->target()),
                                   cv, ARR_MIN_END, ps_x1, ps_y1))
      {
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(prev2->target()), cv, ARR_MAX_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END)));

        res = SMALLER;
      }
      else {
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(prev2->target()), cv, ARR_MIN_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END)));
        CGAL_assertion
          (m_topol_traits.are_equal
           (_vertex(prev1->target()), cv, ARR_MAX_END,
            m_geom_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END),
            m_geom_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END)));

        res = LARGER;
      }
    }
  }

  // Check if e1 and e2 are on the same connected component.
  DHalfedge* p_prev1 = _halfedge(prev1);
  DHalfedge* p_prev2 = _halfedge(prev2);

  // Note that in this case we may create a new face.
  bool new_face_created = false;
  bool swapped_predecessors = false;
  DHalfedge* new_he =
    _insert_at_vertices(p_prev1, cv,
                        (res == SMALLER ? ARR_LEFT_TO_RIGHT : ARR_RIGHT_TO_LEFT),
                        p_prev2->next(), new_face_created,
                        swapped_predecessors);

  if (new_face_created)
    // Comment EBEB 2012-10-21: Here we allow swapping, as there might be
    // a local minima (or other needs), and thus new_he can lie on an inner CCB.
    // In fact we cannot expect new_he to lie on an inner or on an outer CCB.

    // In case a new face has been created (pointed by the new halfedge we
    // obtained), we have to examine the holes and isolated vertices in the
    // existing face (pointed by the twin halfedge) and move the relevant
    // holes and isolated vertices into the new face.
    _relocate_in_new_face(new_he);

  // Return a handle to the new halfedge directed from prev1's target to
  // prev2's target. Note that this may be the twin halfedge of the one
  // returned by _insert_at_vertices();
  if (swapped_predecessors) new_he = new_he->opposite();

  return (Halfedge_handle(new_he));
}

//-----------------------------------------------------------------------------
// Replace the point associated with the given vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Vertex_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
modify_vertex(Vertex_handle vh, const Point_2& p)
{
  CGAL_precondition_msg
    (! vh->is_at_open_boundary(),
     "The modified vertex must not lie on open boundary.");
  CGAL_precondition_msg(m_geom_traits->equal_2_object()(vh->point(), p),
                        "The new point is different from the current one.");

  // Modify the vertex.
  _modify_vertex(_vertex(vh), p);

  // Return a handle to the modified vertex.
  return vh;
}

//-----------------------------------------------------------------------------
// Remove an isolated vertex from the interior of a given face.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Face_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
remove_isolated_vertex(Vertex_handle v)
{
  CGAL_precondition(v->is_isolated());

  // Get the face containing v.
  DVertex* p_v = _vertex(v);
  DIso_vertex* iv = p_v->isolated_vertex();
  DFace* p_f = iv->face();
  Face_handle f = Face_handle(p_f);

  // Notify the observers that we are about to remove a vertex.
  _notify_before_remove_vertex(v);

  // Remove the isolated vertex from the face that contains it.
  p_f->erase_isolated_vertex(iv);
  _dcel().delete_isolated_vertex(iv);

  // Delete the vertex.
  _delete_point(p_v->point());
  _dcel().delete_vertex(p_v);

  // Notify the observers that the vertex has been removed.
  _notify_after_remove_vertex();

  // Return a handle for the face that used to contain the deleted vertex.
  return f;
}

//-----------------------------------------------------------------------------
// Replace the x-monotone curve associated with the given edge.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
modify_edge(Halfedge_handle e, const X_monotone_curve_2& cv)
{
  CGAL_precondition_msg(! e->is_fictitious(),
                        "The edge must be a valid one.");
  CGAL_precondition_msg(m_geom_traits->equal_2_object()(e->curve(), cv),
                        "The new curve is different from the current one.");

  // Modify the edge.
  _modify_edge(_halfedge(e), cv);

  // Return a handle to the modified halfedge.
  return e;
}

//-----------------------------------------------------------------------------
// Split a given edge into two, and associate the given x-monotone
// curves with the split edges.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
split_edge(Halfedge_handle e,
           const X_monotone_curve_2& cv1, const X_monotone_curve_2& cv2)
{
  CGAL_precondition_msg(! e->is_fictitious(), "The edge must be a valid one.");

  // Find the point where we split the halfedge, and determine which curve
  // should be associated with which pair of split halfedges.
  DHalfedge* he1 = _halfedge(e);
  DHalfedge* he2 = he1->opposite();
  DVertex* source = he2->vertex();

  /* The halfedge we return and e must have a common source vertex.
   * There are 4 cases:
   */

  // 1. o---cv1---o---cv2---o
  //    o---------e-------->o
  if (_are_equal(source, cv1, ARR_MIN_END)) {
    const Point_2& p = m_geom_traits->construct_max_vertex_2_object()(cv1);
    CGAL_postcondition_code
      (const Point_2& q = m_geom_traits->construct_min_vertex_2_object()(cv2));
    CGAL_precondition(m_geom_traits->equal_2_object()(p, q));
    CGAL_precondition(_are_equal(he1->vertex(), cv2, ARR_MAX_END));
    return (Halfedge_handle(_split_edge(he1, p, cv1, cv2)));
  }

  // 2. o---cv2---o---cv1---o
  //    o<--------e---------o
  if (_are_equal(source, cv1, ARR_MAX_END)) {
    const Point_2& p = m_geom_traits->construct_min_vertex_2_object()(cv1);
    CGAL_postcondition_code
      (const Point_2& q = m_geom_traits->construct_max_vertex_2_object()(cv2));
    CGAL_precondition(m_geom_traits->equal_2_object()(p, q));
    CGAL_precondition(_are_equal(he1->vertex(), cv2, ARR_MIN_END));
    return (Halfedge_handle(_split_edge(he1, p, cv1, cv2)));
  }

  // 3. o---cv2---o---cv1---o
  //    o---------e-------->o
  if (_are_equal(source, cv2, ARR_MIN_END)) {
    const Point_2& p = m_geom_traits->construct_max_vertex_2_object()(cv2);
    CGAL_postcondition_code
      (const Point_2& q = m_geom_traits->construct_min_vertex_2_object()(cv1));
    CGAL_precondition(m_geom_traits->equal_2_object()(p, q));
    CGAL_precondition(_are_equal(he1->vertex(), cv1, ARR_MAX_END));
    return (Halfedge_handle(_split_edge(he1, p, cv2, cv1)));
  }

  // 4. o---cv1---o---cv2---o
  //    o<--------e---------o
  CGAL_precondition(_are_equal(source, cv2, ARR_MAX_END));
  const Point_2& p = m_geom_traits->construct_min_vertex_2_object()(cv2);
  CGAL_postcondition_code
    (const Point_2& q = m_geom_traits->construct_max_vertex_2_object()(cv1));
  CGAL_precondition(m_geom_traits->equal_2_object()(p, q));
  CGAL_precondition(_are_equal(he1->vertex(), cv1, ARR_MIN_END));
  return (Halfedge_handle(_split_edge(he1, p, cv2, cv1)));
}

//-----------------------------------------------------------------------------
// Merge two edges to form a single edge, and associate the given x-monotone
// curve with the merged edge.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Halfedge_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
merge_edge(Halfedge_handle e1, Halfedge_handle e2,
           const X_monotone_curve_2& cv)
{
  CGAL_precondition_msg(! e1->is_fictitious() && ! e2->is_fictitious(),
                        "The edges must be a valid.");

  // Assign pointers to the existing halfedges, such that we have:
  //
  //            he1      he3
  //         -------> ------->
  //       (.)      (.)v     (.)
  //         <------- <-------
  //            he2      he4
  //
  DHalfedge* _e1 = _halfedge(e1);
  DHalfedge* _e2 = _halfedge(e2);
  DHalfedge* he1 = nullptr;
  DHalfedge* he2 = nullptr;
  DHalfedge* he3 = nullptr;
  DHalfedge* he4 = nullptr;

  if (_e1->vertex() == _e2->opposite()->vertex()) {
    he1 = _e1;
    he2 = he1->opposite();
    he3 = _e2;
    he4 = he3->opposite();
  }
  else if (_e1->opposite()->vertex() == _e2->opposite()->vertex()) {
    he2 = _e1;
    he1 = he2->opposite();
    he3 = _e2;
    he4 = he3->opposite();
  }
  else if (_e1->vertex() == _e2->vertex()) {
    he1 = _e1;
    he2 = he1->opposite();
    he4 = _e2;
    he3 = he4->opposite();
  }
  else if (_e1->opposite()->vertex() == _e2->vertex()) {
    he2 = _e1;
    he1 = he2->opposite();
    he4 = _e2;
    he3 = he4->opposite();
  }
  else {
    CGAL_precondition_msg(false,
                          "The input edges do not share a common vertex.");
    return Halfedge_handle();
  }

  // The vertex we are about to delete is now he1's target vertex.
  // Make sure that he1 and he4 are the only halfedges directed to v.
  DVertex* v = he1->vertex();

  CGAL_precondition_msg
    (! v->has_null_point(),
     "The vertex removed by the merge must not lie on open boundary.");
  CGAL_precondition_msg
    (he1->next()->opposite() == he4 &&
     he4->next()->opposite() == he1,
     "The degree of the deleted vertex is greater than 2.");

  // Make sure the curve ends match the end vertices of the merged edge.
  CGAL_precondition_msg
    ((_are_equal(he2->vertex(), cv, ARR_MIN_END) &&
      _are_equal(he3->vertex(), cv, ARR_MAX_END)) ||
     (_are_equal(he3->vertex(), cv, ARR_MIN_END) &&
      _are_equal(he2->vertex(), cv, ARR_MAX_END)),
     "The endpoints of the merged curve must match the end vertices.");

  // Keep pointers to the components that contain two halfedges he3 and he2,
  // pointing at the end vertices of the merged halfedge.
  DInner_ccb* ic1 = (he3->is_on_inner_ccb()) ? he3->inner_ccb() : nullptr;
  DOuter_ccb* oc1 = (ic1 == nullptr) ? he3->outer_ccb() : nullptr;

  DInner_ccb* ic2 = (he4->is_on_inner_ccb()) ? he4->inner_ccb() : nullptr;
  DOuter_ccb* oc2 = (ic2 == nullptr) ? he4->outer_ccb() : nullptr;

  // Notify the observers that we are about to merge an edge.
  _notify_before_merge_edge(e1, e2, cv);

  // As he1 and he2 will evetually represent the merged edge, while he3 and he4
  // will be deleted, check if the deleted halfedges are represantatives of a
  // the CCBs they belong to. If so, replace he3 by he1 and he4 by he2. Note
  // that as we just change the component representatives, we do not have to
  // notify the observers on the change.
  if (oc1 != nullptr && oc1->halfedge() == he3)
    oc1->set_halfedge(he1);
  else if (ic1 != nullptr && ic1->halfedge() == he3)
    ic1->set_halfedge(he1);

  if (oc2 != nullptr && oc2->halfedge() == he4)
    oc2->set_halfedge(he2);
  else if (ic2 != nullptr && ic2->halfedge() == he4)
    ic2->set_halfedge(he2);

  // If he3 is the incident halfedge to its target, replace it by he1.
  if (he3->vertex()->halfedge() == he3)
    he3->vertex()->set_halfedge(he1);

  // Disconnect he3 and he4 from the edge list.
  if (he3->next() == he4) {
    // he3 and he4 form an "antenna", so he1 and he2 must be connected
    // together.
    he1->set_next(he2);
  }
  else {
    he1->set_next(he3->next());
    he4->prev()->set_next(he2);
  }

  // Note that he1 (and its twin) is going to represent the merged edge while
  // he3 (and its twin) is going to be removed. We therefore associate the
  // merged curve with he1 and delete the curve associated with he3.
  he1->curve() = cv;
  _delete_curve(he3->curve());

  // Set the properties of the merged edge.
  he1->set_vertex(he3->vertex());

  // Notify the observers that we are about to delete a vertex.
  _notify_before_remove_vertex(Vertex_handle(v));

  // Delete the point associated with the merged vertex.
  _delete_point(v->point());

  // Delete the merged vertex.
  _dcel().delete_vertex(v);

  // Notify the observers that the vertex has been deleted.
  _notify_after_remove_vertex();

  // Delete the redundant halfedge pair.
  _dcel().delete_edge(he3);

  // Create a handle for one of the merged halfedges.
  Halfedge_handle hh(he1);

  // Notify the observers that the edge has been merge.
  _notify_after_merge_edge(hh);

  // Return a handle for one of the merged halfedges.
  return hh;
}

//-----------------------------------------------------------------------------
// Remove an edge from the arrangement.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::Face_handle
Arrangement_on_surface_2<GeomTraits, TopTraits>::
remove_edge(Halfedge_handle e, bool remove_source, bool remove_target)
{
  // Comment EBEB 2012-08-06: this has become a simple forwarding function
  // the intelligence of whether to swap he with he->opposite()
  // has been moved to _remove_edge itself, as additional computed
  // data is reused there

  CGAL_precondition_msg(! e->is_fictitious(),
                        "The edge must be a valid one.");

  DHalfedge* he1 = _halfedge(e);
  DFace* f = _remove_edge(he1, remove_source, remove_target);
  return Face_handle(f);
}

//-----------------------------------------------------------------------------
// Protected member functions (for internal use).
//-----------------------------------------------------------------------------

//-----------------------------------------------------------------------------
// Locate the place for the given curve around the given vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DHalfedge*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_locate_around_vertex(DVertex* v,
                      const X_monotone_curve_2& cv, Arr_curve_end ind) const
{
  // Check if the given curve-end has boundary conditions.
  const Arr_parameter_space ps_x =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ind);
  const Arr_parameter_space ps_y =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ind);

  if ((ps_x != ARR_INTERIOR) || (ps_y != ARR_INTERIOR))
    // Use the topology-traits class to locate the predecessor halfedge for
    // cv around the given vertex.
    return m_topol_traits.locate_around_boundary_vertex(v, cv, ind, ps_x, ps_y);

  // In case of a non-boundary vertex, we look for the predecessor around v.
  // Get the first incident halfedge around v and the next halfedge.
  DHalfedge* first = v->halfedge();
  DHalfedge* curr = first;

  if (curr == nullptr) return nullptr;

  DHalfedge* next = curr->next()->opposite();

  // In case there is only one halfedge incident to v, return this halfedge.
  if (curr == next) return curr;

  // Otherwise, we traverse the halfedges around v until we find the pair
  // of adjacent halfedges between which we should insert cv.
  typename Traits_adaptor_2::Is_between_cw_2 is_between_cw =
    m_geom_traits->is_between_cw_2_object();

  bool eq_curr, eq_next;
  while (! is_between_cw(cv, (ind == ARR_MIN_END),
                         curr->curve(),
                         (curr->direction() == ARR_RIGHT_TO_LEFT),
                         next->curve(),
                         (next->direction() == ARR_RIGHT_TO_LEFT),
                         v->point(), eq_curr, eq_next))
  {
    // If cv equals one of the curves associated with the halfedges, it is
    // an illegal input curve, as it already exists in the arrangement.
    if (eq_curr || eq_next) return nullptr;

    // Move to the next pair of incident halfedges.
    curr = next;
    next = curr->next()->opposite();

    // If we completed a full traversal around v without locating the
    // place for cv, it follows that cv overlaps and existing curve.
    if (curr == first) return nullptr;
  }

  // Return the halfedge we have located.
  return curr;
}

//-----------------------------------------------------------------------------
// Compute the distance (in halfedges) between two halfedges.
//
template <typename GeomTraits, typename TopTraits>
unsigned int
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_halfedge_distance(const DHalfedge* e1, const DHalfedge* e2) const
{
  CGAL_precondition(e1 != e2);
  if (e1 == e2) return (0);

  // Traverse the halfedge chain from e1 until reaching e2.
  const DHalfedge* curr = e1->next();
  unsigned int dist = 1;

  while (curr != e2) {
    // If we have returned to e1, e2 is not reachable from e1.
    if (curr == e1) {
      CGAL_error();
      return (0);
    }

    curr = curr->next();
    ++dist;
  }

  // We have located e2 along the boundary of e1's component - return the
  // distance (number of halfedges) between e1 and e2.
  return (dist);
}

//-----------------------------------------------------------------------------
//Compare the length of the induced paths from e1 to e2 and from e2 to e1.
// return SMALLER if e1 to e2 is shorter, EQUAL if paths lengths are equal,
//  o/w LARGER
//
template <typename GeomTraits, typename TopTraits>
Comparison_result
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compare_induced_path_length(const DHalfedge* e1, const DHalfedge* e2) const
{
  CGAL_precondition(e1 != e2);
  if (e1 == e2) return EQUAL;

  // Traverse the halfedge chain from e1 until reaching e2.
  const DHalfedge* curr1 = e1->next();
  // Traverse the halfedge chain from e2 until reaching e1.
  const DHalfedge* curr2 = e2->next();

  while ((curr1 != e2) && (curr2 != e1)) {
    // If we have returned to e1, e2 is not reachable from e1.
    if (curr1 == e1) {
      CGAL_error();
      return EQUAL;
    }

    // If we have returned to e2, e1 is not reachable from e2.
    if (curr2 == e2) {
      CGAL_error();
      return EQUAL;
    }

    curr1 = curr1->next();
    curr2 = curr2->next();
  }

  // Return SMALLER if e1 to e2 is shorter than e2 to e1,
  //  EQUAL if their lengths are equal, or LARGER if e2 to e1 is longer.
  Comparison_result res =
    (curr1 == e2) ? ((curr2 != e1) ? SMALLER : EQUAL) : LARGER;

  CGAL_postcondition_code(int dist1 = _halfedge_distance(e1,e2));
  CGAL_postcondition_code(int dist2 = _halfedge_distance(e2,e1));
  CGAL_postcondition(((dist1 < dist2) && (res == SMALLER)) ||
                     ((dist1 == dist2) && (res == EQUAL)) ||
                     ((dist1 > dist2) && (res == LARGER)));

  return res;
}

//-----------------------------------------------------------------------------
// Move a given outer CCB from one face to another.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_move_outer_ccb(DFace* from_face, DFace* to_face, DHalfedge* he)
{
  // Get the DCEL record that represents the outer CCB.
  DOuter_ccb* oc = he->outer_ccb();

  CGAL_assertion(oc->face() == from_face);

  // Notify the observers that we are about to move an outer CCB.
  Ccb_halfedge_circulator circ = (Halfedge_handle(he))->ccb();

  _notify_before_move_outer_ccb(Face_handle(from_face), Face_handle(to_face),
                                circ);

  // Remove the hole from the current face.
  from_face->erase_outer_ccb(oc);

  // Modify the component that represents the hole.
  oc->set_face(to_face);
  to_face->add_outer_ccb(oc, he);

  // Notify the observers that we have moved the outer CCB.
  _notify_after_move_outer_ccb(circ);
}

//-----------------------------------------------------------------------------
// Move a given inner CCB (hole) from one face to another.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_move_inner_ccb(DFace* from_face, DFace* to_face, DHalfedge* he)
{
  // Get the DCEL record that represents the inner CCB.
  DInner_ccb* ic = he->inner_ccb();

  CGAL_assertion(ic->face() == from_face);

  // Notify the observers that we are about to move an inner CCB.
  Ccb_halfedge_circulator   circ = (Halfedge_handle(he))->ccb();

  _notify_before_move_inner_ccb(Face_handle(from_face), Face_handle(to_face),
                                circ);

  // Remove the hole from the current face.
  from_face->erase_inner_ccb(ic);

  // Modify the component that represents the hole.
  ic->set_face(to_face);
  to_face->add_inner_ccb(ic, he);

  // Notify the observers that we have moved the inner CCB.
  _notify_after_move_inner_ccb(circ);
}

//-----------------------------------------------------------------------------
// Move all inner CCBs (holes) from one face to another.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_move_all_inner_ccb(DFace* from_face, DFace* to_face)
{
  // Comment EFEF 2015-09-28: The following loop and the loop at the end of this
  // function should be replaced with a pair of notifiers, respectively,
  // function_notify_before_move_all_inner_ccb();
  // function_notify_after_move_all_inner_ccb();
  DInner_ccb_iter ic_it = from_face->inner_ccbs_begin();
  while (ic_it != from_face->inner_ccbs_end()) {
    DHalfedge* he = *ic_it++;
    Ccb_halfedge_circulator circ = (Halfedge_handle(he))->ccb();
    _notify_before_move_inner_ccb(Face_handle(from_face), Face_handle(to_face),
                                  circ);
  }
  ic_it = to_face->splice_inner_ccbs(*from_face);
  while (ic_it != to_face->inner_ccbs_end()) {
    DHalfedge* he = *ic_it++;
    Ccb_halfedge_circulator circ = (Halfedge_handle(he))->ccb();
    _notify_after_move_inner_ccb(circ);
  }
}

//-----------------------------------------------------------------------------
// Insert the given vertex as an isolated vertex inside the given face.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_insert_isolated_vertex(DFace* f, DVertex* v)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: _insert_isolated_vertex (internal)" << std::endl;
  if (!v->has_null_point())
    Rcpp::Rcout << "v->point: " << v->point() << std::endl;
  Rcpp::Rcout << "face   : " << f << std::endl;
#endif

  Face_handle fh(f);
  Vertex_handle vh(v);

  // Notify the observers that we are about to insert an isolated vertex
  // inside f.
  _notify_before_add_isolated_vertex(fh, vh);

  // Create an isolated vertex-information object,
  DIso_vertex* iv = _dcel().new_isolated_vertex();

  // Set a pointer to the face containing the vertex.
  iv->set_face(f);

  // Initiate a new hole inside the given face.
  f->add_isolated_vertex(iv, v);

  // Associate the information with the vertex.
  v->set_isolated_vertex(iv);

  // Notify the observers that we have formed a new isolated vertex.
  _notify_after_add_isolated_vertex(vh);
}

//-----------------------------------------------------------------------------
// Move a given isolated vertex from one face to another.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_move_isolated_vertex(DFace* from_face, DFace* to_face, DVertex* v)
{
  // Get the DCEL isolated-vertex record.
  DIso_vertex* iv = v->isolated_vertex();

  // Notify the observers that we are about to move an isolated vertex.
  Vertex_handle vh(v);

  _notify_before_move_isolated_vertex(Face_handle(from_face),
                                      Face_handle(to_face), vh);

  // Set the new face is the isolated vertex-information object.
  iv->set_face(to_face);

  // Move the isolated vertex from the first face to the other.
  from_face->erase_isolated_vertex(iv);
  to_face->add_isolated_vertex(iv, v);

  // Notify the observers that we have moved the isolated vertex.
  _notify_after_move_isolated_vertex(vh);
}

//-----------------------------------------------------------------------------
// Move all isolated vertices from one face to another.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_move_all_isolated_vertices(DFace* from_face, DFace* to_face)
{
  // Comment EFEF 2015-09-28: The following loop and the loop at the end of this
  // function should be replaced with a pair of notifiers, respectively,
  // function_notify_before_move_all_isolated_vertices();
  // function_notify_after_move_all_isolated_vertices();
  DIso_vertex_iter iv_it = from_face->isolated_vertices_begin();
  while (iv_it != from_face->isolated_vertices_end()) {
    DVertex* v = &(*iv_it++);
    Vertex_handle vh(v);
    _notify_before_move_isolated_vertex(Face_handle(from_face),
                                        Face_handle(to_face),
                                        vh);
  }
  iv_it = to_face->splice_isolated_vertices(*from_face);
  while (iv_it != to_face->isolated_vertices_end()) {
    DVertex* v = &(*iv_it++);
    Vertex_handle vh(v);
    _notify_after_move_isolated_vertex(vh);
  }
}

//-----------------------------------------------------------------------------
// Create a new vertex and associate it with the given point.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DVertex*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_create_vertex(const Point_2& p)
{
  // Notify the observers that we are about to create a new vertex.
  Point_2* p_p = _new_point(p);

  _notify_before_create_vertex(*p_p);

  // Create a new vertex and associate it with the given point.
  DVertex* v = _dcel().new_vertex();

  v->set_point(p_p);
  v->set_boundary(ARR_INTERIOR, ARR_INTERIOR);

  // Notify the observers that we have just created a new vertex.
  Vertex_handle vh(v);
  _notify_after_create_vertex(vh);

  return v;
}

// Create a new vertex on boundary
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DVertex*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_create_boundary_vertex(const Point_2& p,
                        Arr_parameter_space ps_x, Arr_parameter_space ps_y)
{
  CGAL_precondition((ps_x != ARR_INTERIOR) || (ps_y != ARR_INTERIOR));

  // Notify the observers that we are about to create a new boundary vertex.
  _notify_before_create_boundary_vertex(p, ps_x, ps_y);

  // Create a new vertex and set its boundary conditions.
  DVertex* v = _dcel().new_vertex();
  v->set_boundary(ps_x, ps_y);
  v->set_point(_new_point(p));

  // Notify the observers that we have just created a new boundary vertex.
  _notify_after_create_boundary_vertex(Vertex_handle(v));

  return v;
}

//-----------------------------------------------------------------------------
// Create a new vertex on boundary
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DVertex*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_create_boundary_vertex(const X_monotone_curve_2& cv, Arr_curve_end ind,
                        Arr_parameter_space ps_x, Arr_parameter_space ps_y)
{
  CGAL_precondition((ps_x != ARR_INTERIOR) || (ps_y != ARR_INTERIOR));

  // Notify the observers that we are about to create a new boundary vertex.
  _notify_before_create_boundary_vertex(cv, ind, ps_x, ps_y);

  // Create a new vertex and set its boundary conditions.
  DVertex* v = _dcel().new_vertex();

  v->set_boundary(ps_x, ps_y);

  // Act according to the boundary type if there is one:
  if (is_open(ps_x, ps_y))
    // The curve-end lies on open boundary so the vertex is not associated
    // with a valid point.
    v->set_point(nullptr);
  else {
    // Create a boundary vertex associated with a valid point.
    Point_2* p_p = (ind == ARR_MIN_END) ?
      _new_point(m_geom_traits->construct_min_vertex_2_object()(cv)) :
      _new_point(m_geom_traits->construct_max_vertex_2_object()(cv));

    v->set_point(p_p);
  }

  // Notify the observers that we have just created a new boundary vertex.
  _notify_after_create_boundary_vertex(Vertex_handle(v));

  return v;
}

//-----------------------------------------------------------------------------
// Locate the DCEL features that will be used for inserting the given point,
// which has a boundary condition, and set a proper vertex there.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DVertex*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_place_and_set_point(DFace* f, const Point_2& p,
                     Arr_parameter_space ps_x, Arr_parameter_space ps_y)
{
  // Use the topology traits to locate the DCEL feature that contains the
  // given point.
  CGAL::Object obj = m_topol_traits.place_boundary_vertex(f, p, ps_x, ps_y);
  DVertex* v;

  // Act according to the result type.
  DHalfedge* fict_he;
  if (CGAL::assign(fict_he, obj)) {
    // The point is located on a fictitious edge.
    // Create a new vertex that corresponds to the point.
    v = _create_boundary_vertex(p, ps_x, ps_y);

    // Split the fictitious halfedge at the newly created vertex.
    // The returned halfedge is the predecessor for the insertion of the curve
    // end around v.
    _notify_before_split_fictitious_edge(Halfedge_handle(fict_he),
                                         Vertex_handle(v));
    DHalfedge* p_pred = m_topol_traits.split_fictitious_edge(fict_he, v);
    _notify_after_split_fictitious_edge(Halfedge_handle(p_pred),
                                        Halfedge_handle((*p_pred)->next()));
  }
  else if (obj.is_empty()) {
    // Create a new vertex that represents the given point.
    v = _create_boundary_vertex(p, ps_x, ps_y);

    // Notify the topology traits on the creation of the boundary vertex.
    m_topol_traits.notify_on_boundary_vertex_creation(v, p, ps_x, ps_y);
  }
  else {
    CGAL_assertion(CGAL::assign(v, obj));
    // The vertex coincides with an existing vertex that represents the point.
    // Do nothing.
  }

  return v;     // return the vertex that represents the point.
}

//-----------------------------------------------------------------------------
// Locate the DCEL features that will be used for inserting the given curve
// end, which has a boundary condition, and set the proper vertex there.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DVertex*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_place_and_set_curve_end(DFace* f,
                         const X_monotone_curve_2& cv, Arr_curve_end ind,
                         Arr_parameter_space ps_x, Arr_parameter_space ps_y,
                         DHalfedge** p_pred)
{
  // Use the topology traits to locate the DCEL feature that contains the
  // given curve end.
  auto obj = m_topol_traits.place_boundary_vertex(f, cv, ind, ps_x, ps_y);
  // Act according to the result type.

  if (! obj) {
    // We have to create a new vertex that represents the given curve end.
    DVertex* v = _create_boundary_vertex(cv, ind, ps_x, ps_y);

    // Notify the topology traits on the creation of the boundary vertex.
    m_topol_traits.notify_on_boundary_vertex_creation(v, cv, ind, ps_x, ps_y);

    // There are no edges incident to v, therefore no predecessor halfedge.
    *p_pred = nullptr;

    // Return the vertex that represents the curve end.
    return v;
  }

  DHalfedge** fict_he_p = std::get_if<DHalfedge*>(&*obj);
  if (fict_he_p != nullptr) {
    DHalfedge* fict_he = *fict_he_p;
    CGAL_assertion(fict_he != nullptr);
    // The curve end is located on a fictitious edge. We first create a new
    // vertex that corresponds to the curve end.
    DVertex* v = _create_boundary_vertex(cv, ind, ps_x, ps_y);

    // Split the fictitious halfedge at the newly created vertex.
    // The returned halfedge is the predecessor for the insertion of the curve
    // end around v.
    _notify_before_split_fictitious_edge(Halfedge_handle(fict_he),
                                         Vertex_handle(v));

    *p_pred = m_topol_traits.split_fictitious_edge(fict_he, v);

    _notify_after_split_fictitious_edge(Halfedge_handle(*p_pred),
                                        Halfedge_handle((*p_pred)->next()));
    return v;
  }
  DVertex** v_p = std::get_if<DVertex*>(&*obj);
  CGAL_assertion(v_p != nullptr);
  DVertex* v = *v_p;
  CGAL_assertion(v != nullptr);
  // In this case we are given an existing vertex that represents the curve
  // end. We now have to locate the predecessor edge for the insertion of cv
  // around this vertex.
  *p_pred = m_topol_traits.locate_around_boundary_vertex(v, cv, ind, ps_x, ps_y);
  return v;
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that both its
// endpoints correspond to free arrangement vertices (newly created vertices
// or existing isolated vertices), so a new inner CCB is formed in the face
// that contains the two vertices.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DHalfedge*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_insert_in_face_interior(DFace* f,
                         const X_monotone_curve_2& cv,
                         Arr_halfedge_direction cv_dir,
                         DVertex* v1, DVertex* v2)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: _insert_in_face_interior (internal)" << std::endl;
  Rcpp::Rcout << "face  : " << f << std::endl;
  Rcpp::Rcout << "cv    : " << cv << std::endl;
  Rcpp::Rcout << "cv_dir: " << cv_dir << std::endl;
  if (!v1->has_null_point())
    Rcpp::Rcout << "v1->point: " << v1->point() << std::endl;
  if (!v2->has_null_point())
    Rcpp::Rcout << "v2->point: " << v2->point() << std::endl;
#endif

  // Notify the observers that we are about to create a new edge.
  _notify_before_create_edge(cv, Vertex_handle(v1), Vertex_handle(v2));

  // Create a pair of twin halfedges connecting the two vertices,
  // and link them together to form a new connected component, a hole in f.
  DHalfedge* he1 = _dcel().new_edge();
  DHalfedge* he2 = he1->opposite();
  DInner_ccb* ic = _dcel().new_inner_ccb();
  X_monotone_curve_2* dup_cv = _new_curve(cv);

  ic->set_face(f);
  he1->set_curve(dup_cv);

  he1->set_next(he2);
  he1->set_vertex(v1);
  he1->set_inner_ccb(ic);

  he2->set_next(he1);
  he2->set_vertex(v2);
  he2->set_inner_ccb(ic);

  // Assign the incident halfedges of the two new vertices.
  v1->set_halfedge(he1);
  v2->set_halfedge(he2);

  // Set the direction of the halfedges
  he2->set_direction(cv_dir);

  // Create a handle to the new halfedge pointing at the curve target.
  Halfedge_handle hh(he2);

  // Notify the observers that we have created a new edge.
  _notify_after_create_edge(hh);

  // Notify the observers that we are about to form a new inner CCB inside f.
  _notify_before_add_inner_ccb(Face_handle(f), hh);

  // Initiate a new inner CCB inside the given face.
  f->add_inner_ccb(ic, he2);

  // Notify the observers that we have formed a new inner CCB.
  _notify_after_add_inner_ccb(hh->ccb());

  return he2;
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, such that one of its
// endpoints corresponds to a given arrangement vertex, given the exact
// place for the curve in the circular list around this vertex. The other
// endpoint corresponds to a free vertex (a newly created vertex or an
// isolated vertex).
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DHalfedge*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_insert_from_vertex(DHalfedge* he_to, const X_monotone_curve_2& cv,
                    Arr_halfedge_direction cv_dir,
                    DVertex* v)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: _insert_from_vertex (internal)" << std::endl;
  if (!he_to->has_null_curve())
    Rcpp::Rcout << "he_to: " << he_to->curve() << std::endl;
  else
    Rcpp::Rcout << "he_to: fictitious" << std::endl;
  Rcpp::Rcout << "f_to: " << (he_to->is_on_inner_ccb() ?
                            he_to->inner_ccb()->face() :
                            he_to->outer_ccb()->face()) << std::endl;
  Rcpp::Rcout << "cv    : " << cv << std::endl;
  Rcpp::Rcout << "cv_dir: " << cv_dir << std::endl;
  if (!v->has_null_point())
    Rcpp::Rcout << "v->point: " << v->point() << std::endl;
#endif

  // Get the incident face of the previous halfedge. Note that this will also
  // be the incident face of the two new halfedges we are about to create.
  DInner_ccb* ic = (he_to->is_on_inner_ccb()) ? he_to->inner_ccb() : nullptr;
  DOuter_ccb* oc = (ic == nullptr) ? he_to->outer_ccb() : nullptr;

  // The first vertex is the one that the he_to halfedge points to.
  // The second vertex is given by v.
  DVertex* v1 = he_to->vertex();
  DVertex* v2 = v;

  // Notify the observers that we are about to create a new edge.
  _notify_before_create_edge(cv, Vertex_handle(v1), Vertex_handle(v2));

  // Create a pair of twin halfedges connecting the two vertices,
  // and associate them with the given curve.
  DHalfedge* he1 = _dcel().new_edge();
  DHalfedge* he2 = he1->opposite();
  X_monotone_curve_2* dup_cv = _new_curve(cv);

  he1->set_curve(dup_cv);

  he1->set_vertex(v1);
  he2->set_vertex(v2);

  // Set the component for the new halfedge pair.
  if (oc != nullptr) {
    // On an outer component:
    he1->set_outer_ccb(oc);
    he2->set_outer_ccb(oc);
  }
  else {
    // On an inner component:
    he1->set_inner_ccb(ic);
    he2->set_inner_ccb(ic);
  }

  // Associate the incident halfedge of the new vertex.
  v2->set_halfedge(he2);

  // Link the new halfedges around the existing vertex v1.
  he2->set_next(he1);
  he1->set_next(he_to->next());

  he_to->set_next(he2);

  // Set the direction of the halfedges
  he2->set_direction(cv_dir);

  // Notify the observers that we have created a new edge.
  _notify_after_create_edge(Halfedge_handle(he2));

  // Return a pointer to the new halfedge whose target is the free vertex v.
  return he2;
}

//-----------------------------------------------------------------------------
// Insert an x-monotone curve into the arrangement, where the end vertices
// are given by the target points of two given halfedges.
// The two halfedges should be given such that in case a new face is formed,
// it will be the incident face of the halfedge directed from the first
// vertex to the second vertex.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DHalfedge*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_insert_at_vertices(DHalfedge* he_to,
                    const X_monotone_curve_2& cv,
                    Arr_halfedge_direction cv_dir,
                    DHalfedge* he_away,
                    bool& new_face,
                    bool& swapped_predecessors,
                    bool allow_swap_of_predecessors /* = true */)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "_insert_at_vertices: " << cv << std::endl;
#endif
  // Comment: This is how the situation looks
  //    ----to--->  >>cv_dir>>  ---away--->
  //               o ===cv=== 0
  //    <-tonext--              <-awaynext-
  // or to be read from right to left ...
  // this way, he_to and he_away lie
  // BEFORE insertion on the same inner ccb and
  // AFTER insertion on the same outer ccb

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: _insert_at_vertices (internal)" << std::endl;

  if (!he_to->has_null_curve())
    Rcpp::Rcout << "he_to: " << he_to->curve() << std::endl;
  else
    Rcpp::Rcout << "he_to: fictitious" << std::endl;
  Rcpp::Rcout << "dir1 : " << he_to->direction() << std::endl;
  Rcpp::Rcout << "f_to : " << (he_to->is_on_inner_ccb() ?
                            he_to->inner_ccb()->face() :
                            he_to->outer_ccb()->face()) << std::endl;
  Rcpp::Rcout << "cv    : " << cv << std::endl;
  Rcpp::Rcout << "cv_dir: " << cv_dir << std::endl;
  if (!he_away->has_null_curve())
    Rcpp::Rcout << "he_away: " << he_away->curve() << std::endl;
  else
    Rcpp::Rcout << "he_away: fictitious" << std::endl;
  Rcpp::Rcout << "dir 2 : " << he_away->direction() << std::endl;
  Rcpp::Rcout << "f_away: " << (he_away->is_on_inner_ccb() ?
                             he_away->inner_ccb()->face() :
                             he_away->outer_ccb()->face()) << std::endl;
#endif

  CGAL_precondition(he_to != nullptr);
  CGAL_precondition(he_away != nullptr);

  // TODO EBEB 2012-10-21 rewrite the code in terms of he_to and he_away instead of prev1 and prev2
  // the remainder of the function we deal with this situation adds he1 and
  // he2 in this way:
  //    ----prev1---> ( --he2--> ) ---p2next--->
  //                  o ===cv=== 0
  //    <---p1next--- ( <--he1-- ) <---prev2----
  DHalfedge* prev1 = he_to;
  DHalfedge* prev2 = he_away->prev();

  CGAL_precondition(prev1 != nullptr);
  CGAL_precondition(prev2 != nullptr);
  CGAL_precondition(prev1 != prev2);

  // in general we do not swap ...
  swapped_predecessors = false;

  // default values for signs
  std::pair<Sign, Sign> signs1(ZERO, ZERO);
  std::pair<Sign, Sign> signs2(ZERO, ZERO);
  // Remark: signs1 and signs2 are only used later when hole1==hole2

  // Comment: This also checks which is the 'cheaper' (previously the
  //          'shorter') way to insert the curve. Now cheaper means checking
  //          less local minima!
  if (allow_swap_of_predecessors) {
    bool swap_predecessors = false;

    // Comment EBEB 2012-08-05 hole1/hole2 appear later as ic1/ic2, but we keep
    // them here, as the usage is rather local to decide swapping
    DInner_ccb* hole1 = (prev1->is_on_inner_ccb()) ? prev1->inner_ccb() : nullptr;
    DInner_ccb* hole2 = (prev2->is_on_inner_ccb()) ? prev2->inner_ccb() : nullptr;

    if ((hole1 == hole2) && (hole1 != nullptr)) {
      // .. only in this special case, we have to check whether swapping should
      // take place

      // EBEB 2012-07-26 the following code enables optimizations:
      // - avoid length-test
      // - search only local minima to find leftmost vertex
      // - re-use of signs of ccbs
      // signs1/2 are only used when hole1 == hole2,
      // thus we have to init them now
      Arr_halfedge_direction cv_dir1 = cv_dir;
      std::list<std::pair<const DHalfedge*, int> > local_mins1;
      signs1 =
        _compute_signs_and_local_minima(prev1, cv, cv_dir1, prev2->next(),
                                        std::back_inserter(local_mins1));

      Arr_halfedge_direction cv_dir2 = (cv_dir == ARR_LEFT_TO_RIGHT) ?
        CGAL::ARR_RIGHT_TO_LEFT : CGAL::ARR_LEFT_TO_RIGHT;
      std::list< std::pair< const DHalfedge*, int > > local_mins2;
      signs2 =
        _compute_signs_and_local_minima(prev2, cv, cv_dir2, prev1->next(),
                                        std::back_inserter(local_mins2));
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << "signs1.x: " << signs1.first << std::endl;
      Rcpp::Rcout << "signs1.y: " << signs1.second << std::endl;
      Rcpp::Rcout << "signs2.x: " << signs2.first << std::endl;
      Rcpp::Rcout << "signs2.y: " << signs2.second << std::endl;
      Rcpp::Rcout << "#local_mins1: " << local_mins1.size() << std::endl;
      Rcpp::Rcout << "#local_mins2: " << local_mins2.size() << std::endl;
#endif

      if (!m_topol_traits.let_me_decide_the_outer_ccb(signs1, signs2,
                                                      swap_predecessors))
      {
        // COMMENT: The previous solution needed O(min(length1, length2)) steps
        //          to determine which path is shorter and the search for the
        //          leftmost vertex on a path needs O(#local_mins) geometric
        //          comparison. This solution saves the initial loop to
        //          determine the shorter path and will only need O(min(#local
        //          _mins1, #local_mins2)) geometric comparisons to determine
        //          the leftmost vertex on a path.

        // If there are no local minima in one the paths, it is expected
        // that the topology traits (or its adapter) do decide the outer ccb.
        CGAL_assertion(local_mins1.size() > 0);
        CGAL_assertion(local_mins2.size() > 0);

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
        Rcpp::Rcout << "decide swap" << std::endl;
#endif

        swap_predecessors =
          !((local_mins1.size() < local_mins2.size()) ?
            (  _defines_outer_ccb_of_new_face(prev1, cv, prev2->next(),
                                              local_mins1.begin(),
                                              local_mins1.end())) :
            (! _defines_outer_ccb_of_new_face(prev2, cv, prev1->next(),
                                              local_mins2.begin(),
                                              local_mins2.end())));
      }

      // perform the swap
      if (swap_predecessors) {
        std::swap(prev1, prev2);
        cv_dir = (cv_dir == ARR_LEFT_TO_RIGHT) ?
          CGAL::ARR_RIGHT_TO_LEFT : CGAL::ARR_LEFT_TO_RIGHT;
        std::swap(signs1, signs2);
        std::swap(local_mins1, local_mins2);

        // and store the value
        swapped_predecessors = true;
      }
    }

    // EBEB: For now, local_mins1/2 are local to this pre-check
    // Idea: Use it later, however, this spoils uses of _insert_at_vertices
    //       where allow_swap = false
    //       On the other hand: this would allow to set representative
    //       halfedge of ccbs to point to minimal vertex

  } // allow_swap_of_predecessors

  // Get the vertices that match cv's endpoints.
  DVertex* v1 = prev1->vertex();
  DVertex* v2 = prev2->vertex();

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: _insert_at_vertices (internal)" << std::endl;

  Rcpp::Rcout << "cv   : " << cv << std::endl;
  if (!prev1->has_null_curve())
    Rcpp::Rcout << "prev1: " << prev1->curve() << std::endl;
  else
    Rcpp::Rcout << "prev1: fictitious" << std::endl;
  Rcpp::Rcout << "dir1 : " << prev1->direction() << std::endl;
  Rcpp::Rcout << "pref: " << (prev1->is_on_inner_ccb() ?
                            prev1->inner_ccb()->face() :
                            prev1->outer_ccb()->face()) << std::endl;
  if (!prev2->has_null_curve())
    Rcpp::Rcout << "prev2: " << prev2->curve() << std::endl;
  else
    Rcpp::Rcout << "prev2: fictitious" << std::endl;
  Rcpp::Rcout << "dir 2: " << prev2->direction() << std::endl;
  Rcpp::Rcout << "pref2: " << (prev2->is_on_inner_ccb() ?
                             prev2->inner_ccb()->face() :
                             prev2->outer_ccb()->face()) << std::endl;
  Rcpp::Rcout << "cv_dir: " << cv_dir << std::endl;
#endif

  // Get the components containing the two previous halfedges and the incident
  // face (which should be the same for the two components).
  DInner_ccb* ic1 = (prev1->is_on_inner_ccb()) ? prev1->inner_ccb() : nullptr;
  DOuter_ccb* oc1 = (ic1 == nullptr) ? prev1->outer_ccb() : nullptr;
  DFace* f = (ic1 != nullptr) ? ic1->face() : oc1->face();
  DInner_ccb* ic2 = (prev2->is_on_inner_ccb()) ? prev2->inner_ccb() : nullptr;
  DOuter_ccb* oc2 = (ic2 == nullptr) ? prev2->outer_ccb() : nullptr;

  CGAL_precondition_code(DFace* f2 = (ic2 != nullptr) ? ic2->face() : oc2->face());

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "ic1: " << ic1 << std::endl;
  Rcpp::Rcout << "ic2: " << ic2 << std::endl;
  Rcpp::Rcout << "oc1: " << oc1 << std::endl;
  Rcpp::Rcout << "oc2: " << oc2 << std::endl;
  Rcpp::Rcout << "f1: " << &(*f) << std::endl;

#if 0
  DHalfedge* curr = prev1;
  if (curr != curr->next()) {
    curr = curr->next();
    while (curr != prev1) {
      if (!curr->has_null_curve())
        Rcpp::Rcout << "curr: " << curr->curve() << std::endl;
      else
        Rcpp::Rcout << "curr: fictitious" << std::endl;
      Rcpp::Rcout << "dir: "
                << (curr->direction() == CGAL::ARR_LEFT_TO_RIGHT ?
                    "L2R" : "R2L")
                << std::endl;
      curr = curr->next();
    }
  } else {
    Rcpp::Rcout << "only prev1" << std::endl;
  }
#endif

  CGAL_precondition_code(Rcpp::Rcout << "f2: " << &(*f2) << std::endl);

#if 0
  curr = prev2;
  if (curr != curr->next()) {
    curr = curr->next();
    while (curr != prev2) {
      if (!curr->has_null_curve())
        Rcpp::Rcout << "curr: " << curr->curve() << std::endl;
      else
        Rcpp::Rcout << "curr: fictitious" << std::endl;
      Rcpp::Rcout << "dir: "
                << (curr->direction() == CGAL::ARR_LEFT_TO_RIGHT ?
                    "L2R" : "R2L")
                << std::endl;
      curr = curr->next();
    }
  } else
    Rcpp::Rcout << "only prev2" << std::endl;
#endif
#endif // CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE

  CGAL_precondition_msg
    (f == f2, "The two halfedges must share the same incident face.");

  // In case the two previous halfedges lie on the same inner component inside
  // the face f, we use the topology-traits class to determine whether we have
  // to create a new face by splitting f, and if so - whether new face is
  // contained in the existing face or just adjacent to it.
  bool split_new_face = true;
  bool is_split_face_contained = false;

  if ((ic1 != nullptr) && (ic1 == ic2)) {

    // EBEB 2012-08-06:
    // This is new code. It relies on the (computed) signs and replaces to
    // trace the ccb again (in particular for torical arrangements)
    // TODO EBEB 2012-08-06:
    // Check what to do here, when allow_swap_of_predecessors = false and thus
    // signs1 and signs2 set to DEFAULT (=ZERO) values.
    // swapping is currently only disabled when _insert_at_vertices is called
    // from Arr_construction_ss_visitor, which however uses the
    // 'swap_predecessors' member of the topology traits' construction helper.
    // So it's questionable whether we can combine the light-weigth swap
    // information with the slightly more expensive sign computations, to keep
    // efficient translated code after compile-time.
    std::pair<bool, bool> res =
      m_topol_traits.face_split_after_edge_insertion(signs1, signs2);

    split_new_face = res.first;
    is_split_face_contained = res.second;

    // The result <false, true> is illegal:
    CGAL_assertion(split_new_face || ! is_split_face_contained);
  }

  // Notify the observers that we are about to create a new edge.
  _notify_before_create_edge(cv, Vertex_handle(v1), Vertex_handle(v2));

  // Create a pair of twin halfedges connecting v1 and v2 and associate them
  // with the given curve.
  DHalfedge* he1 = _dcel().new_edge();
  DHalfedge* he2 = he1->opposite();
  X_monotone_curve_2* dup_cv = _new_curve(cv);

  he1->set_curve(dup_cv);

  he1->set_vertex(v1);
  he2->set_vertex(v2);

  // Connect the new halfedges to the existing halfedges around the two
  // incident vertices.
  he1->set_next(prev1->next());
  he2->set_next(prev2->next());

  prev1->set_next(he2);
  prev2->set_next(he1);

  he2->set_direction(cv_dir);

  // Check the various cases of insertion (in the design document: the
  // various sub-cases of case 3 in the insertion procedure).
  if (((ic1 != nullptr) || (ic2 != nullptr)) && (ic1 != ic2)) {
    // In case we have to connect two disconnected components, no new face
    // is created.
    new_face = false;

    // Check whether both halfedges are inner components (holes) in the same
    // face, or whether one is a hole and the other is on the outer boundary
    // of the face.
    Face_handle fh(f);

    if ((ic1 != nullptr) && (ic2 != nullptr)) {
      // In this case (3.1) we have to connect to inner CCBs (holes) inside f.
      // Notify the observers that we are about to merge two holes in the face.
      _notify_before_merge_inner_ccb(fh,
                                     (Halfedge_handle(prev1))->ccb(),
                                     (Halfedge_handle(prev2))->ccb(),
                                     Halfedge_handle(he1));

      // Remove the inner component prev2 belongs to, and unite it with the
      // inner component that prev1 belongs to.
      f->erase_inner_ccb(ic2);

      // Set the merged component for the two new halfedges.
      he1->set_inner_ccb(ic1);
      he2->set_inner_ccb(ic1);

      if (m_sweep_mode)
      {
        // Inner CCB are obtained using Halfedge::inner_ccb() which
        // performs path reduction and always return valid iCCB
        CGAL_assertion(ic1->is_valid());
        CGAL_assertion(ic2->is_valid());
        ic2->set_next(ic1);
      }
      else
      {
        // Make all halfedges along ic2 to point to ic1.
        DHalfedge* curr;
        for (curr = he2->next(); curr != he1; curr = curr->next())
          curr->set_inner_ccb(ic1);

        // Delete the redundant inner CCB.
        _dcel().delete_inner_ccb(ic2);
      }

      // Notify the observers that we have merged the two inner CCBs.
      _notify_after_merge_inner_ccb(fh, (Halfedge_handle(he1))->ccb());
    }
    else {
      // In this case (3.2) we connect a hole (inner CCB) with an outer CCB
      // of the face that contains it. We remove the hole and associate the
      // pair of new halfedges with the outer boundary of the face f.
      DInner_ccb* del_ic;
      DOuter_ccb* oc;
      DHalfedge* ccb_first;
      DHalfedge* ccb_last;

      if (ic1 != nullptr) {
        // We remove the inner CCB ic1 and merge in with the outer CCB oc2.
        del_ic = ic1;
        oc = oc2;
        ccb_first = he1->next();
        ccb_last = he2;
      }
      else {
        // We remove the inner CCB ic2 and merge in with the outer CCB oc1.
        del_ic = ic2;
        oc = oc1;
        ccb_first = he2->next();
        ccb_last = he1;
      }

      he1->set_outer_ccb(oc);
      he2->set_outer_ccb(oc);

      // Notify the observers that we are about to remove an inner CCB from
      // the face.
      _notify_before_remove_inner_ccb(fh, (Halfedge_handle(ccb_first))->ccb());

      // Remove the inner CCB from the face, as we have just connected it to
      // the outer boundary of its incident face.
      f->erase_inner_ccb(del_ic);

      // Make all halfedges along the inner CCB to point to the outer CCB of f.
      DHalfedge* curr;
      for (curr = ccb_first; curr != ccb_last; curr = curr->next())
        curr->set_outer_ccb(oc);

      // Delete the redundant hole.
      _dcel().delete_inner_ccb(del_ic);

      // Notify the observers that we have removed an inner ccb.
      _notify_after_remove_inner_ccb(fh);
    }
  }
  else if (! split_new_face) {
    // RWRW: NEW!
    CGAL_assertion((ic1 == ic2) && (ic1 != nullptr));

    // Handle the special case where we close an inner CCB, such that
    // we form two outer CCBs of the same face.
    Face_handle fh(f);

    // Notify the obserers we are about to remove an inner CCB from f.
    _notify_before_remove_inner_ccb(fh, (Halfedge_handle(he1))->ccb());

    // Erase the inner CCB from the incident face and delete the
    // corresponding component.
    f->erase_inner_ccb(ic1);

    _dcel().delete_inner_ccb(ic1);

    // Notify the observers that the inner CCB has been removed.
    _notify_after_remove_inner_ccb(fh);

    // Handle the first split outer CCB (the one containing he1):
    // Notify the obserers we are about to add an outer CCB to f.
    _notify_before_add_outer_ccb(fh, Halfedge_handle(he1));

    // Create a new outer CCB that for the face f, and make he1 the
    // representative halfedge of this component.
    DOuter_ccb* f_oc1 = _dcel().new_outer_ccb();

    f->add_outer_ccb(f_oc1, he1);
    f_oc1->set_face(f);
    he1->set_outer_ccb(f_oc1);

    // Set the component of all halfedges that used to belong to he1's CCB.
    DHalfedge* curr;

    for (curr = he1->next(); curr != he1; curr = curr->next())
      curr->set_outer_ccb(f_oc1);

    // Notify the observers that we have added an outer CCB to f.
    _notify_after_add_outer_ccb((Halfedge_handle(he1))->ccb());

    // Handle the second split outer CCB (the one containing he2):
    // Notify the obserers we are about to add an outer CCB to f.
    _notify_before_add_outer_ccb(fh, Halfedge_handle(he2));

    // Create a new outer CCB that for the face f, and make he2 the
    // representative halfedge of this component.
    DOuter_ccb* f_oc2 = _dcel().new_outer_ccb();

    f->add_outer_ccb(f_oc2, he2);
    f_oc2->set_face(f);
    he2->set_outer_ccb(f_oc2);

    // Set the component of all halfedges that used to belong to he2's CCB.
    for (curr = he2->next(); curr != he2; curr = curr->next())
      curr->set_outer_ccb(f_oc2);

    // Notify the observers that we have added an outer CCB to f.
    _notify_after_add_outer_ccb((Halfedge_handle(he2))->ccb());

    // Mark that in this case no new face is created:
    new_face = false;
  }
  else if ((ic1 == ic2) && (oc1 == oc2)) {
    // In this case we created a pair of halfedge that connect halfedges that
    // already belong to the same component. This means we have to create a
    // new face by splitting the existing face f.
    // Notify the observers that we are about to split a face.
    Face_handle fh(f);

    _notify_before_split_face(fh, Halfedge_handle(he1));

    // Create the new face and create a single outer component which should
    // point to he2.
    DFace* new_f = _dcel().new_face();
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
    Rcpp::Rcout << "new face: " << new_f << std::endl;
#endif

    DOuter_ccb* new_oc = _dcel().new_outer_ccb();

    new_face = true;
    new_f->add_outer_ccb(new_oc, he2);
    new_oc->set_face(new_f);

    // Set the components of the new halfedge he2, which should be the new
    // outer comoponent of the new face.
    // Note that there are several cases for setting he1's component, so we
    // do not do it yet.
    he2->set_outer_ccb(new_oc);

    // Set the component of all halfedges that used to belong to he2's CCB.
    DHalfedge* curr;

    for (curr = he2->next(); curr != he2; curr = curr->next())
      curr->set_outer_ccb(new_oc);

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
    Rcpp::Rcout << "(=> prev1=" << &(*prev1) << ") he2= " << &(*he2)
              << "  defines new outer CCB" << std::endl;
    Rcpp::Rcout << "he2dir  : " << he2->direction() << std::endl;
    Rcpp::Rcout << "prev1->face(): " << (prev1->is_on_inner_ccb() ?
                                       prev1->inner_ccb()->face() :
                                       prev1->outer_ccb()->face())
              << std::endl;
    Rcpp::Rcout << "signs1: " << signs1.first  << "," << signs1.second
              << std::endl;
#endif

    // Check whether the two previous halfedges lie on the same innder CCB
    // or on the same outer CCB (distinguish case 3.3 and case 3.4).
    bool   is_hole;

    if (ic1 != nullptr) {
      // In this case (3.3) we have too distinguish two sub-cases.
      if (is_split_face_contained) {
        // Comment: This is true for all non-identification topologies

        // The halfedges prev1 and prev2 belong to the same inner component
        // (hole) inside the face f, such that the new edge creates a new
        // face that is contained in f (case 3.3.1).
        is_hole = true;

        // In this case, he1 lies on an inner CCB of f.
        he1->set_inner_ccb(ic1);

        // Note that the current representative of the inner CCB may not
        // belong to the hole any more. In this case we replace the hole
        // representative by he1.
        if (! ic1->halfedge()->is_on_inner_ccb())
          ic1->set_halfedge(he1);

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
        Rcpp::Rcout << "(=> prev2=" << &(*prev2) << ") he1= " << &(*he1)
                  << "  defines new inner CCB" << std::endl;
        Rcpp::Rcout << "he1dir  : " << he1->direction() << std::endl;
        Rcpp::Rcout << "prev2->face(): " << (prev2->is_on_inner_ccb() ?
                                           prev2->inner_ccb()->face() :
                                           prev2->outer_ccb()->face())
                  << std::endl;
        Rcpp::Rcout << "signs2: " << signs2.first  << "," << signs2.second
                  << std::endl;
#endif
      }
      else {
        // Comment: This case can only occur in identification topologies

        // The new face we have created should be adjacent to the existing
        // face (case 3.3.2).
        is_hole = false;

        // Notify the obserers we are about to add an outer CCB to f.
        _notify_before_add_outer_ccb(fh, Halfedge_handle(he1));

        // Create a new outer CCB that for the face f, and make he1 the
        // representative halfedge of this component.
        DOuter_ccb* f_oc = _dcel().new_outer_ccb();

        f->add_outer_ccb(f_oc, he1);
        f_oc->set_face(f);
        he1->set_outer_ccb(f_oc);

        // Set the component of all halfedges that used to belong to he1's
        // CCB.
        for (curr = he1->next(); curr != he1; curr = curr->next())
          curr->set_outer_ccb(f_oc);

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
        Rcpp::Rcout << "(=> prev2=" << &(*prev2) << ") he1= " << &(*he1) << "  defines new outer CCB" << std::endl;
        Rcpp::Rcout << "he1dir  : " << he1->direction() << std::endl;
        Rcpp::Rcout << "prev2->face(): " << (prev2->is_on_inner_ccb() ?
                                           prev2->inner_ccb()->face() :
                                           prev2->outer_ccb()->face())
                  << std::endl;
        Rcpp::Rcout << "signs2: " << signs2.first  << "," << signs2.second
                  << std::endl;
#endif

        // Notify the observers that we have added an outer CCB to f.
        _notify_after_add_outer_ccb((Halfedge_handle(he1))->ccb());

        // Go over all other outer CCBs of f and check whether they should be
        // moved to be outer CCBs of the new face.
        DOuter_ccb_iter  oc_it = f->outer_ccbs_begin();
        DOuter_ccb_iter  oc_to_move;


        while (oc_it != f->outer_ccbs_end()) {
          // Use the topology traits to determine whether the representative
          // of the current outer CCB should belong to the same face as he2
          // (which is on the outer boundary of the new face).
          bool increment = true;
          if (*oc_it != he1) {

            // he2 is supposed to be a perimetric path and so all of the oc_its,
            // we only have to detect which one. We do so by comparing signs of
            // ccbs:
            // IDEA EBEB 2012-07-28
            // store signs of CCB with CCB in DCEL and use them here
            // *oc_it is already closed, so we do a full round
            // (default = false)
            std::pair<Sign, Sign> signs_oc =
              _compute_signs(*oc_it, Has_identified_sides());

            bool move = false;

            // TODO EBEB 2013-07-15 refactor into own function
            // TODO EBEB 2012-08-07 this either compares signs in left-right
            // direction OR signs in bottom-top direction, which will probably
            // not work for torus!
            if ((signs2.first != CGAL::ZERO) && (signs_oc.first != CGAL::ZERO))
            {
              if (signs2.first != signs_oc.first) move = true;
            }
            else if ((signs2.second != CGAL::ZERO) &&
                     (signs_oc.second != CGAL::ZERO))
            {
              if (signs2.second != signs_oc.second) move = true;
            }

            if (move) {
              // We increment the itrator before moving the outer CCB, because
              // this operation invalidates the iterator.
              increment = false;
              oc_to_move = oc_it;
              ++oc_it;

              _move_outer_ccb(f, new_f, *oc_to_move);
            }
          }

          if (increment) ++oc_it;
        }
      }
    }
    else {
      // In this case the face f is simply split into two (case 3.4).
      is_hole = false;

      // In this case, he1 lies on an outer CCB of f.
      he1->set_outer_ccb(oc1);

      // As the outer component of the existing face f may associated with
      // one of the halfedges along the boundary of the new face, we set it
      // to be he1.
      oc1->set_halfedge(he1);
    }

    // Check whether we should mark the original face and the new face as
    // bounded or as unbounded faces.
    if (! f->is_unbounded())
      // The original face is bounded, so the new face split from it is
      // obviously bounded.
      new_f->set_unbounded(false);
    else if (is_hole)
      // The new face is a hole inside the original face, so it must be
      // bounded.
      new_f->set_unbounded(false);
    else {
      // Use the topology traits to determine whether each of the split
      // faces is unbounded. Note that if the new face is bounded, then f
      // obviously remains unbounded and there is no need for further checks.
      new_f->set_unbounded(m_topol_traits.is_unbounded(new_f));

      if (new_f->is_unbounded())
        f->set_unbounded(m_topol_traits.is_unbounded(f));
    }

    // Notify the observers that we have split the face.
    _notify_after_split_face(fh, Face_handle(new_f), is_hole);
  }
  else {
    CGAL_assertion((oc1 != nullptr) && (oc2 != nullptr) && (oc1 != oc2));

    // In case prev1 and prev2 belong to different outer CCBs of the same
    // face f (case 3.5), we have to merge this ccbs into one. Note that we
    // do not create a new face.
    new_face = false;

    // Notify the observers that we are about to merge two outer CCBs.
    Face_handle fh(f);

    _notify_before_merge_outer_ccb(fh,
                                   (Halfedge_handle(prev1))->ccb(),
                                   (Halfedge_handle(prev2))->ccb(),
                                   Halfedge_handle(he1));

    // Remove the outer component prev2 belongs to, and unite it with the
    // outer component that prev1 belongs to.
    f->erase_outer_ccb(oc2);

    // Set the merged component for the two new halfedges.
    he1->set_outer_ccb(oc1);
    he2->set_outer_ccb(oc1);

    // Make all halfedges along oc2 to point to oc1.
    DHalfedge* curr;

    for (curr = he2->next(); curr != he1; curr = curr->next())
      curr->set_outer_ccb(oc1);

    // Delete the redundant outer CCB.
    _dcel().delete_outer_ccb(oc2);

    // Notify the observers that we have merged the two CCBs.
    _notify_after_merge_outer_ccb(fh, (Halfedge_handle(he1))->ccb());
  }

  // Notify the observers that we have created a new edge.
  _notify_after_create_edge(Halfedge_handle(he2));

  // TODO EBEB 2012-08-08 add postcondition that checks sanity
#if 0
  {
    DHalfedge* he1 = he2->opposite();
    DInner_ccb* ic1 = (he1->is_on_inner_ccb()) ? he1->inner_ccb() : nullptr;
    DOuter_ccb* oc1 = (ic1 == nullptr) ? he1->outer_ccb() : nullptr;
    DFace* f1 = (ic1 != nullptr) ? ic1->face() : oc1->face();
    DInner_ccb* ic2 = (he2->is_on_inner_ccb()) ? he2->inner_ccb() : nullptr;
    DOuter_ccb* oc2 = (ic2 == nullptr) ? he2->outer_ccb() : nullptr;
    DFace* f2 = (ic2 != nullptr) ? ic2->face() : oc2->face();
    CGAL_postcondition((ic1 != ic2) || (f1 == f2));
  }
#endif

  // Return the halfedge directed from v1 to v2.
  return he2;
}

//-----------------------------------------------------------------------------
// Relocate all inner CCBs (holes) to their proper position,
// immediately after a face has split due to the insertion of a new halfedge.
//
template <typename GeomTraits, typename TopTraits>
void  Arrangement_on_surface_2<GeomTraits, TopTraits>::
_relocate_inner_ccbs_in_new_face(DHalfedge* new_he)
{
  // The given halfedge points to the new face, while its twin points to the
  // old face (the one that has just been split).
  DFace* new_face = (new_he->is_on_inner_ccb()) ?
    new_he->inner_ccb()->face() : new_he->outer_ccb()->face();
  DHalfedge* opp_he = new_he->opposite();
  const bool opp_on_inner_ccb = opp_he->is_on_inner_ccb();
  DFace* old_face = opp_on_inner_ccb ? opp_he->inner_ccb()->face() :
    opp_he->outer_ccb()->face();

  CGAL_assertion(new_face != old_face);

  // Examine the inner CCBs inside the existing old face and move the relevant
  // ones into the new face.
  DInner_ccb_iter ic_it = old_face->inner_ccbs_begin();
  while (ic_it != old_face->inner_ccbs_end()) {
    // In case the new edge represents the current component in the old face
    // (note we take the opposite halfedge, as it is incident to the old face),
    // then the new face already forms a hole in the old face, and we do not
    // need to move it.
    CGAL_assertion((*ic_it)->is_on_inner_ccb());

    if (opp_on_inner_ccb && ((*ic_it)->inner_ccb() == opp_he->inner_ccb())) {
      ++ic_it;
      continue;
    }

    // Check whether the current inner CCB is inside new face (we actually
    // check if a representative vertex is located in the new face).
    if (m_topol_traits.is_in_face(new_face, (*ic_it)->vertex()->point(),
                                  (*ic_it)->vertex()))
    {
      // We store the current iterator which get then incremented before it
      // gets moved, as the move operation invalidates the iterator.
      DInner_ccb_iter ic_to_move = ic_it;
      ++ic_it;
      _move_inner_ccb(old_face, new_face, *ic_to_move); // move the hole
    }
    else
      ++ic_it;
  }
}

//-----------------------------------------------------------------------------
// Relocate all isolated vertices to their proper position,
// immediately after a face has split due to the insertion of a new halfedge.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_relocate_isolated_vertices_in_new_face(DHalfedge* new_he)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2: _relocate_isolated_vertices_in_new_face" << std::endl;
#endif
  // The given halfedge points to the new face, while its twin points to the
  // old face (the one that has just been split).
  DFace* new_face = (new_he->is_on_inner_ccb()) ?
    new_he->inner_ccb()->face() :
    new_he->outer_ccb()->face();
  DHalfedge* opp_he = new_he->opposite();
  DFace* old_face = (opp_he->is_on_inner_ccb()) ?
    opp_he->inner_ccb()->face() :
    opp_he->outer_ccb()->face();

  CGAL_assertion(new_face != old_face);

  // Examine the isolated vertices inside the existing old face and move the
  // relevant ones into the new face.
  DIso_vertex_iter    iv_it;
  DIso_vertex_iter    iv_to_move;

  iv_it = old_face->isolated_vertices_begin();
  while (iv_it != old_face->isolated_vertices_end()) {
    // Check whether the isolated vertex lies inside the new face.
    if (m_topol_traits.is_in_face(new_face, iv_it->point(), &(*iv_it))) {
      // We increment the isolated vertices itrator before moving the vertex,
      // because this operation invalidates the iterator.
      iv_to_move  = iv_it;
      ++iv_it;

      // Move the isolated vertex.
      _move_isolated_vertex(old_face, new_face, &(*iv_to_move));
    }
    else
      ++iv_it;
  }
}

//-----------------------------------------------------------------------------
// Relocate all holes (inner CCBs) and isolated vertices to their proper
// position, immediately after a face has split due to the insertion of a new
// halfedge.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_relocate_in_new_face(DHalfedge* new_he)
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "Aos_2 _relocate_in_new_face" << std::endl;
  Rcpp::Rcout << "HeCv: " << new_he->curve() << std::endl;
  Rcpp::Rcout << "HeDi: " << new_he->direction() << std::endl;
#endif
  _relocate_inner_ccbs_in_new_face(new_he);
  _relocate_isolated_vertices_in_new_face(new_he);
}

//-----------------------------------------------------------------------------
// Replace the point associated with the given vertex.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_modify_vertex(DVertex* v, const Point_2& p)
{
  // Notify the observers that we are about to modify a vertex.
  Vertex_handle vh(v);
  _notify_before_modify_vertex(vh, p);

  // Modify the point associated with the vertex.
  v->point() = p;

  // Notify the observers that we have modified the vertex.
  _notify_after_modify_vertex(vh);
}

//-----------------------------------------------------------------------------
// Replace the x-monotone curve associated with the given edge.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_modify_edge(DHalfedge* he, const X_monotone_curve_2& cv)
{
  // Notify the observers that we are about to modify an edge.
  Halfedge_handle e(he);
  _notify_before_modify_edge(e, cv);

  // Modify the curve associated with the halfedge.
  he->curve() = cv;

  // Notify the observers that we have modified the edge.
  _notify_after_modify_edge(e);
}

//-----------------------------------------------------------------------------
// Check if the given vertex represents one of the ends of a given curve.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_are_equal(const DVertex* v,
           const X_monotone_curve_2& cv, Arr_curve_end ind) const
{
  // In case the given curve end has boundary conditions, use the topology
  // traits to determine whether it is equivalent to v.
  const Arr_parameter_space ps_x =
    m_geom_traits->parameter_space_in_x_2_object()(cv, ind);
  const Arr_parameter_space ps_y =
    m_geom_traits->parameter_space_in_y_2_object()(cv, ind);

  if ((ps_x != ARR_INTERIOR) || (ps_y != ARR_INTERIOR))
    return (m_topol_traits.are_equal(v, cv, ind, ps_x, ps_y));

  // Otherwise, the curve end is a valid endpoint. Check that v is also
  // associated with a valid point that equals this endpoint.
  if (v->has_null_point()) return false;

  return (ind == ARR_MIN_END) ?
    (m_geom_traits->equal_2_object()
     (m_geom_traits->construct_min_vertex_2_object()(cv), v->point())) :
    (m_geom_traits->equal_2_object()
     (m_geom_traits->construct_max_vertex_2_object()(cv), v->point()));
}

//-----------------------------------------------------------------------------
// Split a given edge into two at a given point, and associate the given
// x-monotone curves with the split edges.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DHalfedge*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_split_edge(DHalfedge* e, const Point_2& p,
            const X_monotone_curve_2& cv1, const X_monotone_curve_2& cv2)
{
  // Allocate a new vertex and associate it with the split point.
  // Obtain the boundary conditions:
  auto ps_x = m_geom_traits->parameter_space_in_x_2_object()(p);
  auto ps_y = m_geom_traits->parameter_space_in_y_2_object()(p);

  DVertex* v(nullptr);
  if ((ps_x == ARR_INTERIOR) && (ps_y == ARR_INTERIOR)) v = _create_vertex(p);
  else {
    v = _create_boundary_vertex(p, ps_x, ps_y);

    // Notify the topology traits on the creation of the boundary vertex.
    m_topol_traits.notify_on_boundary_vertex_creation(v, p, ps_x, ps_y);
  }

  // Split the edge from the given vertex.
  return (_split_edge(e, v, cv1, cv2));
}

//-----------------------------------------------------------------------------
// Split a given edge into two at a given vertex, and associate the given
// x-monotone curves with the split edges.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DHalfedge*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_split_edge(DHalfedge* e, DVertex* v,
            const X_monotone_curve_2& cv1, const X_monotone_curve_2& cv2)
{
  // Get the split halfedge and its twin, its source and target.
  DHalfedge* he1 = e;
  DHalfedge* he2 = he1->opposite();
  DInner_ccb* ic1 = (he1->is_on_inner_ccb()) ? he1->inner_ccb() : nullptr;
  DOuter_ccb* oc1 = (ic1 == nullptr) ? he1->outer_ccb() : nullptr;
  DInner_ccb* ic2 = (he2->is_on_inner_ccb()) ? he2->inner_ccb() : nullptr;
  DOuter_ccb* oc2 = (ic2 == nullptr) ? he2->outer_ccb() : nullptr;

  // Notify the observers that we are about to split an edge.
  _notify_before_split_edge(Halfedge_handle(e), Vertex_handle(v), cv1, cv2);

  // Allocate a pair of new halfedges.
  DHalfedge* he3 = _dcel().new_edge();
  DHalfedge* he4 = he3->opposite();

  // Connect the new halfedges:
  //
  //            he1      he3
  //         -------> ------->
  //       (.)      (.)v     (.)
  //         <------- <-------
  //            he2      he4
  //
  v->set_halfedge(he4);

  if (he1->next() != he2) {
    // Connect e3 between e1 and its successor.
    he3->set_next(he1->next());

    // Insert he4 between he2 and its predecessor.
    he2->prev()->set_next(he4);
  }
  else
    // he1 and he2 form an "antenna", so he4 becomes he3's successor.
    he3->set_next(he4);

  if (oc1 != nullptr)
    he3->set_outer_ccb(oc1);
  else
    he3->set_inner_ccb(ic1);

  he3->set_vertex(he1->vertex());
  he4->set_vertex(v);
  he4->set_next(he2);

  if (oc2 != nullptr)
    he4->set_outer_ccb(oc2);
  else
    he4->set_inner_ccb(ic2);

  if (he1->vertex()->halfedge() == he1)
    // If he1 is the incident halfedge to its target, he3 replaces it.
    he1->vertex()->set_halfedge(he3);

  // Update the properties of the twin halfedges we have just split.
  he1->set_next(he3);
  he1->set_vertex(v);

  // The direction of he3 is the same as he1's (and the direction of he4 is
  // the same as he2).
  he3->set_direction(he1->direction());

  // Associate cv1 with he1 (and its twin). We allocate a new curve for cv2
  // and associate it with he3 (and its twin).
  X_monotone_curve_2* dup_cv2 = _new_curve(cv2);

  he1->curve() = cv1;
  he3->set_curve(dup_cv2);

  // Notify the observers that we have split an edge into two.
  _notify_after_split_edge(Halfedge_handle(he1), Halfedge_handle(he3));

  // Return a handle for one of the existing halfedge that is incident to the
  // split point.
  return he1;
}

template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compute_indices(Arr_parameter_space /* ps_x_curr */,
                 Arr_parameter_space /* ps_y_curr */,
                 Arr_parameter_space /* ps_x_next */,
                 Arr_parameter_space /* ps_y_next */,
                 int& /* x_index */, int& /* y_index */,
                 Arr_false) const
{ /* nothing if no identification */ }

template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compute_indices(Arr_parameter_space ps_x_curr, Arr_parameter_space ps_y_curr,
                 Arr_parameter_space ps_x_next, Arr_parameter_space ps_y_next,
                 int& x_index, int& y_index,  Arr_true) const
{
  // If we cross the identification curve in x, then we must update the
  // x_index. Note that a crossing takes place in the following cases:
  //                .                                  .
  //                .                                  .
  //                .                                  .
  //                . v    he                   he     . v
  //       <-------(.)<---------             -------->(.)------->
  //                .                                  .
  //       (BEFORE) .    (AFTER)              (BEFORE) .  (AFTER)
  //       x_index-1.    x_index              x_index  .  x_index+1
  //
  if ((ps_x_curr == ARR_LEFT_BOUNDARY) && (ps_x_next == ARR_RIGHT_BOUNDARY)) {
    CGAL_assertion(is_identified(Left_side_category()) &&
                   is_identified(Right_side_category()));
    --x_index; // in "negative" u-direction
  }
  else if ((ps_x_curr == ARR_RIGHT_BOUNDARY) &&
           (ps_x_next == ARR_LEFT_BOUNDARY))
  {
    CGAL_assertion(is_identified(Left_side_category()) &&
                   is_identified(Right_side_category()));
    ++x_index; // in "positive" u-direction
  }

  // Check if we cross the identification curve in y.
  if ((ps_y_curr == ARR_BOTTOM_BOUNDARY) && (ps_y_next == ARR_TOP_BOUNDARY)) {
    CGAL_assertion(is_identified(Bottom_side_category()) &&
                   is_identified(Top_side_category()));
    --y_index; // in "negative" v-direction
  }
  else if ((ps_y_curr == ARR_TOP_BOUNDARY) &&
           (ps_y_next == ARR_BOTTOM_BOUNDARY))
  {
    CGAL_assertion(is_identified(Bottom_side_category()) &&
                   is_identified(Top_side_category()));
    ++y_index; // in "positive" v-direction
  }
}

// Computes signs and locale minima of an open path to be closed by a
// newly inserted curve.
//
// Precondition The OutputIterator must be a back inserter.
// Precondition The traversed ccb is an inner ccb; thus, it cannot be
//              on an open boundary.
// Postcondition If nullptr is a local minimum, it is inserted first.
//                No other local minima can be nullptr.
template <typename GeomTraits, typename TopTraits>
template <typename OutputIterator>
std::pair<Sign, Sign>
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compute_signs_and_local_minima(const DHalfedge* he_to,
                                const X_monotone_curve_2& cv,
                                Arr_halfedge_direction cv_dir,
                                const DHalfedge* he_away,
                                OutputIterator local_mins_it) const
{
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "he_to: " << he_to->opposite()->vertex()->point()
            << " => " << he_to->vertex()->point() << std::endl;
  Rcpp::Rcout << "cv: " << cv << std::endl;
  Rcpp::Rcout << "cv_dir: " << cv_dir << std::endl;
  Rcpp::Rcout << "he_away: " << he_away->opposite()->vertex()->point()
            << " => " << he_away->vertex()->point() << std::endl;
#endif

  // We go over the sequence of vertices, starting from he_away's target
  // vertex, until reaching he_to's source vertex, and find the leftmost
  // one. Note that we do this carefully, keeping track of the number of
  // times we crossed the identification curve in x or in y (if they exist).
  // Note that the path must not be incident to any vertex on open boundary.
  auto parameter_space_in_x = m_geom_traits->parameter_space_in_x_2_object();
  auto parameter_space_in_y = m_geom_traits->parameter_space_in_y_2_object();

  // TODO 2012-09-20 check "correction" here too (as in "other" function of this kind
  int x_index = 0;
  int y_index = 0;

  // Obtain the parameter space pair of cv.
  Arr_curve_end cv_to_end =
    (cv_dir == ARR_LEFT_TO_RIGHT) ? ARR_MIN_END : ARR_MAX_END;
  Arr_parameter_space ps_x_cv_to = parameter_space_in_x(cv, cv_to_end);
  Arr_parameter_space ps_y_cv_to = parameter_space_in_y(cv, cv_to_end);
  Arr_curve_end cv_away_end =
    (cv_dir == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
  Arr_parameter_space ps_x_cv_away = parameter_space_in_x(cv, cv_away_end);
  Arr_parameter_space ps_y_cv_away = parameter_space_in_y(cv, cv_away_end);

  // Obtain the parameter space pair of he_to and he_away
  Arr_curve_end he_to_tgt_end =
    (he_to->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
  Arr_parameter_space ps_x_he_to =
    parameter_space_in_x(he_to->curve(), he_to_tgt_end);
  Arr_parameter_space ps_y_he_to =
    parameter_space_in_y(he_to->curve(), he_to_tgt_end);
  Arr_curve_end he_away_src_end =
    (he_away->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MIN_END : ARR_MAX_END;
  Arr_parameter_space ps_x_he_away =
    parameter_space_in_x(he_away->curve(), he_away_src_end);
  Arr_parameter_space ps_y_he_away =
    parameter_space_in_y(he_away->curve(), he_away_src_end);
  Arr_curve_end he_away_tgt_end =
    (he_away->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
  Arr_parameter_space ps_x_he_away_tgt =
    parameter_space_in_x(he_away->curve(), he_away_tgt_end);
  Arr_parameter_space ps_y_he_away_tgt =
    parameter_space_in_y(he_away->curve(), he_away_tgt_end);

  Arr_parameter_space ps_x_curr, ps_y_curr;
  Arr_parameter_space ps_x_next, ps_y_next;
  Arr_parameter_space ps_x_save, ps_y_save;

  ps_x_curr = ps_x_cv_away;
  ps_y_curr = ps_y_cv_away;
  ps_x_next = ps_x_he_away;
  ps_y_next = ps_y_he_away;
  ps_x_save = ps_x_he_away_tgt;
  ps_y_save = ps_y_he_away_tgt;

  CGAL_assertion(!is_open(ps_x_curr, ps_y_curr));
  CGAL_assertion(!is_open(ps_x_next, ps_y_next));

  if ((cv_dir == ARR_RIGHT_TO_LEFT) &&
      (he_away->direction() == ARR_LEFT_TO_RIGHT)) {
    const DHalfedge* null_he = nullptr;
    *local_mins_it++ = std::make_pair(null_he, x_index);
  }

  _compute_indices(ps_x_curr, ps_y_curr, ps_x_next, ps_y_next,
                   x_index, y_index, Has_identified_sides());

  const DHalfedge* he = he_away;
  while (he != he_to) {
    ps_x_curr = ps_x_save;
    ps_y_curr = ps_y_save;
    CGAL_assertion(!is_open(ps_x_curr, ps_y_curr));

    Arr_curve_end he_next_src_end, he_next_tgt_end;
    if (he->next()->direction() == ARR_LEFT_TO_RIGHT) {
      he_next_src_end = ARR_MIN_END;
      he_next_tgt_end = ARR_MAX_END;
    }
    else {
      he_next_src_end = ARR_MAX_END;
      he_next_tgt_end = ARR_MIN_END;
    }

    ps_x_next = parameter_space_in_x(he->next()->curve(), he_next_src_end);
    ps_y_next = parameter_space_in_y(he->next()->curve(), he_next_src_end);
    CGAL_assertion(!is_open(ps_x_next, ps_y_next));

    ps_x_save = parameter_space_in_x(he->next()->curve(), he_next_tgt_end);
    ps_y_save = parameter_space_in_y(he->next()->curve(), he_next_tgt_end);

    // If the halfedge is directed from right to left and its successor is
    // directed from left to right, the target vertex might be the smallest:
    if ((he->direction() == ARR_RIGHT_TO_LEFT) &&
        (he->next()->direction() == ARR_LEFT_TO_RIGHT))
      *local_mins_it++  = std::make_pair(he, x_index);

    _compute_indices(ps_x_curr, ps_y_curr, ps_x_next, ps_y_next,
                     x_index, y_index, Has_identified_sides());

    // Move to the next halfedge.
    he = he->next();
  }

  ps_x_curr = ps_x_he_to;
  ps_y_curr = ps_y_he_to;
  ps_x_next = ps_x_cv_to;
  ps_y_next = ps_y_cv_to;

  CGAL_assertion(!is_open(ps_x_curr, ps_y_curr));
  CGAL_assertion(!is_open(ps_x_next, ps_y_next));

  if ((he_to->direction() == ARR_RIGHT_TO_LEFT) &&
      (cv_dir == ARR_LEFT_TO_RIGHT))
    *local_mins_it++  = std::make_pair(he_to, x_index);

  _compute_indices(ps_x_curr, ps_y_curr, ps_x_next, ps_y_next, x_index, y_index,
                   Has_identified_sides());

  return (std::make_pair(CGAL::sign(x_index), CGAL::sign(y_index)));
}

// Computes the signs of a closed ccb (loop) when deleting he_anchor and its
// opposite belonging to different faces for the case where non of the
// boundaries is identified, thus, return the pair (ZERO, ZERO)
template <typename GeomTraits, typename TopTraits>
std::pair<Sign, Sign>
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compute_signs(const DHalfedge* /* he_anchor */, Arr_false) const
{ return (std::make_pair(ZERO, ZERO)); }

  // Computes the signs of a closed ccb (loop) when deleting he_anchor and its
// opposite belonging to different faces.
template <typename GeomTraits, typename TopTraits>
std::pair<Sign, Sign>
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compute_signs(const DHalfedge* he_anchor, Arr_true) const
{
  // We go over the sequence of vertices, starting from he_before's target
  // vertex, until reaching he_after's source vertex, and find the leftmost
  // one. Note that we do this carefully, keeping track of the number of
  // times we crossed the identification curve in x or in y (if they exist).
  // Note that the path must not be incident to any vertex on open boundary.
  typename Traits_adaptor_2::Parameter_space_in_x_2 parameter_space_in_x =
    m_geom_traits->parameter_space_in_x_2_object();
  typename Traits_adaptor_2::Parameter_space_in_y_2 parameter_space_in_y =
    m_geom_traits->parameter_space_in_y_2_object();
  // typename Traits_adaptor_2::Compare_y_at_x_right_2 compare_y_at_x_right_2 =
  //   m_geom_traits->compare_y_at_x_right_2_object();

  // IDEA EBEB 2012-07-28 store indices of local_minima with CCB in DCEL:
  // - determine values upon insertion of a curve
  // - or if this is not possible, perform the following computation
  //   on-demand only

  // init with edges at first link
  // assuming that he_anchor has been removed
  const DHalfedge* he_curr = he_anchor;
  CGAL_assertion(! he_curr->has_null_curve());
  const DHalfedge* he_next = he_anchor->next();
  // init edge where loop should end
  const DHalfedge* he_end = he_anchor;

  int x_index = 0;
  int y_index = 0;

  // obtain the parameter space pair of he_curr
  Arr_curve_end he_curr_tgt_end =
    (he_curr->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
  Arr_parameter_space ps_x_save =
    parameter_space_in_x(he_curr->curve(), he_curr_tgt_end);
  Arr_parameter_space ps_y_save =
    parameter_space_in_y(he_curr->curve(), he_curr_tgt_end);

  Arr_parameter_space ps_x_curr, ps_y_curr;
  Arr_parameter_space ps_x_next, ps_y_next;

  // start loop
  do {
    ps_x_curr = ps_x_save;
    ps_y_curr = ps_y_save;
    CGAL_assertion(!is_open(ps_x_curr, ps_y_curr));

    Arr_curve_end he_next_src_end =
      (he_next->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MIN_END : ARR_MAX_END;
    ps_x_next = parameter_space_in_x(he_next->curve(), he_next_src_end);
    ps_y_next = parameter_space_in_y(he_next->curve(), he_next_src_end);
    CGAL_assertion(!is_open(ps_x_next, ps_y_next));

    Arr_curve_end he_next_tgt_end =
      (he_next->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
    ps_x_save = parameter_space_in_x(he_next->curve(), he_next_tgt_end);
    ps_y_save = parameter_space_in_y(he_next->curve(), he_next_tgt_end);

    _compute_indices(ps_x_curr, ps_y_curr, ps_x_next, ps_y_next,
                     x_index, y_index, Has_identified_sides());

    // iterate
    he_curr = he_next;
    he_next = he_next->next();
  } while (he_curr != he_end);

  // Return the leftmost vertex and its x_index (with respect to he_before).
  return (std::make_pair(CGAL::sign(x_index), CGAL::sign(y_index)));
}

// Computes the halfedge that points at the smallest vertex in a closed ccb
// when deleting he_anchor and its opposite belonging to same face
// (loop-about-to-split).
template <typename GeomTraits, typename TopTraits>
std::pair<std::pair<Sign, Sign>,
          const typename Arrangement_on_surface_2<GeomTraits,
                                                  TopTraits>::DHalfedge*>
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_compute_signs_and_min(const DHalfedge* he_anchor,
                       Arr_parameter_space& ps_x_min,
                       Arr_parameter_space& ps_y_min,
                       int& index_min) const
{
  // Initialize
  const DHalfedge* he_min = nullptr;
  ps_x_min = ARR_INTERIOR;
  ps_y_min = ARR_INTERIOR;
  index_min = 0;

  // We go over the sequence of vertices, starting from he_before's target
  // vertex, until reaching he_after's source vertex, and find the leftmost
  // one. Note that we do this carefully, keeping track of the number of
  // times we crossed the identification curve in x or in y (if they exist).
  // Note that the path must not be incident to any vertex on open boundary.
  typename Traits_adaptor_2::Parameter_space_in_x_2 parameter_space_in_x =
    m_geom_traits->parameter_space_in_x_2_object();
  typename Traits_adaptor_2::Parameter_space_in_y_2 parameter_space_in_y =
    m_geom_traits->parameter_space_in_y_2_object();

  // init with edges at first link.
  // assuming that he_anchor has been removed
  const DHalfedge* he_curr = he_anchor->opposite()->prev();
  const DHalfedge* he_next = he_anchor->next();
  // init edge where loop should end
  const DHalfedge* he_end = he_anchor->opposite();

  int x_index = 0;
  int y_index = 0;

  // obtain the parameter space pair of he_curr
  Arr_parameter_space ps_x_save, ps_y_save;
  if (he_curr->has_null_curve()) {
    ps_x_save = he_curr->vertex()->parameter_space_in_x();
    ps_y_save = he_curr->vertex()->parameter_space_in_y();
  }
  else {
    Arr_curve_end he_curr_tgt_end =
      (he_curr->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
    ps_x_save = parameter_space_in_x(he_curr->curve(), he_curr_tgt_end);
    ps_y_save = parameter_space_in_y(he_curr->curve(), he_curr_tgt_end);
  }

  // TODO EBEB 2012-09-20 check whether this fix is correct
  // EBEB 2012-08-22 the 'start' of one (out of two) loops might
  // be directed towards the identification.
  // In this cases, we have to adapt the index:
  int x_correction = 0;
  if (ps_x_save == ARR_RIGHT_BOUNDARY) {
    x_correction--;
  }

  Arr_parameter_space ps_x_curr, ps_y_curr;
  Arr_parameter_space ps_x_next, ps_y_next;

  // Start loop
  do {
    ps_x_curr = ps_x_save;
    ps_y_curr = ps_y_save;

    if (he_next->has_null_curve()) {
      ps_x_next = he_next->opposite()->vertex()->parameter_space_in_x();
      ps_y_next = he_next->opposite()->vertex()->parameter_space_in_y();
      ps_x_save = he_next->vertex()->parameter_space_in_x();
      ps_y_save = he_next->vertex()->parameter_space_in_y();
    }
    else {
      Arr_curve_end he_next_src_end =
        (he_next->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MIN_END : ARR_MAX_END;
      ps_x_next = parameter_space_in_x(he_next->curve(), he_next_src_end);
      ps_y_next = parameter_space_in_y(he_next->curve(), he_next_src_end);

      Arr_curve_end he_next_tgt_end =
        (he_next->direction() == ARR_LEFT_TO_RIGHT) ? ARR_MAX_END : ARR_MIN_END;
      ps_x_save = parameter_space_in_x(he_next->curve(), he_next_tgt_end);
      ps_y_save = parameter_space_in_y(he_next->curve(), he_next_tgt_end);
    }

    // If the halfedge is directed from right to left and its successor is
    // directed from left to right, the target vertex might be the smallest:
    if ((he_curr->direction() == ARR_RIGHT_TO_LEFT) &&
        (he_next->direction() == ARR_LEFT_TO_RIGHT))
    {
      const int index_curr = x_index + x_correction;

      // Test the halfedge incident to the leftmost vertex.
      // Note that we may visit the same vertex several times.

      if ((he_min == nullptr) ||
          (index_curr < index_min) ||
          ((index_curr == index_min) &&
           ((he_curr->vertex() != he_min->vertex()) &&
            _is_smaller(he_curr, ps_x_curr, ps_y_curr,
                        he_min, ps_x_min, ps_y_min,
                        Are_all_sides_oblivious_category()))))
      {
        index_min = index_curr;
        ps_x_min = ps_x_curr;
        ps_y_min = ps_y_curr;
        he_min = he_curr;
      }
    }

    _compute_indices(ps_x_curr, ps_y_curr, ps_x_next, ps_y_next,
                     x_index, y_index, Has_identified_sides());

    // iterate
    he_curr = he_next;
    he_next = he_next->next();

    CGAL_assertion(he_curr != he_anchor);

  } while (he_next != he_end);

  // Return the leftmost vertex and the signs.
  return std::make_pair(std::make_pair(CGAL::sign(x_index), CGAL::sign(y_index)), he_min);
}

/* This is the implementation for the case where all 4 boundary sides are
 * oblivious.
 */
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_smaller(const DHalfedge* he1,
            Arr_parameter_space /* ps_x1 */, Arr_parameter_space /* ps_y1 */,
            const DHalfedge* he2,
            Arr_parameter_space /* ps_x2 */, Arr_parameter_space /* ps_y2 */,
            Arr_all_sides_oblivious_tag) const
{
  CGAL_precondition(he1->direction() == ARR_RIGHT_TO_LEFT);
  CGAL_precondition(he2->direction() == ARR_RIGHT_TO_LEFT);
  CGAL_precondition(he1->vertex() != he2->vertex());
  return
    (m_geom_traits->compare_xy_2_object()(he1->vertex()->point(),
                                          he2->vertex()->point()) == SMALLER);
}

/* This is a wrapper for the case where any boundary side is not
 * necessarily oblivious.
 */
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_smaller(const DHalfedge* he1,
            Arr_parameter_space ps_x1, Arr_parameter_space ps_y1,
            const DHalfedge* he2,
            Arr_parameter_space ps_x2, Arr_parameter_space ps_y2,
            Arr_not_all_sides_oblivious_tag tag) const
{
  CGAL_precondition(he1->direction() == ARR_RIGHT_TO_LEFT);
  CGAL_precondition(he2->direction() == ARR_RIGHT_TO_LEFT);
  CGAL_precondition(he1->vertex() != he2->vertex());

  /* If he1 points to a vertex on the left or the bottom boundary, then it
   * is the smaller.
   */
  if ((ps_x1 == ARR_LEFT_BOUNDARY) || (ps_y1 == ARR_BOTTOM_BOUNDARY))
    return true;

  /* If he2 points to a vertex on the left or the bottom boundary, then it
   * is the smaller.
   */
  if ((ps_x2 == ARR_LEFT_BOUNDARY) || (ps_y2 == ARR_BOTTOM_BOUNDARY))
    return false;

  return _is_smaller(he1->curve(), he1->vertex()->point(), ps_x1, ps_y1,
                     he2->curve(), he2->vertex()->point(), ps_x2, ps_y2, tag);
}

/* This is the implementation for the case where all 4 boundary sides are
 * oblivious.
 */
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_smaller(const X_monotone_curve_2& /* cv1 */, const Point_2& p1,
            Arr_parameter_space /* ps_x1 */, Arr_parameter_space /* ps_y1 */,
            const X_monotone_curve_2& /* cv2 */, const Point_2& p2,
            Arr_parameter_space /* ps_x2 */, Arr_parameter_space /* ps_y2 */,
            Arr_all_sides_oblivious_tag) const
{
  CGAL_precondition(! m_geom_traits->equal_2_object()(p1, p2));
  return (m_geom_traits->compare_xy_2_object()(p1, p2) == SMALLER);
}

/*! This is the implementation for the case where any boundary side is not
 * necessarily oblivious.
 * This can be further refined as the combination of LEFT and LEFT can occur
 * only when the right and left boundary sides are identified.
 */
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_smaller(const X_monotone_curve_2& cv1, const Point_2& p1,
            Arr_parameter_space ps_x1, Arr_parameter_space ps_y1,
            const X_monotone_curve_2& cv2, const Point_2& p2,
            Arr_parameter_space ps_x2, Arr_parameter_space ps_y2,
            Arr_not_all_sides_oblivious_tag) const
{
  CGAL_precondition(! m_geom_traits->equal_2_object()(p1, p2));

  if (ps_x2 == ARR_INTERIOR) {
    if (ps_x1 == ARR_INTERIOR) {
      if (ps_y2 == ARR_INTERIOR) {
        if (ps_y1 == ARR_INTERIOR)
          return (m_geom_traits->compare_xy_2_object()(p1,p2) == SMALLER);

        // ps1 == {INTERIOR, !INTERIOR}, ps2 == {INTERIOR,INTERIOR},
        Comparison_result res =
          m_geom_traits->compare_x_on_boundary_2_object()(p2, cv1, ARR_MIN_END);
        return
          (res == EQUAL) ? (ps_y1 == ARR_BOTTOM_BOUNDARY) : (res == LARGER);
      }

      if (ps_y1 == ARR_INTERIOR) {
        // ps1 == {INTERIOR,INTERIOR}, ps2 == {INTERIOR,!INTERIOR}
        Comparison_result res =
          m_geom_traits->compare_x_on_boundary_2_object()(p1, cv2, ARR_MIN_END);
        return (res == EQUAL) ? (ps_y2 == ARR_TOP_BOUNDARY) : (res == SMALLER);
      }

      // ps1 == {INTERIOR,!INTERIOR}, ps2 == {INTERIOR,!INTERIOR}
      Comparison_result res =
        m_geom_traits->compare_x_on_boundary_2_object()(cv1, ARR_MIN_END,
                                                        cv2, ARR_MIN_END);
      return (res == EQUAL) ?
        ((ps_y1 == ARR_BOTTOM_BOUNDARY) && (ps_y2 == ARR_TOP_BOUNDARY)) :
        (res == SMALLER);
    }

    // ps_x2 == ARR_INTERIOR, ps_x == ARR_LEFT_BOUNDARY
    CGAL_assertion(ps_x1 == ARR_LEFT_BOUNDARY);
    return true;
  }
  if (ps_x1 == ARR_INTERIOR)
    // ps_x2 == ARR_LEFT_BOUNDARY, ps_x == ARR_INTERIOR
    return false;

  // ps_x2 == ARR_LEFT_BOUNDARY, ps_x == ARR_LEFT_BOUNDARY
  Comparison_result res =
    m_geom_traits->compare_y_on_boundary_2_object()(p1, p2);
  return (res == SMALLER);
}

/* This is the implementation for the case where all 4 boundary sides are
 * oblivious.
 */
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_smaller_near_right(const X_monotone_curve_2& cv1,
                       const X_monotone_curve_2& cv2,
                       const Point_2& p,
                       Arr_parameter_space /* ps_x */,
                       Arr_parameter_space /* ps_y */,
                       Arr_all_sides_oblivious_tag) const
{
  return
    (m_geom_traits->compare_y_at_x_right_2_object()(cv1, cv2, p) == SMALLER);
}

/*! This is the implementation for the case where any one of the 4 boundary
 * sides can be of any type.
 */
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_smaller_near_right(const X_monotone_curve_2& cv1,
                       const X_monotone_curve_2& cv2,
                       const Point_2& p,
                       Arr_parameter_space ps_x, Arr_parameter_space ps_y,
                       Arr_not_all_sides_oblivious_tag) const
{
  CGAL_precondition((ps_x == ARR_INTERIOR) || (ps_x == ARR_LEFT_BOUNDARY));
  CGAL_precondition((ps_y == ARR_INTERIOR) || (ps_x == ARR_BOTTOM_BOUNDARY));

  if ((ps_x == ARR_INTERIOR) && (ps_y == ARR_INTERIOR))
    return
      (m_geom_traits->compare_y_at_x_right_2_object()(cv1, cv2, p) == SMALLER);
  return
    (m_geom_traits->compare_y_near_boundary_2_object()(cv1, cv2, ARR_MIN_END) ==
     SMALLER);
}

//-----------------------------------------------------------------------------
// Determine whether a given subsequence (halfedge, curve, halfedge)
// lies in the interior of a new face we are about to create
// Comment: This is how the situation looks
//    ----to--->  >>cv_dir>>  ---away--->
//               o ===cv=== 0
//    <-tonext--              <-awaynext-
// or to be read from right to left ... this way, he_to and he_away lie
// BEFORE insertion on the same inner ccb and
// AFTER insertion on the same outer ccb
//
// Precondition: The range of local minima [lm_begin,lm_end) is greater than 0.
//   That is, there is at least one local minimum, which might be the leftend
//   of cv itself.
// Precondition: If the leftend of cv is a local minimum, it must be the first
//   in the range.
template <typename GeomTraits, typename TopTraits>
template <typename InputIterator>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_defines_outer_ccb_of_new_face(const DHalfedge* he_to,
                               const X_monotone_curve_2& cv,
                               const DHalfedge* he_away,
                               InputIterator lm_begin,
                               InputIterator lm_end) const
{
  // Rcpp::Rcout << "_defines_outer_ccb_of_new_face" << std::endl;
  // Search for the leftmost vertex among the local minima
  auto parameter_space_in_x = m_geom_traits->parameter_space_in_x_2_object();
  auto parameter_space_in_y = m_geom_traits->parameter_space_in_y_2_object();

  // check all reported local minima
  InputIterator lm_it = lm_begin;

  int index_min = lm_it->second;
  const DHalfedge* he_min = lm_it->first;
  const DVertex* v_min =
    (he_min == nullptr) ? he_away->opposite()->vertex() : he_min->vertex();
  const X_monotone_curve_2* cv_min =
    (he_min == nullptr) ? &cv : &(he_min->curve());
  Arr_parameter_space ps_x_min = parameter_space_in_x(*cv_min, ARR_MIN_END);
  Arr_parameter_space ps_y_min = parameter_space_in_y(*cv_min, ARR_MIN_END);

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "1 set global min to " << *cv_min << std::endl;
#endif

  for (++lm_it; lm_it != lm_end; ++lm_it) {
    const DHalfedge* he = lm_it->first;
    CGAL_assertion(he->direction() == CGAL::ARR_RIGHT_TO_LEFT);
    int index = lm_it->second;
    Arr_parameter_space ps_x_he_min =
      parameter_space_in_x(he->curve(), ARR_MIN_END);
    Arr_parameter_space ps_y_he_min =
      parameter_space_in_y(he->curve(), ARR_MIN_END);

    // If the following condition is met, the vertex is indeed the smallest:
    // The current x_index is smaller than the x_index of the smallest
    // recorded, or
    // The current x_index is equivalent to the recorded x_index, and
    //   - No smallest has bin recorded so far, or
    //   - The current target vertex and the recorded vertex are the same and
    //       * The current curve is smaller than the recorded curve, or
    //   - The current curve end is smaller than the recorded curve end.
    // smaller than its source, so we should check whether it is also smaller
    // Note that we compare the vertices lexicographically: first by the
    // indices, then by x, then by y.

    if ((index < index_min) ||
        ((index == index_min) &&
         ((v_min == he->vertex()) ?
          _is_smaller_near_right(he->curve(), *cv_min,
                                 v_min->point(), ps_x_min, ps_y_min,
                                 Are_all_sides_oblivious_category()) :
          _is_smaller(he->curve(), he->vertex()->point(),
                      ps_x_he_min, ps_y_he_min,
                      *cv_min, v_min->point(), ps_x_min, ps_y_min,
                      Are_all_sides_oblivious_category()))))
    {
      index_min = index;
      cv_min = &(he->curve());
      ps_x_min = ps_x_he_min;
      ps_y_min = ps_y_he_min;
      he_min = he;
      v_min = he->vertex();
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << "2 set global min to " << *cv_min << std::endl;
#endif
    }
  }

  CGAL_assertion(v_min != nullptr);
  CGAL_assertion(!v_min->has_null_point());

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
  Rcpp::Rcout << "v_min: " << v_min->point() << std::endl;
  Rcpp::Rcout << "he_min: ";
  if (he_min)
    Rcpp::Rcout << he_min->opposite()->vertex()->point()
              << " => " << he_min->vertex()->point();
  else Rcpp::Rcout << "nullptr";
  Rcpp::Rcout << std::endl;
#endif

  CGAL_assertion(! he_min || (he_min->direction() == ARR_RIGHT_TO_LEFT));

  // Note that the curves of the leftmost edge and its successor are defined
  // to the right of the leftmost vertex. We compare them to the right of this
  // point to determine whether he_to (the curve) and he_away are incident to
  // the hole to be created or not.
  const X_monotone_curve_2& cv_next = (he_min == nullptr) ?
    he_away->curve() : ((he_min == he_to) ? cv : he_min->next()->curve());
  return _is_above(*cv_min, cv_next, v_min->point(), ps_y_min,
                   Top_or_bottom_sides_category());
}

// Is the first given x-monotone curve above the second given?
// This function is invoked when the bottom and top boundaries are neither
// identified nor contracted
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_above(const X_monotone_curve_2& xcv1, const X_monotone_curve_2& xcv2,
          const Point_2& point,
          Arr_parameter_space /* ps_y1 */,
          Arr_boundary_cond_tag) const
{
  return (m_geom_traits->compare_y_at_x_right_2_object()(xcv1, xcv2, point) ==
          LARGER);
}

// Is the first given x-monotone curve above the second given?
// This function is invoked when the bottom and top boundaries are identified
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_above(const X_monotone_curve_2& xcv1, const X_monotone_curve_2& xcv2,
          const Point_2& point,
          Arr_parameter_space ps_y1,
          Arr_has_identified_side_tag) const
{
  // Check whether the vertex lies on the identification curve in y,
  // in which case special care must be taken.
  if ((ps_y1 == ARR_BOTTOM_BOUNDARY) || (ps_y1 == ARR_TOP_BOUNDARY)) {
    // TODO EBEB 2010-10-08 is this code really executed or should it be part
    // of top traits?

    // Both current and next curves are incident to the identification curve.
    // As v_min is the leftmost vertex, we know that their left ends must have
    // a boundary condition of type identification in y.
    Arr_parameter_space ps_y2 =
      m_geom_traits->parameter_space_in_y_2_object()(xcv2, ARR_MIN_END);

    // Check if the curves lie on opposite sides of the identification curve.
    if ((ps_y1 == ARR_BOTTOM_BOUNDARY) && (ps_y2 == ARR_TOP_BOUNDARY))
      // In this case the current curve is "above" the next one to the right
      // of v_min, in a cyclic order around the identification curve.
      return true;

    if ((ps_y1 == ARR_TOP_BOUNDARY) && (ps_y2 == ARR_BOTTOM_BOUNDARY))
      // In this case the current curve is "below" the next one to the right
      // of v_min, in a cyclic order around the identification curve.
      return false;

    // If both curves are on the same side of the identification curve, we
    // continue to compare them to the right of v_min.
    CGAL_assertion(((ps_y1 == ARR_BOTTOM_BOUNDARY) &&
                    (ps_y2 == ARR_BOTTOM_BOUNDARY)) ||
                   ((ps_y1 == ARR_TOP_BOUNDARY) &&
                    (ps_y2 == ARR_TOP_BOUNDARY)));
  }

  return _is_above(xcv1, xcv2, point, ps_y1, Arr_all_sides_oblivious_tag());
}

// Is the first given x-monotone curve above the second given?
// This function is invoked when the bottom or top boundaries are contracted
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_above(const X_monotone_curve_2& xcv1, const X_monotone_curve_2& xcv2,
          const Point_2& point,
          Arr_parameter_space ps_y1,
          Arr_has_contracted_side_tag) const
{
  // Check whether the leftmost vertex is a contraction point in y,
  // in which case special care must be taken.
  if (((ps_y1 == ARR_TOP_BOUNDARY) && is_contracted(Bottom_side_category())) ||
      ((ps_y1 == ARR_BOTTOM_BOUNDARY) && is_contracted(Top_side_category())))
  {
    // Compare the horizontal position of the two curve-ends at the point
    // of contraction.
    typename Traits_adaptor_2::Compare_x_curve_ends_2 cmp_x_curve_ends =
      m_geom_traits->compare_x_curve_ends_2_object();
    typename Traits_adaptor_2::Parameter_space_in_x_2 ps_x_op =
      m_geom_traits->parameter_space_in_x_2_object();

    Arr_parameter_space ps_x1 = ps_x_op(xcv1, ARR_MIN_END);
    Arr_parameter_space ps_x2 = ps_x_op(xcv2, ARR_MIN_END);
    Comparison_result x_res = (ps_x1 != ps_x2) ?
      ((ps_x1 == ARR_LEFT_BOUNDARY) ? SMALLER :
       ((ps_x1 == ARR_RIGHT_BOUNDARY) ? LARGER :
        ((ps_x2 == ARR_LEFT_BOUNDARY) ? LARGER : SMALLER))) :
      ((ps_x1 != ARR_INTERIOR) ? EQUAL :
       cmp_x_curve_ends(xcv1, ARR_MIN_END, xcv2, ARR_MIN_END));

    // Observe that if x_res == EQUAL the given subsequence is always exterior.
    return (((ps_y1 == ARR_BOTTOM_BOUNDARY) && (x_res == SMALLER)) ||
            ((ps_y1 == ARR_TOP_BOUNDARY) && (x_res == LARGER)));
  }

  return _is_above(xcv1, xcv2, point, ps_y1, Arr_all_sides_oblivious_tag());
}


//-----------------------------------------------------------------------------
// Remove a pair of twin halfedges from the arrangement.
// In case the removal causes the creation of a new hole, the given halfedge
// should point at this hole.
//
template <typename GeomTraits, typename TopTraits>
typename Arrangement_on_surface_2<GeomTraits, TopTraits>::DFace*
Arrangement_on_surface_2<GeomTraits, TopTraits>::
_remove_edge(DHalfedge* e, bool remove_source, bool remove_target)
{
  // Obtain the pair of twin edges to be removed, the connected components they
  // belong to and their incident faces.
  DHalfedge* he1 = e;
  DHalfedge* he2 = e->opposite();
  DInner_ccb* ic1 = (he1->is_on_inner_ccb()) ? he1->inner_ccb() : nullptr;
  DOuter_ccb* oc1 = (ic1 == nullptr) ? he1->outer_ccb() : nullptr;
  DFace* f1 = (oc1 != nullptr) ? oc1->face() : ic1->face();
  DInner_ccb* ic2 = (he2->is_on_inner_ccb()) ? he2->inner_ccb() : nullptr;
  DOuter_ccb* oc2 = (ic2 == nullptr) ? he2->outer_ccb() : nullptr;
  DFace* f2 = (oc2 != nullptr) ? oc2->face() : ic2->face();

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
#if 0
  Rcpp::Rcout << "before swap" << std::endl;
  Rcpp::Rcout << "he1c: " << he1->curve() <<  ", " << he1->direction()
            << std::endl;
  Rcpp::Rcout << "he2c: " << he2->curve() <<  ", " << he2->direction()
            << std::endl;
  Rcpp::Rcout << "he1: " << he1 << std::endl;
  Rcpp::Rcout << "he2: " << he2 << std::endl;
  Rcpp::Rcout << "ic1: " << ic1 << std::endl;
  Rcpp::Rcout << "ic2: " << ic2 << std::endl;
  Rcpp::Rcout << "oc1: " << oc1 << std::endl;
  Rcpp::Rcout << "oc2: " << oc2 << std::endl;
  Rcpp::Rcout << "f1 : " << f1 << std::endl;
  Rcpp::Rcout << "f2 : " << f2 << std::endl;
#endif
#endif

  // will be used for "_hole_creation_on_edge_removal"
  std::pair< CGAL::Sign, CGAL::Sign > signs1(ZERO, ZERO);
  std::pair< CGAL::Sign, CGAL::Sign > signs2(ZERO, ZERO);

  bool swap_he1_he2 = false;
  if (f1 != f2) {
    // If f1 != f2, the removal of he1 (and its twin halfedge) will cause
    // the two incident faces to merge. Thus, swapping is not needed. However,
    // it is more efficient to retain the face that has a larger number of
    // inner ccbs.

    // f1 is the face to be kept by default. If f2 has more holes it is more
    // efficient to kept f2
    if (f1->number_of_inner_ccbs() < f2->number_of_inner_ccbs())
      swap_he1_he2 = true;
  }
  else {
    // If f1 == f2 (same_face-case), then we consider two loops that occur when
    // he1 and he2 get removed; if f1 != f2, then he1 and he2 separates the two
    // faces that will be merged upon their removal---here both he1 and he2
    // belong to a full cycle, and THAT IS WHY we give the f1 == f2 test to
    // determine whether end of loop should be he1->opposite() and
    // he2->opposite(), respectively.

    // If the removal of he1 (and its twin halfedge) form an "antenna", there
    // is neither a need to compute signs and nor swapping of the halfedges
    if ((he1->next() != he2) && (he2->next() != he1)) {

      // In this case one of the following can happen: (a) a new hole will be
      // created by the removal of the edge (case 3.2.1 of the removal
      // procedure), or (b) an outer CCB will be split into two (case 3.2.2).
      // We begin by locating the leftmost vertex along the path from he1 to its
      // twin he2 and the leftmost vertex point along the path from the twin to
      // he1 (both paths do not include he1 and he2 themselves).

      // Comment EFEF 2013-05-31: if we ever find the need to use signs1 and
      // signs2 out of this scope (for the non-planar case), the code must be
      // dispatched, so that the planar case is not affected.

      // Compute signs of ccbs for he1 and he2 used later for
      // _hole_creation_on_edge_removal

      // Compute the signs and minimum along ccb of he1:
      Arr_parameter_space ps_x_min1, ps_y_min1;
      int index_min1;
      std::pair<std::pair<Sign, Sign>, const DHalfedge*> res1 =
        _compute_signs_and_min(he1, ps_x_min1, ps_y_min1, index_min1);
      signs1 = res1.first;
      const DHalfedge* he_min1 = res1.second;

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << "signs1.x: " << signs1.first << std::endl;
      Rcpp::Rcout << "signs1.y: " << signs1.second << std::endl;
      if (! he_min1->has_null_curve())
        Rcpp::Rcout << "he_min1: " << he_min1->curve() << std::endl;
      else Rcpp::Rcout << "he_min1 fictitious" << std::endl;
#endif

      // Compute the signs and minimum along ccb of he2:
      Arr_parameter_space ps_x_min2, ps_y_min2;
      int index_min2;
      std::pair<std::pair<Sign, Sign>, const DHalfedge*> res2 =
        _compute_signs_and_min(he2, ps_x_min2, ps_y_min2, index_min2);
      signs2 = res2.first;
      const DHalfedge* he_min2 = res2.second;

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << "signs2.x: " << signs2.first << std::endl;
      Rcpp::Rcout << "signs2.y: " << signs2.second << std::endl;
      if (! he_min2->has_null_curve())
        Rcpp::Rcout << "he_min2: " << he_min2->curve() << std::endl;
      else Rcpp::Rcout << "he_min2 fictitious" << std::endl;
#endif

      // TODO EBEB 2012-07-29
      // is this the right thing to do for torus, or let TopTraits decide?
      bool is_perimetric1 = signs1.first || signs1.second;
      bool is_perimetric2 = signs2.first || signs2.second;

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << std::endl
                << "index 1: " << index_min1
                << ", ps_x_min1: " << ps_x_min1
                << ", ps_y_min1: " << ps_y_min1
                << ", is_perimetric1: " << is_perimetric1
                << std::endl;

      Rcpp::Rcout << "index 2: " << index_min2
                << ", ps_x_min2: " << ps_x_min2
                << ", ps_y_min2: " << ps_y_min2
                << ", is_perimetric2: " << is_perimetric2
                << std::endl;
#endif

      if (is_perimetric1 || is_perimetric2) {
#if 1 // this is old code
        swap_he1_he2 =
          (! is_perimetric1) ? false :
          ((! is_perimetric2) ? true : false);
          // We are in case (a) and he1 is directed to the new hole to be
          // created or
          // We are in case (a) and he2 is directed to the new hole to be
          // created.
        // Both paths are perimetric; thus, we are in case (b).
#else // THIS IS NEW CODE 2012-08-06 which is much easier to read
        swap_he1_he2 = !is_perimetric2;
#endif
      }
      else {
        // const DVertex* v_min1 = he_min1->vertex(); const DVertex* v_min2 =
        // he_min2->vertex(); Both paths from he1 to he2 and back from he2 to
        // he1 are not perimetric, so we are in case (a). As we want to
        // determine which halfedge points to the new hole to be created (he1
        // or he2), we have to compare the two leftmost vertices
        // lexicographically, first by the indices then by x and y. v_min2
        // lies to the left of v_min1 if and only if he1 points at the hole we
        // are about to create.
        //
        //         +---------------------+
        //         |                     |
        //         |   he1    +----+     |
        //         +--------->+    |     |
        //         |          +----+     |
        //         |      v_min1         |
        //         |                     |
        //  v_min2 +---------------------+
        //
        // Note that if one of the paths we have examined ends at a boundary
        // side of the parameter space (and only of the paths may end at a
        // boundary side of the parameter space), then the other path becomes
        // a hole in a face bounded by the parameter-space boundary.

        // TODO EBEB 2012-08-22 check whether this fix is correct
        // EBEB 2012-08-22 the 'start' of the two loops might lie
        // on different sides of the identification, which is only
        // problematic when either he1 or he2 points to the
        // identification. In these cases, we have to adapt the indices:
        typename Traits_adaptor_2::Parameter_space_in_x_2
          parameter_space_in_x =
          m_geom_traits->parameter_space_in_x_2_object();

        Arr_curve_end he1_tgt_end =
          (he1->direction() == ARR_LEFT_TO_RIGHT ? ARR_MAX_END : ARR_MIN_END);
        Arr_parameter_space ps_x_he1_tgt =
          parameter_space_in_x(he1->curve(), he1_tgt_end);
        if (ps_x_he1_tgt == ARR_RIGHT_BOUNDARY) index_min2 -= 1;

        Arr_curve_end he2_tgt_end =
          (he2->direction() == ARR_LEFT_TO_RIGHT ? ARR_MAX_END : ARR_MIN_END);
        Arr_parameter_space ps_x_he2_tgt =
          parameter_space_in_x(he2->curve(), he2_tgt_end);
        if (ps_x_he2_tgt == ARR_RIGHT_BOUNDARY) index_min1 -= 1;

        swap_he1_he2 =
          (index_min1 > index_min2) ? false :
          ((index_min1 < index_min2) ? true :
           _is_smaller(he_min1, ps_x_min1, ps_y_min1,
                       he_min2, ps_x_min2, ps_y_min2,
                       Are_all_sides_oblivious_category()));
      }
    }
  }
  // swapping?
  if (swap_he1_he2) {
    // swap all entries
    std::swap(he1, he2);
    std::swap(ic1, ic2);
    std::swap(oc1, oc2);
    std::swap(f1 , f2);
    // not needed below here std::swap(local_mins1, local_mins2);
    std::swap(signs1, signs2);
  }

#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
#if 0
  Rcpp::Rcout << "after swap" << std::endl;
  Rcpp::Rcout << "he1c: " << he1->curve() <<  ", " << he1->direction()
            << std::endl;
  Rcpp::Rcout << "he1c: " << he2->curve() <<  ", " << he2->direction()
            << std::endl;
  Rcpp::Rcout << "he1: " << he1 << std::endl;
  Rcpp::Rcout << "he2: " << he2 << std::endl;
  Rcpp::Rcout << "ic1: " << ic1 << std::endl;
  Rcpp::Rcout << "ic2: " << ic2 << std::endl;
  Rcpp::Rcout << "oc1: " << oc1 << std::endl;
  Rcpp::Rcout << "oc2: " << oc2 << std::endl;
  Rcpp::Rcout << "f1 : " << f1 << std::endl;
  Rcpp::Rcout << "f2 : " << f2 << std::endl;
#endif
#endif

  // Now the real removal starts.
  DHalfedge* prev1 = nullptr;
  DHalfedge* prev2 = nullptr;

  // Notify the observers that we are about to remove an edge.
  Halfedge_handle  hh(e);

  _notify_before_remove_edge(hh);

  // Check if the two incident faces are equal, in which case no face will be
  // merged and deleted (and a hole may be created).
  if (f1 == f2) {
    // Check whether the two halfedges are successors along the face boundary.
    if ((he1->next() == he2) && (he2->next() == he1)) {
      CGAL_assertion((ic1 != nullptr) && (ic1 == ic2));

      // The two halfedges form a "singleton" hole inside the incident face
      // (case 1 of the removal procedure, as detailed in the design document),
      // so we simply have to remove it.

      // Notify before the removal of this hole (inner CCB).
      // Erase the inner CCB from the incident face.
      // Delete the corresponding component.
      // Notify after the removal.
      Face_handle fh(f1);
      _notify_before_remove_inner_ccb(fh, (Halfedge_handle(he1))->ccb());
      f1->erase_inner_ccb(ic1);
      _dcel().delete_inner_ccb(ic1);
      _notify_after_remove_inner_ccb(fh);

      DVertex* v1 = he1->vertex();
      DVertex* v2 = he2->vertex();

      _delete_curve(he1->curve());
      _dcel().delete_edge(he1);

#ifndef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
      _notify_after_remove_edge();
#endif

      // Remove the end-vertices, if necessary.
      if (remove_target) {
        if ((v1->parameter_space_in_x() != ARR_INTERIOR) ||
            (v1->parameter_space_in_y() != ARR_INTERIOR))
        {
          v1->set_halfedge(nullptr);    // disconnect the end vertex
          _remove_vertex_if_redundant(v1, f1);
        }
        else {
          // Delete the he1's target vertex and its associated point.
          _notify_before_remove_vertex(Vertex_handle(v1));

          _delete_point(v1->point());
          _dcel().delete_vertex(v1);

          _notify_after_remove_vertex();
        }
      }
      else
        // The remaining target vertex now becomes an isolated vertex inside
        // the containing face:
        _insert_isolated_vertex(f1, v1);

      if (remove_source) {
        if ((v2->parameter_space_in_x() != ARR_INTERIOR) ||
            (v2->parameter_space_in_y() != ARR_INTERIOR))
        {
          v2->set_halfedge(nullptr);    // disconnect the end vertex
          _remove_vertex_if_redundant(v2, f1);
        }
        else {
          // Delete the he1's source vertex and its associated point.
          _notify_before_remove_vertex(Vertex_handle(v2));

          _delete_point(v2->point());
          _dcel().delete_vertex(v2);

          _notify_after_remove_vertex();
        }
      }
      else
        // The remaining source vertex now becomes an isolated vertex inside
        // the containing face:
        _insert_isolated_vertex(f1, v2);

#ifdef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
      _notify_after_remove_edge();
#endif

      // Return the face that used to contain the hole.
      return f1;
    }

    if ((he1->next() == he2) || (he2->next() == he1)) {
      CGAL_assertion((oc1 == oc2) && (ic1 == ic2));

      // In this case the two halfedges form an "antenna" (case 2).
      // Make he1 point at the tip of this "antenna" (swap the pointer if
      // necessary).
      bool remove_tip_vertex = remove_target;

      if (he2->next() == he1) {
        he1 = he2;
        he2 = he1->opposite();
        remove_tip_vertex = remove_source;
      }

      // Remove the two halfedges from the boundary chain by connecting
      // he1's predecessor with he2's successor.
      prev1 = he1->prev();
      prev1->set_next(he2->next());

      // In case the halfedges to be deleted are represantatives of their
      // CCB (note that noth should belong to the same CCB, be it an outer
      // CCB or an inner one), make prev1 the components representative.
      if ((oc1 != nullptr) &&
          ((oc1->halfedge() == he1) || (oc1->halfedge() == he2)))
        oc1->set_halfedge(prev1);
      else if ((ic1 != nullptr) &&
               ((ic1->halfedge() == he1) || (ic1->halfedge() == he2)))
        ic1->set_halfedge(prev1);

      // In case he2 is the representative halfedge of its target vertex,
      // replace it by prev1 (which also points at this vertex).
      if (he2->vertex()->halfedge() == he2)
        he2->vertex()->set_halfedge(prev1);

      DVertex* v1 = he1->vertex();
      DVertex* v2 = he2->vertex();

      _delete_curve(he1->curve());
      _dcel().delete_edge(he1);

#ifndef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
      _notify_after_remove_edge();
#endif

      // Try to temove the base vertex, in case it has boundary conditions.
      if ((v2->parameter_space_in_x() != ARR_INTERIOR) ||
          (v2->parameter_space_in_y() != ARR_INTERIOR))
        _remove_vertex_if_redundant(v2, f1);

      // Remove the redundant tip vertex, if necessary.
      if (remove_tip_vertex) {
        if ((v1->parameter_space_in_x() != ARR_INTERIOR) ||
            (v1->parameter_space_in_y() != ARR_INTERIOR))
        {
          v1->set_halfedge(nullptr);    // disconnect the end vertex
          _remove_vertex_if_redundant(v1, f1);
        }
        else {
          // Delete the vertex that forms the tip of the "antenna".
          _notify_before_remove_vertex(Vertex_handle(v1));

          _delete_point(v1->point());
          _dcel().delete_vertex(v1);

          _notify_after_remove_vertex();
        }
      }
      else
        // The remaining "antenna" tip now becomes an isolated vertex inside
        // the containing face:
        _insert_isolated_vertex(f1, v1);

#ifdef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
      _notify_after_remove_edge();
#endif

      // Return the incident face.
      return f1;
    }

    // In this case the degree of both end-vertices is at least 2, so we
    // can use the two predecessor halfedges of he1 and he2.
    bool        add_inner_ccb = false;

    prev1 = he1->prev();
    prev2 = he2->prev();

    if ((ic1 != nullptr) && (ic1 == ic2)) {

      // If both halfedges lie on the same inner component (hole) inside the
      // face (case 3.1), we have to split this component into two holes.
      //
      //    +-----------------------------+
      //    |           prev1             |
      //    |   +----+ /    +----+        |
      //    |   |    +......+    |        |
      //    |   +----+      +----+        |
      //    |                             |
      //    +-----------------------------+
      //
      // Notify the observers we are about to split an inner CCB.
      _notify_before_split_inner_ccb(Face_handle(f1),
                                     (Halfedge_handle
                                      (*(ic1->iterator())))->ccb(),
                                     Halfedge_handle(he1));

      // We first make prev1 the new representative halfedge of the first
      // inner CCB.
      ic1->set_halfedge(prev1);

      // Create a new component that represents the new hole we split.
      DInner_ccb* new_ic = _dcel().new_inner_ccb();
      f1->add_inner_ccb(new_ic, prev2);
      new_ic->set_face(f1);

      // Associate all halfedges along the hole boundary with the new inner
      // component.
      DHalfedge* curr;
      for (curr = he1->next(); curr != he2; curr = curr->next())
        curr->set_inner_ccb(new_ic);

      // Notify the observers that the hole has been split.
      _notify_after_split_inner_ccb(Face_handle(f1),
                                    (Halfedge_handle(prev1))->ccb(),
                                    (Halfedge_handle(prev2))->ccb());
    }
    else if (oc1 != oc2) {
      // RWRW: NEW!
      CGAL_assertion((oc1 != nullptr) && (oc2 != nullptr));

      // In case both halfedges he1 and he2 are incident to the same face
      // but lie on different outer CCBs of this face, removing this pair of
      // halfedge causes the two components two merge and to become an
      // inner CCB in the face.
      // We first remove the outer CCB oc1 from f, and inform the observers
      // on doing so.
      Face_handle fh(f1);

      _notify_before_remove_outer_ccb(fh, (Halfedge_handle(he1))->ccb());
      f1->erase_outer_ccb(oc1);
      _dcel().delete_outer_ccb(oc1);
      _notify_after_remove_outer_ccb(fh);

      // We now remove the outer CCBs oc2 from f, and inform the observers
      // on doing so.
      _notify_before_remove_outer_ccb(fh, (Halfedge_handle(he2))->ccb());
      f2->erase_outer_ccb(oc2);
      _dcel().delete_outer_ccb(oc2);
      _notify_after_remove_outer_ccb(fh);

      // Mark that we should eventually add a new inner CCB inside the face.
      add_inner_ccb = true;
    }
    else {
      CGAL_assertion((oc1 != nullptr) && (oc1 == oc2));

      // If both halfedges are incident to the same outer CCB of their
      // face (case 3.2), we have to distinguish two sub-cases:
      // TODO EBEB 2012-07-30 replace with signs
      if (_hole_creation_on_edge_removal(signs1, signs2, true)) {
        // We have to create a new hole in the interior of the incident face
        // (case 3.2.1):
        //
        //    +-----------------------------+
        //    | prev1                       |
        //    v         +----+              |
        //    +........>+    |              |
        //    |   he1   +----+              |
        //    |                             |
        //    +-----------------------------+
        //
        // Note that it is guaranteed that he1 points at this new hole, while
        // he2 points at the boundary of the face that contains this hole.
        // First notify the observers we are about to form a new inner
        // CCB inside f1.
        _notify_before_add_inner_ccb(Face_handle(f1),
                                     Halfedge_handle(he1->next()));

        // Create a new component that represents the new hole.
        DInner_ccb* new_ic = _dcel().new_inner_ccb();

        f1->add_inner_ccb(new_ic, he1->next());
        new_ic->set_face(f1);

        // Associate all halfedges along the hole boundary with the new inner
        // component.
        DHalfedge* curr;
        for (curr = he1->next(); curr != he2; curr = curr->next())
          curr->set_inner_ccb(new_ic);

        // As the outer CCB of f1 may be represented by any of the
        // halfedges in between he1 -> ... -> he2 (the halfedges in between
        // represent the outer boundary of the new hole that is formed),
        // We represent the outer boundary of f1 by prev1, which definitely
        // stays on the outer boundary.
        oc1->set_halfedge(prev1);

        // Notify the observers that a new hole has been formed.
        Ccb_halfedge_circulator hccb = (Halfedge_handle(he1->next()))->ccb();

        _notify_after_add_inner_ccb(hccb);
      }
      else {
        // We have to split the outer CCB into two outer components
        // (case 3.2.2), such that the number of outer CCBs of the face is
        // incremented.
        //
        //    +----------------------------+
        //    |                            |
        //    |            prev1           |
        //    +<........+<.................|
        //    |         |                  |
        //    |         |                  |
        //    |         |                  |
        //    |         |                  |
        //    |  prev2  |                  |
        //    +........>+..................|
        //    |                            |
        //    +----------------------------+
        //

        // First we notify the observers that we are about to split an outer
        // component.
        _notify_before_split_outer_ccb(Face_handle(f1),
                                       Halfedge_handle(he1)->ccb(),
                                       Halfedge_handle(he1));

        // Create a new outer component.
        DOuter_ccb* new_oc = _dcel().new_outer_ccb();

        f1->add_outer_ccb(new_oc, he1->next());
        new_oc->set_face(f1);

        // Associate all halfedges from he1 until he2 with the new CCB.
        DHalfedge* curr;

        for (curr = he1->next(); curr != he2; curr = curr->next())
          curr->set_outer_ccb(new_oc);

        // As the outer CCB of f1 may be represented by any of the
        // halfedges in between he1 -> ... -> he2 (the halfedges in between
        // are on the new outer CCB we have just created), we represent the
        // former outer CCB by prev1, which definitely stays on it.
        oc1->set_halfedge(prev1);

        // Notify the observers that a new outer CCB has been formed.
        _notify_after_split_outer_ccb(Face_handle(f1),
                                      Halfedge_handle(he1->next())->ccb(),
                                      Halfedge_handle(prev1)->ccb());
      }
    }

    // Disconnect the two halfedges we are about to delete from the edge list.
    prev1->set_next(he2->next());
    prev2->set_next(he1->next());

    // If one of these edges is an incident halfedge of its target vertex,
    // replace it by the appropriate predecessor.
    if (he1->vertex()->halfedge() == he1)
      he1->vertex()->set_halfedge(prev2);

    if (he2->vertex()->halfedge() == he2)
      he2->vertex()->set_halfedge(prev1);

    DVertex* v1 = he1->vertex();
    DVertex* v2 = he2->vertex();

    _delete_curve(he1->curve());
    _dcel().delete_edge(he1);

    // RWRW: NEW!
    // In case we have to create a new inner CCB inside the face (new removal
    // case), do it now.
    if (add_inner_ccb) {
      // Notify the observers that we are about to create a new inner CCB
      // inside the merged face.
      Halfedge_handle hh(prev1);

      _notify_before_add_inner_ccb(Face_handle(f1), hh);

      // Initiate a new inner CCB inside the given face.
      DInner_ccb* new_ic = _dcel().new_inner_ccb();

      f1->add_inner_ccb(new_ic, prev1);
      new_ic->set_face(f1);

      // Set the innser CCB of the halfedges along the component boundary.
      DHalfedge* curr = prev1;

      do {
        curr->set_inner_ccb(new_ic);
        curr = curr->next();
      } while (curr != prev1);

      // Notify the observers that we have formed a new inner CCB.
      _notify_after_add_inner_ccb(hh->ccb());
    }

#ifndef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
    _notify_after_remove_edge();
#endif

    // Remove the end vertices, in case they become redundant.
    // We defer the removal of the end vertices (in case they are indeed
    // redundant) to occur after the observer is notified that the edge has
    // been removed, to match the case where the end vertices are associated
    // with concrete points, and need to be removed as they became isolated
    // vertices, and the user has requested the removal of isolated end vertices.
    if ((v1->parameter_space_in_x() != ARR_INTERIOR) ||
        (v1->parameter_space_in_y() != ARR_INTERIOR))
      _remove_vertex_if_redundant(v1, f1);

    if ((v2->parameter_space_in_x() != ARR_INTERIOR) ||
        (v2->parameter_space_in_y() != ARR_INTERIOR))
      _remove_vertex_if_redundant(v2, f1);

#ifdef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
    _notify_after_remove_edge();
#endif

    // Return the incident face.
    return f1;
  }

  CGAL_assertion(f1 != f2);

  // The two incident faces are not the same - in this case, the edge we are
  // about to delete separates these two faces. We therefore have to delete
  // one of these faces and merge it with the other face.
  // First notify the observers we are about to merge the two faces.
  _notify_before_merge_face(Face_handle(f1), Face_handle(f2),
                            Halfedge_handle(he1));

  // We begin by checking whether one of the faces is a hole inside the other
  // face.
  DHalfedge* curr;

  prev1 = he1->prev();
  prev2 = he2->prev();

  CGAL_assertion((ic1 == nullptr) || (ic2 == nullptr));

  if ((ic1 == nullptr) && (ic2 == nullptr)) {
    bool add_inner_ccb = false;

    // Comment EFEF 2013-05-31: if we ever find the need to use signs1 and
    // signs2 out of this scope (for the non-planar case), the code must be
    // dispatched, so that the planar case is not affected.

    // Compute the signs of the ccbs for he1 and he2.
    // The signs are computed here, a sub case of (f1 != f2), in addition to
    // the case (f1 == f2) above. This way unnecessary computations of the
    // signs are avoided.

    // EFEF 2013-07-29. The call to _compute_signs() is dispatched.
    // Currently, only 2 cases are supported, namely, the case where non of
    // the boundaries are identified and the case where at least one pair of
    // opposite boundaries are identified. However, the code for the latter
    // assumes that non of the (other) boundaries is open. A 3rd version
    // that supports the remaining case, (for example the cylinder), should
    // be developed and used.
    signs1 = _compute_signs(he1, Has_identified_sides());
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << "signs1.x: " << signs1.first << std::endl;
      Rcpp::Rcout << "signs1.y: " << signs1.second << std::endl;
#endif

      signs2 = _compute_signs(he2, Has_identified_sides());
#if CGAL_ARRANGEMENT_ON_SURFACE_INSERT_VERBOSE
      Rcpp::Rcout << "signs2.x: " << signs2.first << std::endl;
      Rcpp::Rcout << "signs2.y: " << signs2.second << std::endl;
#endif

    // Both halfedges lie on the outer boundary of their incident faces
    // (case 3.4). We have to distinguish two possible sub-cases.
    // TODO EBEB 2012-07-30 replace with signs
    if (_hole_creation_on_edge_removal(signs1, signs2, false)) {
      // We have to remove the outer CCBs of f1 and f2 that he1 and he2 lie
      // on, and create a new hole in the merged face (case 3.4.2).
      // We first remove the outer CCB oc1 from f1, and inform the observers
      // on doing so.
      _notify_before_remove_outer_ccb(Face_handle(f1),
                                      (Halfedge_handle(he1))->ccb());

      f1->erase_outer_ccb(oc1);
      _dcel().delete_outer_ccb(oc1);

      _notify_after_remove_outer_ccb(Face_handle(f1));

      // We now remove the outer CCBs oc2 from f2, and inform the observers
      // on doing so.
      _notify_before_remove_outer_ccb(Face_handle(f2),
                                      (Halfedge_handle(he2))->ccb());

      f2->erase_outer_ccb(oc2);
      _dcel().delete_outer_ccb(oc2);

      _notify_after_remove_outer_ccb(Face_handle(f2));

      // Mark that we should eventually add a new inner CCB in the merged face.
      add_inner_ccb = true;
    }
    else {
      // f1 and f2 are two adjacent faces (case 3.4.1), so we simply merge
      // them.
      // We first set the connected component of f2's outer-boundary halfedges
      // to be the same as f1's outer component.
      for (curr = he2->next(); curr != he2; curr = curr->next())
        curr->set_outer_ccb(oc1);
    }

    _move_all_inner_ccb(f2, f1);        // move all inner CCBs from f2 to f1

    // In case he1, which is about to be deleted, is a representative
    // halfedge of outer component of f1, we replace it by its predecessor.
    if (oc1->halfedge() == he1)
      oc1->set_halfedge(prev1);

    _move_all_isolated_vertices(f2, f1); // move all iso vertices from f2 to f1

    // If he1 or he2 are the incident halfedges to their target vertices,
    // we replace them by the appropriate predecessors.
    if (he1->vertex()->halfedge() == he1)
      he1->vertex()->set_halfedge(prev2);

    if (he2->vertex()->halfedge() == he2)
      he2->vertex()->set_halfedge(prev1);

    // Disconnect the two halfedges we are about to delete from the edge
    // list.
    prev1->set_next(he2->next());
    prev2->set_next(he1->next());

    // If the face f2 we have just merged with f1 is unbounded, then the
    // merged face is also unbounded.
    if (f2->is_unbounded())
      f1->set_unbounded(true);

    // Delete the face f2.
    _dcel().delete_face(f2);

    // Notify the observers that the faces have been merged.
    _notify_after_merge_face(Face_handle(f1));

    DVertex* v1 = he1->vertex();
    DVertex* v2 = he2->vertex();

    _delete_curve(he1->curve());
    _dcel().delete_edge(he1);

    // In case we have to create a new inner CCB inside the merged face
    // (case 3.4.1), do it now.
    if (add_inner_ccb) {
      // Notify the observers that we are about to create a new inner CCB
      // inside the merged face.
      Halfedge_handle hh(prev1);

      _notify_before_add_inner_ccb(Face_handle(f1), hh);

      // Initiate a new inner CCB inside the given face.
      DInner_ccb* new_ic = _dcel().new_inner_ccb();

      f1->add_inner_ccb(new_ic, prev1);
      new_ic->set_face(f1);

      // Set the innser CCB of the halfedges along the component boundary.
      curr = prev1;
      do {
        curr->set_inner_ccb(new_ic);
        curr = curr->next();
      } while (curr != prev1);

      // Notify the observers that we have formed a new inner CCB.
      _notify_after_add_inner_ccb(hh->ccb());
    }

#ifndef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
    _notify_after_remove_edge();
#endif

    // Remove the end vertices, in case they become redundant.
    // We defer the removal of the end vertices (in case they are indeed
    // redundant) to occur after the observer is notified that the edge has
    // been removed, to match the case where the end vertices are associated
    // with concrete points, and need to be removed as they became isolated
    // vertices, and the user has requested the removal of isolated end vertices.
    if ((v1->parameter_space_in_x() != ARR_INTERIOR) ||
        (v1->parameter_space_in_y() != ARR_INTERIOR))
      _remove_vertex_if_redundant(v1, f1);

    if ((v2->parameter_space_in_x() != ARR_INTERIOR) ||
        (v2->parameter_space_in_y() != ARR_INTERIOR))
      _remove_vertex_if_redundant(v2, f1);

#ifdef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
    _notify_after_remove_edge();
#endif

    // Return the merged face.
    return f1;
  }

  // In this case we merge a face with another face that now forms a hole
  // inside it (case 3.3). We first make sure that f1 contains the hole f2, so
  // we can merge f2 with it (we swap roles between the halfedges if
  // necessary).
  if (ic2 != nullptr) {
    he1 = he2;
    he2 = he1->opposite();

    ic1 = ic2;
    ic2 = nullptr;

    oc2 = oc1;
    oc1 = nullptr;

    DFace* tf = f1;
    f1 = f2;
    f2 = tf;

    prev1 = he1->prev();
    prev2 = he2->prev();
  }

  // By removing the edge we open a closed face f2 contained in f1. By doing
  // this, the outer boundary of f2 unites with the hole boundary that ic1
  // represents. We therefore have to set the component of all halfedges
  // along the boundary of f2 to be ic1.
  for (curr = he2->next(); curr != he2; curr = curr->next())
    curr->set_inner_ccb(ic1);

  _move_all_inner_ccb(f2, f1);          // move the inner CCBs from f2 to f1
  _move_all_isolated_vertices(f2, f1);  // move all iso vertices from f2 to f1

  // Notice that f2 will be merged with f1, but its boundary will still be
  // a hole inside this face. In case he1 is a representative of this hole,
  // replace it by its predecessor.
  if (ic1->halfedge() == he1)
    ic1->set_halfedge(prev1);

  // If he1 or he2 are the incident halfedges to their target vertices,
  // we replace them by the appropriate predecessors.
  if (he1->vertex()->halfedge() == he1)
    he1->vertex()->set_halfedge(prev2);

  if (he2->vertex()->halfedge() == he2)
    he2->vertex()->set_halfedge(prev1);

  // Disconnect the two halfedges we are about to delete from the edge
  // list.
  prev1->set_next(he2->next());
  prev2->set_next(he1->next());

  // If the face f2 we have just merged with f1 is unbounded, then the merged
  // face is also unbounded.
  if (f2->is_unbounded())
    f1->set_unbounded(true);

  // Delete the face f2.
  _dcel().delete_face(f2);

  // Notify the observers that the faces have been merged.
  _notify_after_merge_face(Face_handle(f1));

  DVertex* v1 = he1->vertex();
  DVertex* v2 = he2->vertex();

  _delete_curve(he1->curve());
  _dcel().delete_edge(he1);

#ifndef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
  _notify_after_remove_edge();
#endif

  // Remove the end vertices, in case they become redundant.
  // We defer the removal of the end vertices (in case they are indeed
  // redundant) to occur after the observer is notified that the edge has
  // been removed, to match the case where the end vertices are associated
  // with concrete points, and need to be removed as they became isolated
  // vertices, and the user has requested the removal of isolated end vertices.
  if ((v1->parameter_space_in_x() != ARR_INTERIOR) ||
      (v1->parameter_space_in_y() != ARR_INTERIOR))
    _remove_vertex_if_redundant(v1, f1);

  if ((v2->parameter_space_in_x() != ARR_INTERIOR) ||
      (v2->parameter_space_in_y() != ARR_INTERIOR))
    _remove_vertex_if_redundant(v2, f1);

#ifdef CGAL_NON_SYMETRICAL_OBSERVER_EDGE_REMOVAL_BACKWARD_COMPATIBILITY
  _notify_after_remove_edge();
#endif

  // Return the merged face.
  return f1;

  // TODO EBEB 2012-08-06 it seems that a torus case is missing
}

// Decide whether a hole is created
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_hole_creation_on_edge_removal(std::pair< CGAL::Sign, CGAL::Sign > signs1,
                               std::pair< CGAL::Sign, CGAL::Sign > signs2,
                               bool same_face) {
  // EBEB 2013-07-16 Remark: For tiled surfaces, this function has to respect the
  // topology of the tiled surface

  // TODO EBEB 2013-07-16 Add code for torus (double identification)
  CGAL::Sign sign1 = signs1.first;
  CGAL::Sign sign2 = signs2.first;

  if (same_face) return true;
  return ((CGAL::ZERO != sign1) && (sign1 == opposite(sign2)));
}

//-----------------------------------------------------------------------------
// Remove a vertex in case it becomes redundant after the deletion of an
// incident edge.
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_remove_vertex_if_redundant(DVertex* v, DFace* f)
{
  CGAL_precondition((v->parameter_space_in_x() != ARR_INTERIOR) ||
                    (v->parameter_space_in_y() != ARR_INTERIOR));

  // In case the vertex has no incident halfedges, remove it if it is
  // redundant. Otherwise, make it an isolated vertex.
  if (v->halfedge() == nullptr) {
    if (m_topol_traits.is_redundant(v)) {
      // Remove the vertex and notify the observers on doing so.
      _notify_before_remove_vertex(Vertex_handle(v));

      m_topol_traits.erase_redundant_vertex(v);

      // Note the topology traits do not free the vertex - we now do it.
      if (! v->has_null_point())
        _delete_point(v->point());
      _dcel().delete_vertex(v);

      _notify_after_remove_vertex();
    }
    else
      // Keep the vertex as an isolated one.
      _insert_isolated_vertex(f, v);
    return;
  }

  // Get the first two incident halfedges of v.
  DHalfedge* he1 = v->halfedge();
  DHalfedge* he2 = he1->next()->opposite();

  if (he2->next()->opposite() != he1)
    // In this case there are more than two incident edges, so v obviously
    // cannot be removed.
    return;

  if (! he1->has_null_curve() || ! he2->has_null_curve())
    // We can only merge fictitious halfedges.
    return;

  // Now check if the vertex is redundant. If it is, remove it by merging
  // its two incident fictitious halfedges.
  if (m_topol_traits.is_redundant(v)) {
    // Use the topology traits to merge the two fictitious halfedges.
    _notify_before_merge_fictitious_edge(Halfedge_handle(he1),
                                         Halfedge_handle(he2));

    he1 = m_topol_traits.erase_redundant_vertex(v);

    _notify_after_merge_fictitious_edge(Halfedge_handle(he1));

    // Note the topology traits do not free the vertex - we now do it.
    _notify_before_remove_vertex(Vertex_handle(v));

    if (! v->has_null_point())
      _delete_point(v->point());
    _dcel().delete_vertex(v);

    _notify_after_remove_vertex();
  }
}

//-----------------------------------------------------------------------------
// Remove an isolated vertex from the interior of a given face (but not from
// the DCEL).
//
template <typename GeomTraits, typename TopTraits>
void Arrangement_on_surface_2<GeomTraits, TopTraits>::
_remove_isolated_vertex(DVertex* v)
{
  // Remove the isolated vertex from the face and delete its record.
  DIso_vertex* iv = v->isolated_vertex();
  DFace* f = iv->face();

  f->erase_isolated_vertex(iv);
  _dcel().delete_isolated_vertex(iv);
}

//---------------------------------------------------------------------------
// Check whether the arrangement is valid. In particular, check the
// validity of each vertex, halfedge, and face.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::is_valid() const
{
  Vertex_const_iterator vit;
  bool is_vertex_valid;
  for (vit = vertices_begin(); vit != vertices_end(); ++vit) {
    is_vertex_valid = _is_valid(vit);
    if (!is_vertex_valid) {
      CGAL_warning_msg(is_vertex_valid, "Invalid vertex.");
      return false;
    }
  }

  Halfedge_const_iterator heit;
  bool is_halfedge_valid;
  for (heit = halfedges_begin(); heit != halfedges_end(); ++heit) {
    is_halfedge_valid = _is_valid(heit);
    if (! is_halfedge_valid) {
      CGAL_warning_msg(is_halfedge_valid, "Invalid halfedge.");
      return false;
    }
  }

  Face_const_iterator     fit;
  bool                    is_face_valid;

  for (fit = faces_begin(); fit != faces_end(); ++fit) {
    is_face_valid = _is_valid(fit);
    if (! is_face_valid) {
      CGAL_warning_msg(is_face_valid, "Invalid face.");
      return false;
    }
  }

  bool  are_vertices_unique = _are_vertices_unique();
  if (! are_vertices_unique) {
    CGAL_warning_msg(are_vertices_unique,
                     "Found two vertices with the same geometric point.");
    return false;
  }

  // If we reached here, the arrangement is valid.
  return true;
}

//---------------------------------------------------------------------------
// Check the validity of a vertex.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_valid(Vertex_const_handle v) const
{
  // Do not check isolated vertices, as they have no incident halfedges.
  if (v->is_isolated()) return true;

  // Make sure that the vertex is the target of all its incident halfedges.
  Halfedge_around_vertex_const_circulator circ = v->incident_halfedges();
  Halfedge_around_vertex_const_circulator start = circ;

  do {
    if (circ->target() != v) return false;
    ++circ;
  } while (circ != start);

  // In case of a non-boundary vertex, make sure the curves are correctly
  // ordered around this vertex.
  if ((v->parameter_space_in_x() == ARR_INTERIOR) &&
      (v->parameter_space_in_y() == ARR_INTERIOR))
  {
    if (! _are_curves_ordered_cw_around_vertrex(v)) return false;
  }

  return true;
}

//---------------------------------------------------------------------------
// Check the validity of a halfedge.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_valid(Halfedge_const_handle he) const
{
  // Check relations with the previous and the next halfedges.
  if (he->prev()->target() != he->source()) return false;

  if (he->target() != he->next()->source()) return false;

  // Check relations with the twin.
  if (he != he->twin()->twin()) return false;

  if (he->source() != he->twin()->target() ||
      he->target() != he->twin()->source())
    return false;

  if (he->direction() == he->twin()->direction()) return false;

  // Stop here in case of a fictitious edge.
  if (he->is_fictitious()) return true;

  // Check that the end points of the curve associated with the halfedge
  // really equal the source and target vertices of this halfedge.
  const X_monotone_curve_2& cv = he->curve();
  const DVertex* source = _vertex(he->source());
  const DVertex* target = _vertex(he->target());
  Comparison_result res = ((source->parameter_space_in_x() == ARR_INTERIOR) &&
                           (source->parameter_space_in_y() == ARR_INTERIOR) &&
                           (target->parameter_space_in_x() == ARR_INTERIOR) &&
                           (target->parameter_space_in_y() == ARR_INTERIOR)) ?
    m_geom_traits->compare_xy_2_object()(source->point(), target->point()) :
    ((he->direction() == ARR_LEFT_TO_RIGHT) ? SMALLER : LARGER);

  if (res == SMALLER) {
    if (he->direction() != ARR_LEFT_TO_RIGHT)
      return false;

    return (_are_equal(_vertex(he->source()), cv, ARR_MIN_END) &&
            _are_equal(_vertex(he->target()), cv, ARR_MAX_END));
  }
  else if (res == LARGER) {
    if (he->direction() != ARR_RIGHT_TO_LEFT)
      return false;

    return (_are_equal(_vertex(he->source()), cv, ARR_MAX_END) &&
            _are_equal(_vertex(he->target()), cv, ARR_MIN_END));
  }

  // In that case, the source and target of the halfedge are equal.
  return false;
}

//---------------------------------------------------------------------------
// Check the validity of a face.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_valid(Face_const_handle f) const
{
  // Check if all outer components of the face refer to f.
  const DFace* p_f = _face(f);
  DOuter_ccb_const_iter oc_it;
  const DHalfedge* he;
  const DOuter_ccb* oc;
  for (oc_it = p_f->outer_ccbs_begin(); oc_it != p_f->outer_ccbs_end(); ++oc_it)
  {
    he = *oc_it;
    if (he->is_on_inner_ccb()) return false;

    oc = he->outer_ccb();
    if (oc->face() != p_f) return false;

    if (! _is_outer_ccb_valid(oc, he)) return false;
  }

  // Check if all inner components of the face refer to f.
  DInner_ccb_const_iter ic_it;
  const DInner_ccb* ic;

  for (ic_it = p_f->inner_ccbs_begin(); ic_it != p_f->inner_ccbs_end(); ++ic_it)
  {
    he = *ic_it;
    if (! he->is_on_inner_ccb()) return false;

    ic = he->inner_ccb();
    if (ic->face() != p_f) return false;

    if (! _is_inner_ccb_valid(ic, he)) return false;
  }

  // Check if all isolated vertices inside the face refer to f.
  DIso_vertex_const_iter iv_it;
  const DVertex* v;
  const DIso_vertex* iv;
  for (iv_it = p_f->isolated_vertices_begin();
       iv_it != p_f->isolated_vertices_end(); ++iv_it)
  {
    v = &(*iv_it);
    if (! v->is_isolated()) return false;

    iv = v->isolated_vertex();
    if (iv->face() != p_f) return false;
  }

  // If we reached here, the face is valid.
  return true;
}

//---------------------------------------------------------------------------
// Check the validity of an outer CCB.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_outer_ccb_valid(const DOuter_ccb* oc, const DHalfedge* first) const
{
  // Make sure that all halfedges along the CCB refer to the same component.
  const DHalfedge* curr = first;
  bool found_rep = false;

  do {
    if (curr->is_on_inner_ccb()) return false;

    if (oc != curr->outer_ccb()) return false;

    if (! found_rep && oc->halfedge() == curr) found_rep = true;

    curr = curr->next();
  } while (curr != first);

  // Return if we found the CCB representative along the outer CCB.
  return found_rep;
}

//---------------------------------------------------------------------------
// Check the validity of an inner CCB.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_is_inner_ccb_valid(const DInner_ccb* ic, const DHalfedge* first) const
{
  // Make sure that all halfedges along the CCB refer to the same component.
  const DHalfedge* curr = first;
  bool found_rep = false;

  do {
    if (! curr->is_on_inner_ccb()) return false;

    if (ic != curr->inner_ccb()) return false;

    if (! found_rep && ic->halfedge() == curr) found_rep = true;

    curr = curr->next();
  } while (curr != first);

  // Return if we found the CCB representative along the inner CCB.
  return found_rep;
}

//---------------------------------------------------------------------------
// Check that all vertices are unique (no two vertices with the same
// geometric point).
//
template <typename GeomTraits, typename TopTraits>
bool
Arrangement_on_surface_2<GeomTraits, TopTraits>::_are_vertices_unique() const
{
  if (number_of_vertices() < 2) return true;

  // Store all points associated with non-boundary vertices.
  std::vector<Point_2>  points_vec(number_of_vertices());
  Vertex_const_iterator vit;
  unsigned int          i = 0;

  for (vit = vertices_begin(); vit != vertices_end(); ++vit) {
    if ((vit->parameter_space_in_x() == ARR_INTERIOR) &&
        (vit->parameter_space_in_y() == ARR_INTERIOR))
    {
      points_vec[i] = vit->point();
      ++i;
    }
  }
  points_vec.resize(i);

  // Sort the vector of points and make sure no two adjacent points in the
  // sorted vector are equal.
  typedef typename Traits_adaptor_2::Compare_xy_2       Compare_xy_2;
  typedef typename Traits_adaptor_2::Equal_2            Equal_2;

  Equal_2       equal = m_geom_traits->equal_2_object();
  Compare_xy_2  compare_xy = m_geom_traits->compare_xy_2_object();
  Compare_to_less<Compare_xy_2> cmp = compare_to_less(compare_xy);

  std::sort(points_vec.begin(), points_vec.end(), cmp);
  for (i = 1; i < points_vec.size(); ++i) {
    if (equal(points_vec[i-1], points_vec[i])) return false;
  }

  return true;
}

//---------------------------------------------------------------------------
// Check that the curves around a given vertex are ordered clockwise.
//
template <typename GeomTraits, typename TopTraits>
bool Arrangement_on_surface_2<GeomTraits, TopTraits>::
_are_curves_ordered_cw_around_vertrex(Vertex_const_handle v) const
{
  if (v->degree() < 3) return true;

  typename Traits_adaptor_2::Is_between_cw_2  is_between_cw =
    m_geom_traits->is_between_cw_2_object();

  Halfedge_around_vertex_const_circulator circ = v->incident_halfedges();
  Halfedge_around_vertex_const_circulator first = circ;
  Halfedge_around_vertex_const_circulator prev, next;
  bool eq1, eq2;

  do {
    prev = circ; --prev;
    next = circ; ++next;

    if (!is_between_cw(circ->curve(), (circ->direction() == ARR_RIGHT_TO_LEFT),
                       prev->curve(), (prev->direction() == ARR_RIGHT_TO_LEFT),
                       next->curve(), (next->direction() == ARR_RIGHT_TO_LEFT),
                       v->point(), eq1, eq2))
      return false;

    if (eq1 || eq2) return false;

    ++circ;
  } while (circ != first);

  return true;
}

} // namespace CGAL

#endif

