// Copyright (c) 2005,2007,2008,2009,2010,2011 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.6.2/Arrangement_on_surface_2/include/CGAL/Arr_point_location/Arr_landmarks_pl_impl.h $
// $Id: Arr_landmarks_pl_impl.h 3674c937f76 2022-11-15T15:21:01+01:00 albert-github
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s)     : Idit Haran   <haranidi@post.tau.ac.il>
//                 Ron Wein     <wein@post.tau.ac.il>
//                 Efi Fogel    <efif@post.tau.ac.il>

#ifndef CGAL_ARR_LANDMARKS_PL_IMPL_H
#define CGAL_ARR_LANDMARKS_PL_IMPL_H

#include <CGAL/license/Arrangement_on_surface_2.h>


/*! \file
 * Member-function definitions for the
 * Arr_landmarks_point_location<Arrangement, Generator> class.
 */

namespace CGAL {

//-----------------------------------------------------------------------------
// Locate the arrangement feature containing the given point.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::result_type
Arr_landmarks_point_location<Arr, Gen>::locate(const Point_2& p) const
{
  // If the arrangement is empty, return its initial (empty and
  // non-fictitious) face.
  if (p_arr->number_of_vertices() == 0) {
    CGAL_assertion(p_arr->number_of_faces() == 1);
    Face_const_handle      fh = p_arr->faces_begin();
    return make_result(fh);
  }
  // Use the generator and to find the closest landmark to the query point.
  result_type    lm_location_obj;
  const Point_2& landmark_point = lm_gen->closest_landmark(p, lm_location_obj);

  // If the query point and the landmark point are equal, return the landmark.
  if (m_traits->equal_2_object()(landmark_point, p))
    return lm_location_obj;

  // Walk from the nearest_vertex to the point p, using walk algorithm,
  // and find the location of the query point p. Note that the set of edges
  // we have crossed so far is initially empty.
  Halfedge_set                   crossed_edges;
  result_type                    out_obj;

  // Locate the arrangement feature that contains the landmark.
  const Vertex_const_handle*   vh;
  const Halfedge_const_handle* hh;
  const Face_const_handle*     fh;
  if ( ( vh = Result().template assign<Vertex_const_handle>(&lm_location_obj) ) )
    out_obj = _walk_from_vertex(*vh, p, crossed_edges);
  else if ( ( hh = Result().template assign<Halfedge_const_handle>(&lm_location_obj) ) )
    out_obj = _walk_from_edge(*hh, landmark_point, p, crossed_edges);
  else if ( ( fh =  Result().template assign<Face_const_handle>(&lm_location_obj) ) )
    out_obj = _walk_from_face(*fh, landmark_point, p, crossed_edges);
  else CGAL_error_msg("lm_location_obj of an unknown type.");

  if ( ( fh = Result().template assign<Face_const_handle>(&out_obj) ) ) {
    // If we reached here, we did not locate the query point in any of the
    // holes inside the current face, so we conclude it is contained in this
    // face.
    // However, we first have to check whether the query point coincides with
    // any of the isolated vertices contained inside this face.
    Isolated_vertex_const_iterator      iso_verts_it;
    typename Traits_adaptor_2::Equal_2  equal = m_traits->equal_2_object();

    for (iso_verts_it = (*fh)->isolated_vertices_begin();
         iso_verts_it != (*fh)->isolated_vertices_end(); ++iso_verts_it)
    {
      if (equal(p, iso_verts_it->point())) {
        Vertex_const_handle  ivh = iso_verts_it;
        return make_result(ivh);
      }
    }
  }

  return out_obj;
}

//-----------------------------------------------------------------------------
// Walk from a given vertex to the query point.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::result_type
Arr_landmarks_point_location<Arr, Gen>::
_walk_from_vertex(Vertex_const_handle nearest_vertex,
                  const Point_2& p,
                  Halfedge_set& crossed_edges) const
{
  Vertex_const_handle vh = nearest_vertex;
  CGAL_assertion_msg(! vh->is_at_open_boundary(),
                     "_walk_from_vertex() from a vertex at infinity.");

  // Check if the query point p coincides with the vertex.
  if (m_traits->equal_2_object()(vh->point(), p))
    return make_result(vh);

  // In case of an isolated vertex, walk to from the face that contains
  // it toward the query point.
  if (vh->is_isolated()) {
    Face_const_handle  fh = vh->face();
    return (_walk_from_face(fh, vh->point(), p, crossed_edges));
  }

  // if we walk from a vertex this means we are crossing the
  // halfedges that form a corridor in which seg is going through
  Halfedge_around_vertex_const_circulator first = vh->incident_halfedges();
  // Create an x-monotone curve connecting the point associated with the
  // vertex vp and the query point p.
  const Point_2&      vp = vh->point();
  X_monotone_curve_2  seg =
    m_traits->construct_x_monotone_curve_2_object()(vp, p);
  const bool          seg_dir_right =
    (m_traits->compare_xy_2_object()(vp, p) == SMALLER);
  Halfedge_around_vertex_const_circulator curr_iter = first;
  Halfedge_around_vertex_const_circulator next_iter = curr_iter;
  ++next_iter;
  typename Traits_adaptor_2::Is_between_cw_2  is_between_cw =
    m_traits->is_between_cw_2_object();
  // Traverse the halfedges around vp until we find the pair of adjacent
  // halfedges such as seg is located clockwise in between them.
  do {
    bool eq_curr_iter, eq_next_iter;
    if (is_between_cw(seg, seg_dir_right, curr_iter->curve(),
                      (curr_iter->direction() == ARR_RIGHT_TO_LEFT),
                      next_iter->curve(),
                      (next_iter->direction() == ARR_RIGHT_TO_LEFT),
                      vp, eq_curr_iter, eq_next_iter))
    {
      // the assumption is that each edge is crossed at most twice
      CGAL_assertion_msg(crossed_edges.count (curr_iter) < 2,
        "crossed_edges should contain each halfedge at most twice.");
      CGAL_assertion_msg(crossed_edges.count (next_iter) < 2,
        "crossed_edges should contain each halfedge at most twice.");
      crossed_edges.insert(curr_iter);
      crossed_edges.insert(curr_iter->twin());
      crossed_edges.insert(next_iter);
      crossed_edges.insert(next_iter->twin());
      break;
    }
    ++curr_iter;
    ++next_iter;
  } while (curr_iter != first);

  // Locate the face around the vertex that contains the curve connecting
  // the vertex and the query point.
  while (true) {
    bool new_vertex;
    result_type obj = _find_face_around_vertex(vh, p, new_vertex);
    if (new_vertex) {
      // We found a vertex closer to p; Continue using this vertex.
      const Vertex_const_handle* p_vh =
        Result().template assign<Vertex_const_handle>(&obj);
      CGAL_assertion(p_vh != nullptr);
      vh = *p_vh;
      continue;
    }

    // If p is located on an edge or on a vertex, return the object
    // that wraps this arrangement feature.
    if (Result().template assign<Halfedge_const_handle>(&obj) ||
        Result().template assign<Vertex_const_handle>(&obj))
      return obj;

    const Face_const_handle* p_fh =
      Result().template assign<Face_const_handle>(&obj);
    if (p_fh)
      // Walk to p from the face we have located:
      return _walk_from_face(*p_fh, vh->point(), p, crossed_edges);

    CGAL_error_msg("_find_face_around_vertex() returned an unknown object.");
  }

  // We should never reach here:
  CGAL_error();
  return default_result();
}

//-----------------------------------------------------------------------------
// Locate an edge around a given vertex that is the predecessor of the curve
// connecting the vertex to the query point in a clockwise order.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::result_type
Arr_landmarks_point_location<Arr, Gen>::
_find_face_around_vertex(Vertex_const_handle vh,
                         const Point_2& p,
                         bool& new_vertex) const
{
  new_vertex = false;

  // Create an x-monotone curve connecting the point associated with the
  // vertex vp and the query point p.
  const Point_2&      vp = vh->point();
  X_monotone_curve_2  seg =
    m_traits->construct_x_monotone_curve_2_object()(vp, p);
  const bool          seg_dir_right =
    (m_traits->compare_xy_2_object()(vp, p) == SMALLER);

  // Get the first incident halfedge around v and the next halfedge.
  Halfedge_around_vertex_const_circulator  first = vh->incident_halfedges();
  Halfedge_around_vertex_const_circulator  curr, next;
  bool                                     equal_curr = false;

  next = curr = first;
  ++next;

  if (next == curr) {
    // The vertex has a single incident edge. Check if its associated
    // curve equals seg next to vp.
    if (seg_dir_right && curr->direction() == ARR_RIGHT_TO_LEFT) {
      // Both curves are defined to the right of vp:
      equal_curr =
        (m_traits->compare_y_at_x_right_2_object()
         (curr->curve(), seg, vp) == EQUAL);
    }
    else if (! seg_dir_right && curr->direction() == ARR_LEFT_TO_RIGHT) {
      // Both curves are defined to the left of vp:
      equal_curr =
        (m_traits->compare_y_at_x_left_2_object()
         (curr->curve(), seg, vp) == EQUAL);
    }

    // In case the curves are not equal, just return the incident face of
    // the single halfedge (note that this is also the incident face of its
    // twin, as v is the tip of an "antenna").
    if (! equal_curr) {
      CGAL_assertion(curr->face() == curr->twin()->face());
      return make_result(curr->face());
    }
  }
  else {
    // Traverse the halfedges around v until we find the pair of adjacent
    // halfedges such as seg is located clockwise in between them.
    typename Traits_adaptor_2::Is_between_cw_2  is_between_cw =
      m_traits->is_between_cw_2_object();
    bool                                        eq_curr, eq_next;

    while (! is_between_cw(seg, seg_dir_right, curr->curve(),
                           (curr->direction() == ARR_RIGHT_TO_LEFT),
                           next->curve(),
                           (next->direction() == ARR_RIGHT_TO_LEFT),
                           vp, eq_curr, eq_next))
    {
      // Break the loop if seg equals one of the halfedges next to v.
      if (eq_curr) {
        equal_curr = true;
        break;
      }

      if (eq_next) {
        curr = next;
        equal_curr = true;
        break;
      }

      // Move to the next pair of incident halfedges.
      curr = next;
      ++next;

      // Guard for an infinitive loop, in case we have completed a full
      // traversal around v without locating a place for seg.
      if (curr == first) {
        CGAL_error_msg("Completed a full cycle around v without locating seg.");
        return default_result();
      }
    }

    // In case seg is not equal to curr's curve, just return the incident face
    // of the halfedge we have located.
    if (! equal_curr)
      return make_result(curr->face());
  }

  // If we reached here, seg overlaps the curve associated with curr next to
  // the vertex v. We first check if p equals the other end-vertex of this
  // halfedge.
  if (m_traits->equal_2_object()(p, curr->source()->point())) {
    // In this case p equals the source point of the edge.
    return make_result(curr->source());
  }

  // Check whether p lies on the curve associated with the edge.
  if (m_traits->is_in_x_range_2_object()(curr->curve(), p) &&
      m_traits->compare_y_at_x_2_object()(p, curr->curve()) == EQUAL)
  {
    // p is located on the interior of the edge.
    Halfedge_const_handle   he = curr;
    return make_result(he);
  }

  // In this case, the source vertex of the current edge is closer
  // to the query point p.
  new_vertex = true;
  return make_result(curr->source());
}

//-----------------------------------------------------------------------------
// Walk from the edge to the query point.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::result_type
Arr_landmarks_point_location<Arr, Gen>::
_walk_from_edge(Halfedge_const_handle eh,
                const Point_2& np,
                const Point_2& p,
                Halfedge_set& crossed_edges) const
{
  CGAL_assertion_msg(! eh->is_fictitious(),
                     "_walk_from_edge() from a fictitious edge.");

  const X_monotone_curve_2& cv = eh->curve() ;
  Comparison_result         res;

  X_monotone_curve_2 seg =
    m_traits->construct_x_monotone_curve_2_object()(np, p);

  // Create an initial set of edges that have been crossed, which currently
  // contains only the halfedge we are currently on (and its twin).
  // the assumption is that each edge is crossed at most twice
  CGAL_assertion_msg(crossed_edges.count (eh) < 2,
    "crossed_edges should contain each halfedge at most twice.");
  crossed_edges.insert(eh);
  crossed_edges.insert(eh->twin());

  // If p equals one of the edge's endpoints, return the vertex
  // that represents this endpoint.
  if (! eh->source()->is_at_open_boundary()) {
    if (m_traits->equal_2_object()(p, eh->source()->point())) {
      Vertex_const_handle vh = eh->source();
      return make_result(vh);
    }
    // if the source point of the halfedge is located on seg then we insert
    // the predecessor of the halfedge and its twin to crossed_edges
    Point_2 temp_p = eh->source()->point();
    if (m_traits->is_in_x_range_2_object()(seg, temp_p)) {
      //we must make sure that eh is not a tip on an "antena"
      if (m_traits->compare_y_at_x_2_object()(temp_p, seg) == EQUAL &&
          eh->prev() != eh->twin())
      {
        // the assumption is that each edge is crossed at most twice
        CGAL_assertion_msg(crossed_edges.count(eh->prev()) < 2,
          "crossed_edges should contain each halfedge at most twice.");
        crossed_edges.insert(eh->prev());
        crossed_edges.insert(eh->prev()->twin());
        return _walk_from_vertex(eh->source(), p, crossed_edges);
      }
    }
  }
  if (! eh->target()->is_at_open_boundary()) {
    if (m_traits->equal_2_object()(p, eh->target()->point())) {
      Vertex_const_handle vh = eh->target();
      return make_result(vh);
    }
    // if the target point of the halfedge is located on seg then we insert
    // the successor of the halfedge and its twin to crossed_edges
    Point_2 temp_p = eh->target()->point();
    if (m_traits->is_in_x_range_2_object()(seg, temp_p)) {
      //we must make sure that eh is not a tip on an "antena"
      if (m_traits->compare_y_at_x_2_object()(temp_p, seg) == EQUAL &&
          eh->next() != eh->twin())
      {
        // the assumption is that each edge is crossed at most twice
        CGAL_assertion_msg(crossed_edges.count(eh->next()) < 2,
          "crossed_edges should contain each halfedge at most twice.");
        crossed_edges.insert(eh->next());
        crossed_edges.insert(eh->next()->twin());
        return _walk_from_vertex(eh->target(), p, crossed_edges);
      }
    }
  }

  // Check whether p is in the x-range of the edge.
  if (m_traits->is_in_x_range_2_object()(cv, p)) {
    // If p is in eh's x_range, then we need to check if it is above or below
    // it, so we can orient the halfedge eh accordingly, such that it will be
    // incident to the face that is most likely to contain p.
    res = m_traits->compare_y_at_x_2_object()(p, cv);

    switch (res) {
     case EQUAL:
      // The edge contains p in its interior:
      return make_result(eh);

     case LARGER:
      // p is above cv: the edge must be oriented from left to right.
      if (eh->direction() == ARR_RIGHT_TO_LEFT)
        eh = eh->twin();
      break;

     case SMALLER:
      // p is below cv: the edge must be oriented from right to left.
      if (eh->direction() == ARR_LEFT_TO_RIGHT)
        eh = eh->twin();
      break;
    }

    // Now walk in the incdient face of eh toward p.
    return (_walk_from_face(eh->face(), np, p, crossed_edges));
  }

  // In this case p is in not in the x-range of eh. We select the end
  // vertex of eh that lies closer to p, and walk from this vertex.
  Vertex_const_handle  vh = eh->source();

  if (vh->is_at_open_boundary()) {
    vh = eh->target();
    CGAL_assertion (! vh->is_at_open_boundary());
  }
  else if (! eh->target()->is_at_open_boundary()) {
    res = m_traits->compare_xy_2_object()(p, eh->source()->point());

    if ((eh->direction() == ARR_LEFT_TO_RIGHT && res == LARGER) ||
        (eh->direction() == ARR_RIGHT_TO_LEFT && res == SMALLER))
      vh = eh->target();
  }

  // Walk from the closer vertex toward the query point p.
  return (_walk_from_vertex(vh, p, crossed_edges));
}

//-----------------------------------------------------------------------------
// In case the arrangement's curve contained in the segment
// from the nearest landmark to the query point
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::result_type
Arr_landmarks_point_location<Arr, Gen>::
_deal_with_curve_contained_in_segment(Halfedge_const_handle he,
                                      bool p_is_left,
                                      const Point_2& p,
                                      Halfedge_set& crossed_edges) const
{
  // in this case we want to walk from to the query point from the nearest
  // vertex either the halfedge's source or target
  Vertex_const_handle  vh;
  bool target_is_left;
  if (m_traits->compare_xy_2_object()
      (he->source()->point(),he->target()->point()) == LARGER)
    target_is_left = true;
  else
    target_is_left = false;
  if (p_is_left) {
    if (target_is_left)
      vh = he->target();
    else
      vh = he->source();
  }
  else {
    if (target_is_left)
      vh = he->source();
    else
      vh = he->target();
  }
  // vh is the closest vertex among the halfedge's end points
  return (_walk_from_vertex(vh, p, crossed_edges));
}

//-----------------------------------------------------------------------------
// Walk from the given face to the query point.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::result_type
Arr_landmarks_point_location<Arr, Gen>::
_walk_from_face(Face_const_handle face,
                const Point_2& np,
                const Point_2& p,
                Halfedge_set& crossed_edges) const
{
  // Construct an x-monotone curve connecting the nearest landmark point np
  // to the query point p and check which CCB intersects this segment.
  X_monotone_curve_2             seg =
    m_traits->construct_x_monotone_curve_2_object()(np, p);
  const bool                     p_is_left =
    (m_traits->compare_xy_2_object()(np, p) == LARGER);

  Inner_ccb_const_iterator       inner_ccb_iter;
  Outer_ccb_const_iterator       outer_ccb_iter;
  const Halfedge_const_handle    invalid_he;
  Halfedge_const_handle          he;
  Face_const_handle              new_face;
  bool                           is_on_edge;
  bool                           is_target;
  bool                           cv_is_contained_in_seg;
  Vertex_const_handle            new_vertex;
  const Vertex_const_handle      invalid_vertex;

  do {
    // Check whether p lies inside the current face (including its holes):
    if (p_arr->topology_traits()->is_in_face(&(*face), p, nullptr))
    {
      // We know that p is located inside the current face, and we check
      // whether it lies inside one of its holes (or on the boundary of
      // its holes).
      cv_is_contained_in_seg = false;
      new_face = face;
      for (inner_ccb_iter = face->inner_ccbs_begin();
           inner_ccb_iter != face->inner_ccbs_end(); ++inner_ccb_iter)
      {
        he = _intersection_with_ccb(*inner_ccb_iter,seg, p, p_is_left,
                                    crossed_edges,is_on_edge, is_target,
                                    cv_is_contained_in_seg,new_vertex);
        if (he == invalid_he && cv_is_contained_in_seg)
        {
          return _deal_with_curve_contained_in_segment(*inner_ccb_iter,
                                                       p_is_left,p,
                                                       crossed_edges);
        }
        if (he != invalid_he) {
          // Check if the query point is located on a vertex or on an edge.
          if (is_target)
            return make_result(he->target());
          else if (is_on_edge)
            return make_result(he);
          if (new_vertex != invalid_vertex) {
            // if we got here it means that a closer vertex then np was found
            return (_walk_from_vertex(new_vertex, p, crossed_edges));
          }
          // Otherwise, cross over he to the incident face of its twin.
          if (face != he->twin()->face()) {
            new_face = he->twin()->face();
            break;
          }
        }
      }

      // Check if we found a new face (hole) containing p. If not, the current
      // face contains p.
      if (new_face == face)
        return make_result(face);

      // Continue from the new face (hole).
      face = new_face;
    }
    else {
      // We know that p is not located inside the current face. We therefore
      // look for an edge on its outer boundary that intersects seg.
      new_face = face;
      for (outer_ccb_iter = face->outer_ccbs_begin();
           outer_ccb_iter != face->outer_ccbs_end(); ++outer_ccb_iter)
      {
        he = _intersection_with_ccb(*outer_ccb_iter,seg, p, p_is_left,
                                    crossed_edges,is_on_edge, is_target,
                                    cv_is_contained_in_seg,new_vertex);
        if (he == invalid_he && cv_is_contained_in_seg) {
          return _deal_with_curve_contained_in_segment(*outer_ccb_iter,
                                                       p_is_left,p,
                                                       crossed_edges);
        }
        if (he != invalid_he) {
          // Check if the query point is located on a vertex or on an edge.
          if (is_target)
            return make_result(he->target());
          else if (is_on_edge)
            return make_result(he);
          if (new_vertex != invalid_vertex) {
            // if we got here it means that a closer vertex then np was found
            return (_walk_from_vertex(new_vertex, p, crossed_edges));
          }
          // Otherwise, cross over he to the incident face of its twin.
          if (face != he->twin()->face()) {
            new_face = he->twin()->face();
            break;
          }
        }
      }

      // Continue from the new face.
      CGAL_assertion(new_face != face);
      face = new_face;
    }
  } while (true);

  // We should never reach here:
  CGAL_error();
  return default_result();
}

//-----------------------------------------------------------------------------
// Find a halfedge on the given CCB that intersects the given x-monotone
// curve, connecting the current landmark to the query point.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::Halfedge_const_handle
Arr_landmarks_point_location<Arr, Gen>::
_intersection_with_ccb(Ccb_halfedge_const_circulator circ,
                       const X_monotone_curve_2& seg,
                       const Point_2& p, bool p_is_left,
                       Halfedge_set& crossed_edges,
                       bool& is_on_edge, bool& is_target,
                       bool& cv_is_contained_in_seg,
                       Vertex_const_handle & new_vertex) const
{
  is_on_edge = false;
  is_target = false;

  // Go over the CCB.
  typename Traits_adaptor_2::Is_in_x_range_2    is_in_x_range =
    m_traits->is_in_x_range_2_object();
  Ccb_halfedge_const_circulator                 curr = circ , temp_circ;
  const Halfedge_const_handle                   invalid_he;
  Halfedge_const_handle                         he;
  bool cv_and_seg_overlap;
  do {
    he = curr;
    // Skip fictitious halfedges.
    if (he->is_fictitious()) {
      ++curr;
      continue;
    }

    // Check if we have already crossed the current halfedge (or its twin).
    // If so, we do not cross it again.
    if (crossed_edges.count(he) != 0) {
      ++curr;
      continue;
    }

    if (m_traits->is_vertical_2_object()(he->curve())) {
      if (is_in_x_range(he->curve(), p)) {
        if (m_traits->compare_y_at_x_2_object()(p, he->curve()) == EQUAL) {
          // special treatment in case the query point is on a vertical curve
          is_on_edge = true;
          return _in_case_p_is_on_edge(he,crossed_edges,p,is_target);
        }
      }
    }

    // Check if the x-range of the curve associated with the current edge
    // does not overlap the x-range of seg, the two curves cannot intersect.
    if (! is_in_x_range(he->curve(), seg)) {
      ++curr;
      continue;
    }
    cv_and_seg_overlap = false;
    cv_is_contained_in_seg = false;
    // Check whether the current curve intersects seg an odd number of times.
    if (_have_odd_intersections(he->curve(), seg, p_is_left,
                                is_on_edge,cv_and_seg_overlap,
                                cv_is_contained_in_seg)
        && !(cv_and_seg_overlap || cv_is_contained_in_seg))
    {
      // Check if the query point lies on the current edge, or whether
      // it lies in its interior.
      if (is_on_edge)
        return _in_case_p_is_on_edge(he,crossed_edges,p,is_target);

      if ((!curr->target()->is_at_open_boundary()) &&
          is_in_x_range(seg, curr->target()->point()))
      {
        // if the target point of curr is located on seg
        // we should walk from it to the query point
        if (m_traits->compare_y_at_x_2_object()
            (curr->target()->point(), seg) == EQUAL)
        {
          new_vertex = curr->target();
        }
      }
      else if ((!curr->source()->is_at_open_boundary()) &&
               is_in_x_range(seg , curr->source()->point() ))
      {
        // if the source point of curr is located on seg
        // we should walk from it to the query point
        if (m_traits->compare_y_at_x_2_object()
            (curr->source()->point() , seg) == EQUAL)
        {
          new_vertex = curr->source();
        }
      }

      // Return the halfedge we found, and mark that we have already crossed
      // it (as well as its twin).
      // the assumption is that each edge is crossed at most twice
      CGAL_assertion_msg(crossed_edges.count(he) < 2,
        "crossed_edges should contain each halfedge at most twice.");
      crossed_edges.insert(he);
      crossed_edges.insert(he->twin());
      return he;
    }
    else if (cv_and_seg_overlap || cv_is_contained_in_seg) {
      // Check if the query point lies on the current edge, or whether
      // it lies in its interior.
      if (cv_is_contained_in_seg) {
        // cv is contained in seg, obviously we crossed it
        // the assumption is that each edge is crossed at most twice
        CGAL_assertion_msg (crossed_edges.count(he) < 2,
          "crossed_edges should contain each halfedge at most twice.");
        crossed_edges.insert(he);
        crossed_edges.insert(he->twin());
        return invalid_he;
      }
      if (is_on_edge)
        return _in_case_p_is_on_edge(he,crossed_edges,p,is_target);
    }
    // Proceed to the next halfedge along the CCB.
    ++curr;
  } while (curr != circ);

  // If we reached here, we did not find any edge intersecting seg.
  return (invalid_he);
}

//-----------------------------------------------------------------------------
// Return the halfedge that contains the query point.
//
template <typename Arr, typename Gen>
typename Arr_landmarks_point_location<Arr, Gen>::Halfedge_const_handle
Arr_landmarks_point_location<Arr, Gen>::
_in_case_p_is_on_edge(Halfedge_const_handle he,
                      Halfedge_set& crossed_edges,
                      const Point_2 & p,
                      bool & is_target) const
{
  // cv and seg overlap, obviously we crossed it
  // the assumption is that each edge is crossed at most twice
  CGAL_assertion_msg(crossed_edges.count(he) < 2,
    "crossed_edges should contain each halfedge at most twice.");
  crossed_edges.insert(he);
  crossed_edges.insert(he->twin());
  // Check if p equals one of the edge end-vertices.
  if (! he->target()->is_at_open_boundary() &&
      m_traits->compare_xy_2_object()(he->target()->point(), p) == EQUAL)
  {
    // p is the target of the current halfedge.
    is_target = true;
  }
  else if (! he->source()->is_at_open_boundary() &&
    m_traits->compare_xy_2_object()(he->source()->point(), p) == EQUAL)
  {
    // Take the twin halfedge, so p equals its target.
    he = he->twin();
    is_target = true;
  }
  // Return the halfedge containing p.
  return he;
}

//-----------------------------------------------------------------------------
// Check whether the given curve intersects a simple segment, which connects
// the current landmark to the query point, an odd number of times.
//
template <typename Arr, typename Gen>
bool Arr_landmarks_point_location<Arr, Gen>::
_have_odd_intersections(const X_monotone_curve_2& cv,
                        const X_monotone_curve_2& seg,
                        bool p_is_left,
                        bool& p_on_curve,
                        bool& cv_and_seg_overlap,
                        bool& cv_is_contained_in_seg) const
{
  typename Traits_adaptor_2::Is_in_x_range_2    is_in_x_range =
    m_traits->is_in_x_range_2_object();
  p_on_curve = false;
  cv_and_seg_overlap = false;
  cv_is_contained_in_seg = false;
  // Use the left and right endpoints of the segment.
  const Point_2& seg_left = m_traits->construct_min_vertex_2_object()(seg);
  const Point_2& seg_right = m_traits->construct_max_vertex_2_object()(seg);
  // Use the left and right endpoints of the segment.
  Point_2 cv_left;
  Point_2 cv_right;
  bool cv_left_is_closed = m_traits->is_closed_2_object()(cv, ARR_MIN_END);
  bool cv_right_is_closed = m_traits->is_closed_2_object()(cv, ARR_MAX_END);
  if (cv_left_is_closed)
    cv_left = m_traits->construct_min_vertex_2_object()(cv);
  if (cv_right_is_closed)
    cv_right = m_traits->construct_max_vertex_2_object()(cv);
  if (cv_left_is_closed && cv_right_is_closed) {
    if (is_in_x_range(seg,cv_left) && is_in_x_range(seg,cv_right)) {
      if ((m_traits->compare_y_at_x_2_object()(cv_left, seg) == EQUAL) &&
          (m_traits->compare_y_at_x_2_object()(cv_right, seg) == EQUAL))
      {
        // cv is contained in seg non of the answer true or false is correct
        // we must set a special flag to distinguish this case
        cv_is_contained_in_seg = true;
        return true;
      }
    }
  }
  // Check if the overlapping x-range of the two curves is trivial.
  // In this case, they cannot cross.
  if (cv_left_is_closed) {
    // Check if the left endpoint of cv has the same x-coordinate as the
    // right endpoint of seg.
    if (m_traits->compare_x_2_object()
        (m_traits->construct_min_vertex_2_object()(cv), seg_right) == EQUAL)
    {
      if (! p_is_left &&
          m_traits->compare_xy_2_object()
          (m_traits->construct_min_vertex_2_object()(cv), seg_right) == EQUAL)
      {
        p_on_curve = true;
        return true;
      }
      else if (m_traits->is_vertical_2_object()(seg)) {
        // Special treatment for vertical segments.
        Comparison_result   res_l =
          m_traits->compare_y_at_x_2_object()(seg_left, cv);
        Comparison_result   res_r =
          m_traits->compare_y_at_x_2_object()(seg_right, cv);
        if ((p_is_left && res_l == EQUAL) || (! p_is_left && res_r == EQUAL)) {
          p_on_curve = true;
          return true;
        }
        return (res_l != res_r);
      }
      return false;
    }
  }
  if (cv_right_is_closed) {
    // Check if the right endpoint of cv has the same x-coordinate as the
    // left endpoint of seg.
    if (m_traits->compare_x_2_object()
        (m_traits->construct_max_vertex_2_object()(cv), seg_left) == EQUAL)
    {
      if (p_is_left &&
          m_traits->compare_xy_2_object()
          (m_traits->construct_max_vertex_2_object()(cv), seg_left) == EQUAL)
      {
        p_on_curve = true;
        return true;
      }
      else if (m_traits->is_vertical_2_object()(seg)) {
        // Special treatment for vertical segments.
        Comparison_result   res_l =
          m_traits->compare_y_at_x_2_object()(seg_left, cv);
        Comparison_result   res_r =
          m_traits->compare_y_at_x_2_object()(seg_right, cv);

        if ((p_is_left && res_l == EQUAL) || (! p_is_left && res_r == EQUAL)) {
          p_on_curve = true;
          return true;
        }
        return (res_l != res_r);
      }
      return false;
    }
  }
  // Compare the two left ends of cv and seg.
  Comparison_result    left_res;
  const Arr_parameter_space  bx_l =
    m_traits->parameter_space_in_x_2_object()(cv, ARR_MIN_END);
  if (bx_l == ARR_LEFT_BOUNDARY) {
    // The left end of cv lies to the left of seg_left:
    // Compare this point to cv.
    left_res = m_traits->compare_y_at_x_2_object()(seg_left, cv);
  }
  else if (bx_l == ARR_RIGHT_BOUNDARY) {
    // The left end of cv lies to the right of seg_left.
    // Compare the left endpoint of cv to seg.
    left_res = m_traits->compare_y_at_x_2_object()
        (m_traits->construct_min_vertex_2_object()(cv), seg);
    left_res = CGAL::opposite(left_res);
  }
  else {
    const Arr_parameter_space  by_l =
      m_traits->parameter_space_in_y_2_object()(cv, ARR_MIN_END);
    if (by_l == ARR_BOTTOM_BOUNDARY)
      // The left end of cv is at y = -oo, so cv obviously lies above it.
      left_res = LARGER;
    else if (by_l == ARR_TOP_BOUNDARY)
      // The left end of cv is at y = +oo, so cv obviously lies below it.
      left_res = SMALLER;
    else {
      // In this case cv has a valid left endpoint: Find the rightmost of
      // these two points and compare it to the other curve.
      Comparison_result res = m_traits->compare_xy_2_object()(cv_left, seg_left);
      if (res != LARGER) {
        left_res = m_traits->compare_y_at_x_2_object()(seg_left, cv);
        if (p_is_left && left_res == EQUAL)
        {
          // In this case the query point p, which is the left endpoint of seg,
          // lies on cv.
          p_on_curve = true;
          return true;
        }
      }
      else {
        left_res = m_traits->compare_y_at_x_2_object()(cv_left, seg);
        left_res = CGAL::opposite(left_res);
      }
    }
  }
  if (left_res == EQUAL) {
    // Compare the two curves to the right of their common left endpoint.
    if (is_in_x_range(cv,seg_left))
      left_res = m_traits->compare_y_at_x_right_2_object()(seg, cv, seg_left);
    else if (is_in_x_range(seg,cv_left))
      left_res = m_traits->compare_y_at_x_right_2_object()(seg, cv, cv_left);
    else
      CGAL_error();
    if (left_res == EQUAL) {
      // RWRW: In this case we have an overlap ...
      // cv and seg overlap non of the answer true or false is correct
      // we must set a special flag to distinguish this case
      if (is_in_x_range(cv,( p_is_left ? seg_left : seg_right)))
        if (m_traits->compare_y_at_x_2_object()
            ((p_is_left ? seg_left : seg_right), cv) == EQUAL)
        {
          p_on_curve = true;
        }
      cv_and_seg_overlap = true;
      return true;
    }
  }
  // Compare the two right ends of cv and seg.
  Comparison_result    right_res;
  const Arr_parameter_space  bx_r =
    m_traits->parameter_space_in_x_2_object()(cv, ARR_MAX_END);
  if (bx_r == ARR_RIGHT_BOUNDARY) {
    // The right end of cv lies to the right of seg_right:
    // Compare this point to cv.
    right_res = m_traits->compare_y_at_x_2_object()(seg_right, cv);
  }
  else if (bx_r == ARR_LEFT_BOUNDARY) {
    // The right end of cv lies to the left of seg_right.
    // Compare the right endpoint of cv to seg.
    right_res = m_traits->compare_y_at_x_2_object()
        (m_traits->construct_max_vertex_2_object()(cv), seg);
    right_res = CGAL::opposite(right_res);
  }
  else {
    const Arr_parameter_space  by_r =
      m_traits->parameter_space_in_y_2_object()(cv, ARR_MAX_END);
    if (by_r == ARR_BOTTOM_BOUNDARY)
      // The right end of cv is at y = -oo, so cv obviously lies above it.
      right_res = LARGER;
    else if (by_r == ARR_TOP_BOUNDARY)
      // The right end of cv is at y = +oo, so cv obviously lies below it.
      right_res = SMALLER;
    else {
      // In this case cv has a valid right endpoint: Find the leftmost of
      // these two points and compare it to the other curve.
      Comparison_result res =
        m_traits->compare_xy_2_object()(cv_right, seg_right);

      if (res != SMALLER) {
        right_res = m_traits->compare_y_at_x_2_object()(seg_right, cv);
        if (! p_is_left && right_res == EQUAL) {
          // In this case the query point p, which is the right endpoint of seg,
          // lies on cv.
          p_on_curve = true;
          return true;
        }
      }
      else {
        right_res = m_traits->compare_y_at_x_2_object()(cv_right, seg);
        right_res = CGAL::opposite(right_res);
      }
    }
  }
  if (right_res == EQUAL) {
    // Compare the two curves to the left of their common right endpoint.
    if (is_in_x_range(cv,seg_right))
      right_res = m_traits->compare_y_at_x_left_2_object()(seg, cv, seg_right);
    else if (is_in_x_range(seg,cv_right))
      right_res = m_traits->compare_y_at_x_left_2_object()(seg, cv, cv_right);
    else
      CGAL_error();
    if (right_res == EQUAL) {
      // RWRW: In this case we have an overlap ...
      // cv and seg overlap non of the answer true or false is correct
      // we must set a special flag to distinguish this case
      if (is_in_x_range(cv, (p_is_left ? seg_left : seg_right)))
        if (m_traits->compare_y_at_x_2_object()
            ((p_is_left ? seg_left : seg_right), cv) == EQUAL)
        {
          p_on_curve = true;
        }
      cv_and_seg_overlap = true;
      return true;
    }
  }
  // The two curves intersect an odd number of times if the comparison
  // results at the two ends are not the same (this indicates that they
  // switch positions).
  return (left_res != right_res);
}

} //namespace CGAL

#endif
