CGAL 5.6.3 - 2D Triangulations

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CGAL::Regular_triangulation_2< Traits, Tds > Class Template Reference
2D Triangulations Reference » Triangulation Classes

`#include <CGAL/Regular_triangulation_2.h>`

## Inherits from

CGAL::Triangulation_2< Traits, Tds >.

## Definition

template<typename Traits, typename Tds>
class CGAL::Regular_triangulation_2< Traits, Tds >

The class `Regular_triangulation_2` is designed to maintain the regular triangulation of a set of weighted points.

Let \( { PW} = \{(p_i, w_i), i = 1, \ldots , n \}\) be a set of weighted points where each \( p_i\) is a point and each \( w_i\) is a scalar called the weight of point \( p_i\). Alternatively, each weighted point \( (p_i, w_i)\) can be regarded as a two dimensional sphere with center \( p_i\) and radius \( r_i=\sqrt{w_i}\).

The power diagram of the set \( { PW}\) is a planar partition such that each cell corresponds to sphere \( (p_i, w_i)\) of \( { PW}\) and is the locus of points \( p\) whose power with respect to \( (p_i, w_i)\) is less than its power with respect to any other sphere \( (p_j, w_j)\) in \( { PW}\). The dual of this diagram is a triangulation whose domain covers the convex hull of the set \( { P}= \{ p_i, i = 1, \ldots , n \}\) of center points and whose vertices are a subset of \( { P}\). Such a triangulation is called a regular triangulation. The three points \( p_i, p_j\) and \( p_k\) of \( { P}\) form a triangle in the regular triangulation of \( { PW}\) iff there is a point \( p\) of the plane whose powers with respect to \( (p_i, w_i)\), \( (p_j, w_j)\) and \( (p_k, w_k)\) are equal and less than the power of \( p\) with respect to any other sphere in \( { PW}\).

Let us defined the power product of two weighted points \( (p_i, w_i)\) and \( (p_j, w_j)\) as:

\[ \Pi(p_i, w_i,p_j, w_j) = p_ip_j ^2 - w_i - w_j . \]

\( \Pi(p_i, w_i,p_j, 0)\) is simply the power of point \( p_j\) with respect to the sphere \( (p_i, w_i)\), and two weighted points are said to be orthogonal if their power product is null. The power circle of three weighted points \( (p_i, w_i)\), \( (p_j, w_j)\) and \( (p_k, w_k)\) is defined as the unique circle \( (\pi, \omega)\) orthogonal to \( (p_i, w_i)\), \( (p_j, w_j)\) and \( (p_k, w_k)\).

The regular triangulation of the sets \( { PW}\) satisfies the following regular property (which just reduces to the Delaunay property when all the weights are null): a triangle \( p_ip_jp_k\) of the regular triangulation of \( { PW}\) is such that the power product of any weighted point \( (p_l, w_l)\) of \( { PW}\) with the power circle of \( (p_i, w_i)\), \( (p_j, w_j)\) is \( (p_k, w_k)\) is positive or null. We call power test of the weighted point \( (p_l, w_l)\) with respect to the face \( p_ip_jp_k\), the predicates testing the sign of the power product of \( (p_l, w_l)\) with respect to the power circle of \( (p_i, w_i)\), \( (p_j, w_j)\) is \( (p_k, w_k)\). This power product is given by the following determinant

\[ \left| \begin{array}{cccc} 1 & x_i & y_i & x_i ^2 + y_i ^2 - w_i \\ 1 & x_j & y_j & x_j ^2 + y_j ^2 - w_j \\ 1 & x_k & y_k & x_k ^2 + y_k ^2 - w_k \\ 1 & x_l & y_l & x_l ^2 + y_l ^2 - w_l \end{array} \right| \]

A pair of neighboring faces \( p_ip_jp_k\) and \( p_ip_jp_l\) is said to be locally regular (with respect to the weights in \( { PW}\)) if the power test of \( (p_l,w_l)\) with respect to \( p_ip_jp_k\) is positive. A classical result of computational geometry establishes that a triangulation of the convex hull of \( { P}\) such that any pair of neighboring faces is regular with respect to \( { PW}\), is a regular triangulation of \( { PW}\).

Alternatively, the regular triangulation of the weighted points set \( { PW}\) can be obtained as the projection on the two dimensional plane of the convex hull of the set of three dimensional points \( { P'}= \{ (p_i,p_i ^2 - w_i ), i = 1, \ldots , n \}\).

The vertices of the regular triangulation of a set of weighted points \( { PW}\) form only a subset of the set of center points of \( { PW}\). Therefore the insertion of a weighted point in a regular triangulation does not necessarily imply the creation of a new vertex. If the new inserted point does not appear as a vertex in the regular triangulation, it is said to be hidden.

Hidden points are stored in special vertices called hidden vertices. A hidden point is considered as hidden by the facet of the triangulation where its point component is located : in fact, the hidden point can appear as vertex of the triangulation only if this facet is removed. Each face of a regular triangulation stores the list of hidden vertices whose points are located in the facet. When a facet is removed, points hidden by this facet are reinserted in the triangulation.

Template Parameters

  Traits | is the geometric traits parameter and must be a model of the concept `RegularTriangulationTraits_2`. The concept `RegularTriangulationTraits_2` refines the concept `TriangulationTraits_2` by adding the type `Weighted_point_2` to describe weighted points and the type `Power_side_of_oriented_power_circle_2` to perform power tests on weighted points.
  Tds | must be a model of `TriangulationDataStructure_2`. The face base of a regular triangulation has to be a model of the concept `RegularTriangulationFaceBase_2`. while the vertex base class has to be a model of `RegularTriangulationVertexBase_2`. CGAL provides a default instantiation for the `Tds` parameter by the class `Triangulation_data_structure_2 < Regular_triangulation_vertex_base_2<Traits>, Regular_triangulation_face_base_2<Traits> >`.

See also
  `CGAL::Triangulation_2<Traits,Tds>`
  `TriangulationDataStructure_2`
  `RegularTriangulationTraits_2`
  `RegularTriangulationFaceBase_2`
  `RegularTriangulationVertexBase_2`
  `CGAL::Regular_triangulation_face_base_2<Traits>`
  `CGAL::Regular_triangulation_vertex_base_2<Traits>`

Examples
  Triangulation_2/info_insert_with_pair_iterator_regular_2.cpp, and Triangulation_2/regular.cpp.

## Types
typedef Traits::Distance | Distance
typedef Traits::Line | Line
typedef Traits::Ray | Ray
typedef Traits::Point_2 | Bare_point
typedef Traits::Weighted_point_2 | Weighted_point
typedef unspecified_type | All_vertices_iterator
| An iterator that allows to enumerate the vertices that are not hidden.
typedef unspecified_type | Finite_vertices_iterator
| An iterator that allows to enumerate the finite vertices that are not hidden.
typedef unspecified_type | Hidden_vertices_iterator
| An iterator that allows to enumerate the hidden vertices.

## Creation
| Regular_triangulation_2 (const Traits &gt=Traits())
| Introduces an empty regular triangulation.
| Regular_triangulation_2 (const Regular_triangulation_2 &rt)
| Copy constructor.
template<class InputIterator >
Regular_triangulation_2< Traits, Tds > | Regular_triangulation_2 (InputIterator first, InputIterator last, Traits gt=Traits())
| Equivalent to constructing an empty triangulation with the optional traits class argument and calling insert(first,last).

## Insertion and Removal
Vertex_handle | insert (const Weighted_point &p, Face_handle f=Face_handle())
| inserts weighted point `p` in the regular triangulation.
Vertex_handle | insert (const Weighted_point &p, Locate_type lt, Face_handle loc, int li)
| insert a weighted point `p` whose bare-point is assumed to be located in `lt,loc,li`.
Vertex_handle | push_back (const Point &p)
| Equivalent to `insert(p)`.
template<class InputIterator >
std::ptrdiff_t | insert (InputIterator first, InputIterator last)
| inserts the weighted points in the range `[first,last)`.
template<class WeightedPointWithInfoInputIterator >
std::ptrdiff_t | insert (WeightedPointWithInfoInputIterator first, WeightedPointWithInfoInputIterator last)
| inserts the weighted points in the range `[first,last)`.
void | remove (Vertex_handle v)
| removes the vertex from the triangulation.

## Queries
template<class OutputItFaces , class OutputItBoundaryEdges , class OutputItHiddenVertices >
CGAL::Triple< OutputItFaces, OutputItBoundaryEdges, OutputItHiddenVertices > | get_conflicts_and_boundary_and_hidden_vertices (const Weighted_point &p, OutputItFaces fit, OutputItBoundaryEdges eit, OutputItHiddenVertices vit, Face_handle start) const
| outputs the faces, boundary edges, and hidden vertices of the conflict zone of point `p` to output iterators.
template<class OutputItFaces , class OutputItBoundaryEdges >
std::pair< OutputItFaces, OutputItBoundaryEdges > | get_conflicts_and_boundary (const Weighted_point &p, OutputItFaces fit, OutputItBoundaryEdges eit, Face_handle start) const
| outputs the faces and boundary edges of the conflict zone of point `p` to output iterators.
template<class OutputItFaces , class OutputItHiddenVertices >
std::pair< OutputItFaces, OutputItHiddenVertices > | get_conflicts_and_hidden_vertices (const Weighted_point &p, OutputItFaces fit, OutputItHiddenVertices vit, Face_handle start) const
| outputs the faces and hidden vertices of the conflict zone of point `p` to output iterators.
template<class OutputItBoundaryEdges , class OutputItHiddenVertices >
std::pair< OutputItBoundaryEdges, OutputItHiddenVertices > | get_boundary_of_conflicts_and_hidden_vertices (const Weighted_point &p, OutputItBoundaryEdges eit, OutputItHiddenVertices vit, Face_handle start) const
| outputs the boundary edges and hidden vertices of the conflict zone of point `p` to output iterators.
template<class OutputItFaces >
OutputItFaces | get_conflicts (const Point &p, OutputItFaces fit, Face_handle start) const
| outputs the faces of the conflict zone of point `p` to output iterators.
template<class OutputItBoundaryEdges >
OutputItBoundaryEdges | get_boundary_of_conflicts (const Point &p, OutputItBoundaryEdges eit, Face_handle start) const
| outputs the boundary edges of the conflict zone of `p` in counterclockwise order where each edge is described through the incident face which is not in conflict with `p`.
template<class OutputItHiddenVertices >
OutputItHiddenVertices | get_hidden_vertices (const Point &p, OutputItHiddenVertices vit, Face_handle start) const
| outputs the hidden vertices of the conflict zone of `p` into an output iterator.
Vertex_handle | nearest_power_vertex (Bare_point p) const
| Returns the vertex of the triangulation which is nearest to `p` with respect to the power distance.

## Access Functions
int | number_of_vertices () const
| returns the number of finite vertices that are not hidden.
int | number_of_hidden_vertices () const
| returns the number of hidden vertices.
Hidden_vertices_iterator | hidden_vertices_begin () const
| starts at an arbitrary hidden vertex.
Hidden_vertices_iterator | hidden_vertices_end () const
| past the end iterator for the sequence of hidden vertices.
Finite_vertices_iterator | finite_vertices_begin () const
| starts at an arbitrary unhidden finite vertex
Finite_vertices_iterator | finite_vertices_end () const
| Past-the-end iterator.
All_vertices_iterator | all_vertices_end () const
| starts at an arbitrary unhidden vertex.
All_vertices_iterator | all_vertices_begin () const
| past the end iterator.

## Dual Power Diagram
The following member functions provide the elements of the dual power diagram.
Point | weighted_circumcenter (const Face_handle &f) const
| returns the center of the circle orthogonal to the three weighted points corresponding to the vertices of face `f`.
Point | dual (const Face_handle &f) const
| same as weighted_circumcenter.
Object | dual (const Edge &e) const
| If both incident faces are finite, returns a segment whose endpoints are the duals of each incident face.
Object | dual (const Edge_circulator &ec) const
| Idem.
Object | dual (const Edge_iterator &ei) const
| Idem.
template<class Stream >
Stream & | draw_dual (Stream &ps)
| output the dual power diagram to stream `ps`.

## Predicates
Oriented_side | power_test (Face_handle f, const Weighted_point &p) const
| Returns the power test of `p` with respect to the power circle associated with `f`.

## Miscellaneous
bool | is_valid (bool verbose=false, int level=0) const
| Tests the validity of the triangulation as a `Triangulation_2` and additionally test the regularity of the triangulation.

## Additional Inherited Members
Public Types inherited from CGAL::Triangulation_2< Traits, Tds >
enum | Locate_type { VERTEX =0 , EDGE , FACE , OUTSIDE_CONVEX_HULL , OUTSIDE_AFFINE_HULL }
| specifies which case occurs when locating a point in the triangulation. More...
typedef Tds::Vertex_handle | Vertex_handle
| handle to a vertex.
typedef Tds::Face_handle | Face_handle
| handle to a face.
typedef Tds::Face_iterator | All_faces_iterator
| iterator over all faces.
typedef Tds::Edge_iterator | All_edges_iterator
| iterator over all edges.
typedef Tds::Vertex_iterator | All_vertices_iterator
| iterator over all vertices.
typedef unspecified_type | Finite_faces_iterator
| iterator over finite faces.
typedef unspecified_type | Finite_edges_iterator
| iterator over finite edges.
typedef unspecified_type | Finite_vertices_iterator
| iterator over finite vertices.
typedef unspecified_type | Point_iterator
| iterator over the points corresponding to the finite vertices of the triangulation.
typedef Iterator_range< unspecified_type > | All_face_handles
| range type for iterating over all faces (including infinite faces), with a nested type `iterator` that has as value type `Face_handle`.
typedef Iterator_range< All_edges_iterator > | All_edges
| range type for iterating over all edges (including infinite ones).
typedef Iterator_range< unspecified_type > | All_vertex_handles
| range type for iterating over all vertices (including the infinite vertex), with a nested type `iterator` that has as value type `Vertex_handle`.
typedef Iterator_range< unspecified_type > | Finite_face_handles
| range type for iterating over finite faces, with a nested type `iterator` that has as value type `Face_handle`.
typedef Iterator_range< Finite_edges_iterator > | Finite_edges
| range type for iterating over finite edges.
typedef Iterator_range< unspecified_type > | Finite_vertex_handles
| range type for iterating over finite vertices, with a nested type `iterator` that has as value type `Vertex_handle`.
typedef Iterator_range< Point_iterator > | Points
| range type for iterating over the points of the finite vertices.
typedef unspecified_type | Line_face_circulator
| circulator over all faces intersected by a line.
typedef unspecified_type | Face_circulator
| circulator over all faces incident to a given vertex.
typedef unspecified_type | Edge_circulator
| circulator over all edges incident to a given vertex.
typedef unspecified_type | Vertex_circulator
| circulator over all vertices incident to a given vertex.
typedef Traits | Geom_traits
| the traits class.
typedef Tds | Triangulation_data_structure
| the triangulation data structure type.
typedef Traits::Point_2 | Point
| the point type.
typedef Traits::Segment_2 | Segment
| the segment type.
typedef Traits::Triangle_2 | Triangle
| the triangle type.
typedef Tds::Vertex | Vertex
| the vertex type.
typedef Tds::Face | Face
| the face type.
typedef Tds::Edge | Edge
| the edge type.
typedef Tds::size_type | size_type
| Size type (an unsigned integral type).
typedef Tds::difference_type | difference_type
| Difference type (a signed integral type).
Public Member Functions inherited from CGAL::Triangulation_2< Traits, Tds >
| Triangulation_2 (const Traits &gt=Traits())
| Introduces an empty triangulation.
| Triangulation_2 (const Triangulation_2 &tr)
| Copy constructor.
template<class InputIterator >
| Triangulation_2 (InputIterator first, InputIterator last, const Traits &gt=Traits())
| Equivalent to constructing an empty triangulation with the optional traits class argument and calling insert(first,last).
Triangulation_2 | operator= (const Triangulation_2< Traits, Tds > &tr)
| Assignment.
void | swap (Triangulation_2 &tr)
| The triangulations `tr` and `*this` are swapped.
void | clear ()
| Deletes all faces and finite vertices resulting in an empty triangulation.
int | dimension () const
| Returns the dimension of the convex hull.
size_type | number_of_vertices () const
| Returns the number of finite vertices.
size_type | number_of_faces () const
| Returns the number of finite faces.
Face_handle | infinite_face () const
| a face incident to the infinite vertex.
Vertex_handle | infinite_vertex () const
| the infinite vertex.
Vertex_handle | finite_vertex () const
| a vertex distinct from the infinite vertex.
const Geom_traits & | geom_traits () const
| Returns a const reference to the triangulation traits object.
const TriangulationDataStructure_2 & | tds () const
| Returns a const reference to the triangulation data structure.
TriangulationDataStructure_2 & | tds ()
| Returns a reference to the triangulation data structure.
bool | is_infinite (Vertex_handle v) const
| `true` iff `v` is the infinite vertex.
bool | is_infinite (Face_handle f) const
| `true` iff face `f` is infinite.
bool | is_infinite (Face_handle f, int i) const
| `true` iff edge `(f,i)` is infinite.
bool | is_infinite (Edge e) const
| `true` iff edge `e` is infinite.
bool | is_infinite (Edge_circulator ec) const
| `true` iff edge `*ec` is infinite.
bool | is_infinite (All_edges_iterator ei) const
| `true` iff edge `*ei` is infinite.
bool | is_edge (Vertex_handle va, Vertex_handle vb)
| `true` if there is an edge having `va` and `vb` as vertices.
bool | is_edge (Vertex_handle va, Vertex_handle vb, Face_handle &fr, int &i)
| as above.
bool | includes_edge (Vertex_handle va, Vertex_handle vb, Vertex_handle &vbb, Face_handle &fr, int &i)
| `true` if the line segment from `va` to `vb` includes an edge `e` incident to `va`.
bool | is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3)
| `true` if there is a face having `v1`, `v2` and `v3` as vertices.
bool | is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3, Face_handle &fr)
| as above.
Face_handle | locate (const Point &query, Face_handle f=Face_handle()) const
| If the point `query` lies inside the convex hull of the points, a face that contains the query in its interior or on its boundary is returned.
Face_handle | inexact_locate (const Point &query, Face_handle start=Face_handle()) const
| Same as `locate()` but uses inexact predicates.
Face_handle | locate (const Point &query, Locate_type &lt, int &li, Face_handle h=Face_handle()) const
| Same as above.
Oriented_side | oriented_side (Face_handle f, const Point &p) const
| Returns on which side of the oriented boundary of `f` lies the point `p`.
Oriented_side | side_of_oriented_circle (Face_handle f, const Point &p)
| Returns on which side of the circumcircle of face `f` lies the point `p`.
void | flip (Face_handle f, int i)
| Exchanges the edge incident to `f` and `f->neighbor(i)` with the other diagonal of the quadrilateral formed by `f` and `f->neighbor(i)`.
Vertex_handle | insert (const Point &p, Face_handle f=Face_handle())
| Inserts point `p` in the triangulation and returns the corresponding vertex.
Vertex_handle | insert (const Point &p, Locate_type lt, Face_handle loc, int li)
| Same as above except that the location of the point `p` to be inserted is assumed to be given by `(lt,loc,i)` (see the description of the `locate` method above.)
Vertex_handle | push_back (const Point &p)
| Equivalent to `insert(p)`.
template<class PointInputIterator >
std::ptrdiff_t | insert (PointInputIterator first, PointInputIterator last)
| Inserts the points in the range `[first,last)` in the given order, and returns the number of inserted points.
template<class PointWithInfoInputIterator >
std::ptrdiff_t | insert (PointWithInfoInputIterator first, PointWithInfoInputIterator last)
| inserts the points in the iterator range `[first,last)` in the given order, and returns the number of inserted points.
void | remove (Vertex_handle v)
| Removes the vertex from the triangulation.
Vertex_handle | move_if_no_collision (Vertex_handle v, const Point &p)
| If there is not already another vertex placed on `p`, the triangulation is modified such that the new position of vertex `v` is `p`, and `v` is returned.
Vertex_handle | move (Vertex_handle v, const Point &p)
| If there is no collision during the move, this function is the same as `move_if_no_collision` .
Vertex_handle | insert_first (const Point &p)
| Inserts the first finite vertex .
Vertex_handle | insert_second (const Point &p)
| Inserts the second finite vertex .
Vertex_handle | insert_in_face (const Point &p, Face_handle f)
| Inserts vertex `v` in face `f`.
Vertex_handle | insert_in_edge (const Point &p, Face_handle f, int i)
| Inserts vertex v in edge `i` of `f`.
Vertex_handle | insert_outside_convex_hull (const Point &p, Face_handle f)
| Inserts a point which is outside the convex hull but in the affine hull.
Vertex_handle | insert_outside_affine_hull (const Point &p)
| Inserts a point which is outside the affine hull.
void | remove_degree_3 (Vertex_handle v)
| Removes a vertex of degree three.
void | remove_second (Vertex_handle v)
| Removes the before last finite vertex.
void | remove_first (Vertex_handle v)
| Removes the last finite vertex.
template<class EdgeIt >
Vertex_handle | star_hole (Point p, EdgeIt edge_begin, EdgeIt edge_end)
| creates a new vertex `v` and use it to star the hole whose boundary is described by the sequence of edges `[edge_begin, edge_end)`.
template<class EdgeIt , class FaceIt >
Vertex_handle | star_hole (Point p, EdgeIt edge_begin, EdgeIt edge_end, FaceIt face_begin, FaceIt face_end)
| same as above, except that the algorithm first recycles faces in the sequence `[face_begin, face_end)` and create new ones only when the sequence is exhausted.
Finite_vertices_iterator | finite_vertices_begin () const
| Starts at an arbitrary finite vertex.
Finite_vertices_iterator | finite_vertices_end () const
| Past-the-end iterator.
Finite_edges_iterator | finite_edges_begin () const
| Starts at an arbitrary finite edge.
Finite_edges_iterator | finite_edges_end () const
| Past-the-end iterator.
Finite_faces_iterator | finite_faces_begin () const
| Starts at an arbitrary finite face.
Finite_faces_iterator | finite_faces_end () const
| Past-the-end iterator.
Point_iterator | points_begin () const
Point_iterator | points_end () const
| Past-the-end iterator.
Finite_vertex_handles | finite_vertex_handles () const
| returns a range of iterators over finite vertices.
Finite_edges | finite_edges () const
| returns a range of iterators over finite edges.
Finite_face_handles | finite_face_handles () const
| returns a range of iterators over finite faces.
Points | points () const
| returns a range of iterators over the points of finite vertices.
All_vertices_iterator | all_vertices_begin () const
| Starts at an arbitrary vertex.
All_vertices_iterator | all_vertices_end () const
| Past-the-end iterator.
All_edges_iterator | all_edges_begin () const
| Starts at an arbitrary edge.
All_edges_iterator | all_edges_end () const
| Past-the-end iterator.
All_faces_iterator | all_faces_begin () const
| Starts at an arbitrary face.
All_faces_iterator | all_faces_end () const
| Past-the-end iterator.
All_vertex_handles | all_vertex_handles () const
| returns a range of iterators over all vertices.
All_edges | all_edges () const
| returns a range of iterators over all edges.
All_face_handles | all_face_handles () const
| returns a range of iterators over all faces.
Line_face_circulator | line_walk (const Point &p, const Point &q, Face_handle f=Face_handle()) const
| This function returns a circulator that allows to visit the faces intersected by the line `pq`.
Face_circulator | incident_faces (Vertex_handle v) const
| Starts at an arbitrary face incident to `v`.
Face_circulator | incident_faces (Vertex_handle v, Face_handle f) const
| Starts at face `f`.
Edge_circulator | incident_edges (Vertex_handle v) const
| Starts at an arbitrary edge incident to `v`.
Edge_circulator | incident_edges (Vertex_handle v, Face_handle f) const
| Starts at the first edge of `f` incident to `v`, in counterclockwise order around `v`.
Vertex_circulator | incident_vertices (Vertex_handle v) const
| Starts at an arbitrary vertex incident to `v`.
Vertex_circulator | incident_vertices (Vertex_handle v, Face_handle f)
| Starts at the first vertex of `f` adjacent to `v` in counterclockwise order around `v`.
Vertex_handle | mirror_vertex (Face_handle f, int i) const
| returns the vertex of the \( i^{th}\) neighbor of `f` that is opposite to `f`.
int | mirror_index (Face_handle f, int i) const
| returns the index of `f` in its \( i^{th}\) neighbor.
Edge | mirror_edge (Edge e) const
| returns the same edge seen from the other adjacent face.
int | ccw (int i) const
| Returns \( i+1\) modulo 3.
int | cw (int i) const
| Returns \( i+2\) modulo 3.
Triangle | triangle (Face_handle f) const
| Returns the triangle formed by the three vertices of `f`.
Segment | segment (Face_handle f, int i) const
| Returns the line segment formed by the vertices `ccw(i)` and `cw(i)` of face `f`.
Segment | segment (const Edge &e) const
| Returns the line segment corresponding to edge `e`.
Segment | segment (const Edge_circulator &ec) const
| Returns the line segment corresponding to edge `*ec`.
Segment | segment (const Edge_iterator &ei) const
| Returns the line segment corresponding to edge `*ei`.
Point | circumcenter (Face_handle f) const
| Compute the circumcenter of the face pointed to by f.
void | set_infinite_vertex (const Vertex_handle &v)
| This is an advanced function.
bool | is_valid (bool verbose=false, int level=0) const
| Checks the combinatorial validity of the triangulation and also the validity of its geometric embedding.
Public Member Functions inherited from CGAL::Triangulation_cw_ccw_2
| Triangulation_cw_ccw_2 ()
| default constructor.
int | ccw (const int i) const
| returns the index of the neighbor or vertex that is next to the neighbor or vertex with index `i` in counterclockwise order around a face.
int | cw (const int i) const
| returns the index of the neighbor or vertex that is next to the neighbor or vertex with index `i` in counterclockwise order around a face.
Related Functions inherited from CGAL::Triangulation_2< Traits, Tds >
ostream & | operator<< (ostream &os, const Triangulation_2< Traits, Tds > &T)
| Inserts the triangulation into the stream `os`.
istream & | operator>> (istream &is, const Triangulation_2< Traits, Tds > &T)
| Reads a triangulation from stream `is` and assigns it to the triangulation.

## Member Function Documentation

## ◆ dual()

template<typename Traits , typename Tds >

Object CGAL::Regular_triangulation_2< Traits, Tds >::dual | ( | const Edge & | e | ) | const

If both incident faces are finite, returns a segment whose endpoints are the duals of each incident face.

If only one incident face is finite, returns a ray whose endpoint is the dual of the finite incident face and supported by the line which is the bisector of the edge's endpoints. If both incident faces are infinite, returns the line which is the bisector of the edge's endpoints otherwise.

## ◆ get_boundary_of_conflicts()

template<typename Traits , typename Tds >
template<class OutputItBoundaryEdges >

OutputItBoundaryEdges CGAL::Regular_triangulation_2< Traits, Tds >::get_boundary_of_conflicts | ( | const Point & | p,
|  | OutputItBoundaryEdges | eit,
|  | Face_handle | start
| ) |  | const

outputs the boundary edges of the conflict zone of `p` in counterclockwise order where each edge is described through the incident face which is not in conflict with `p`.

The function returns the resulting output iterator.

## ◆ get_boundary_of_conflicts_and_hidden_vertices()

template<typename Traits , typename Tds >
template<class OutputItBoundaryEdges , class OutputItHiddenVertices >

std::pair< OutputItBoundaryEdges, OutputItHiddenVertices > CGAL::Regular_triangulation_2< Traits, Tds >::get_boundary_of_conflicts_and_hidden_vertices | ( | const Weighted_point & | p,
|  | OutputItBoundaryEdges | eit,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the boundary edges and hidden vertices of the conflict zone of point `p` to output iterators.

See `get_conflicts_and_boundary_and_hidden_vertices()` for details. The function returns in a `std::pair` the resulting output iterators.

## ◆ get_conflicts()

template<typename Traits , typename Tds >
template<class OutputItFaces >

OutputItFaces CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts | ( | const Point & | p,
|  | OutputItFaces | fit,
|  | Face_handle | start
| ) |  | const

outputs the faces of the conflict zone of point `p` to output iterators.

The function returns the resulting output iterator.

Precondition
  `dimension()==2`.

## ◆ get_conflicts_and_boundary()

template<typename Traits , typename Tds >
template<class OutputItFaces , class OutputItBoundaryEdges >

std::pair< OutputItFaces, OutputItBoundaryEdges > CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts_and_boundary | ( | const Weighted_point & | p,
|  | OutputItFaces | fit,
|  | OutputItBoundaryEdges | eit,
|  | Face_handle | start
| ) |  | const

outputs the faces and boundary edges of the conflict zone of point `p` to output iterators.

See `get_conflicts_and_boundary_and_hidden_vertices()` for details.

The function returns in a `std::pair` the resulting output iterators.

Precondition
  `dimension()==2`.

## ◆ get_conflicts_and_boundary_and_hidden_vertices()

template<typename Traits , typename Tds >
template<class OutputItFaces , class OutputItBoundaryEdges , class OutputItHiddenVertices >

CGAL::Triple< OutputItFaces, OutputItBoundaryEdges, OutputItHiddenVertices > CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts_and_boundary_and_hidden_vertices | ( | const Weighted_point & | p,
|  | OutputItFaces | fit,
|  | OutputItBoundaryEdges | eit,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the faces, boundary edges, and hidden vertices of the conflict zone of point `p` to output iterators.

Template Parameters

  OutputItFaces | is an output iterator with `Face_handle` as value type.
  OutputItBoundaryEdges | is an output iterator with `Edge` as value type.
  OutputItHiddenVertices | is an output iterator with `Vertex_handle` as value type.

This member function outputs in the container pointed to by `fit` the faces which are in conflict with point `p`, i.e., the faces whose power circles have negative power wrt. `p`. It outputs in the container pointed to by `eit` the boundary of the zone in conflict with `p`. It inserts the vertices that would be hidden by `p` into the container pointed to by `vit`. The boundary edges of the conflict zone are output in counter-clockwise order and each edge is described through its incident face which is not in conflict with `p`. The function returns in a `CGAL::Triple` the resulting output iterators.

Precondition
  `dimension()==2`.

## ◆ get_conflicts_and_hidden_vertices()

template<typename Traits , typename Tds >
template<class OutputItFaces , class OutputItHiddenVertices >

std::pair< OutputItFaces, OutputItHiddenVertices > CGAL::Regular_triangulation_2< Traits, Tds >::get_conflicts_and_hidden_vertices | ( | const Weighted_point & | p,
|  | OutputItFaces | fit,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the faces and hidden vertices of the conflict zone of point `p` to output iterators.

See `get_conflicts_and_boundary_and_hidden_vertices()` for details. The function returns in a `std::pair` the resulting output iterators.

Precondition
  `dimension()==2`.

## ◆ get_hidden_vertices()

template<typename Traits , typename Tds >
template<class OutputItHiddenVertices >

OutputItHiddenVertices CGAL::Regular_triangulation_2< Traits, Tds >::get_hidden_vertices | ( | const Point & | p,
|  | OutputItHiddenVertices | vit,
|  | Face_handle | start
| ) |  | const

outputs the hidden vertices of the conflict zone of `p` into an output iterator.

The function returns the resulting output iterator.

## ◆ insert() [1/4]

template<typename Traits , typename Tds >

Vertex_handle CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | const Weighted_point & | p,
|  | Face_handle | f = `Face_handle()`
| ) |  |

inserts weighted point `p` in the regular triangulation.

If the point `p` does not appear as a vertex of the triangulation, the returned vertex is a hidden vertex. If given the parameter `f` is used as an hint for the place to start the location process of point `p`.

## ◆ insert() [2/4]

template<typename Traits , typename Tds >

Vertex_handle CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | const Weighted_point & | p,
|  | Locate_type | lt,
|  | Face_handle | loc,
|  | int | li
| ) |  |

insert a weighted point `p` whose bare-point is assumed to be located in `lt,loc,li`.

See the description of member function `Triangulation_2::locate()`.

## ◆ insert() [3/4]

template<typename Traits , typename Tds >
template<class InputIterator >

std::ptrdiff_t CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | InputIterator | first,
|  | InputIterator | last
| ) |  |

inserts the weighted points in the range `[first,last)`.

It returns the difference of the number of vertices between after and before the insertions (it may be negative due to hidden points). Note that this function is not guaranteed to insert the weighted points following the order of `InputIterator`, as `spatial_sort()` is used to improve efficiency.

Template Parameters

  InputIterator | must be an input iterator with the value type `Weighted_point` .

## ◆ insert() [4/4]

template<typename Traits , typename Tds >
template<class WeightedPointWithInfoInputIterator >

std::ptrdiff_t CGAL::Regular_triangulation_2< Traits, Tds >::insert | ( | WeightedPointWithInfoInputIterator | first,
|  | WeightedPointWithInfoInputIterator | last
| ) |  |

inserts the weighted points in the range `[first,last)`.

It returns the difference of the number of vertices between after and before the insertions (it may be negative due to hidden points). Note that this function is not guaranteed to insert the weighted points following the order of `WeightedPointWithInfoInputIterator`, as `spatial_sort` is used to improve efficiency. Given a pair `(p,i)`, the vertex `v` storing `p` also stores `i`, that is `v.point() == p` and `v.info() == i`. If several pairs have the same point, only one vertex is created, one of the objects of type `Vertex::Info` will be stored in the vertex.

Precondition
  `Vertex` must be model of the concept `TriangulationVertexBaseWithInfo_2`.

Template Parameters

  WeightedPointWithInfoInputIterator | must be an input iterator with value type `std::pair<Weighted_point,Vertex::Info>`.

## ◆ is_valid()

template<typename Traits , typename Tds >

bool CGAL::Regular_triangulation_2< Traits, Tds >::is_valid | ( | bool | verbose = `false`,
|  | int | level = `0`
| ) |  | const

Tests the validity of the triangulation as a `Triangulation_2` and additionally test the regularity of the triangulation.

This method is useful to debug regular triangulation algorithms implemented by the user.

## ◆ nearest_power_vertex()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Regular_triangulation_2< Traits, Tds >::nearest_power_vertex | ( | Bare_point | p | ) | const

Returns the vertex of the triangulation which is nearest to `p` with respect to the power distance.

This means that the power of the query point `p` with respect to the weighted point in the nearest vertex is smaller than the power of `p` with respect to the weighted point in any other vertex. Ties are broken arbitrarily. The default constructed handle is returned if the triangulation is empty.

## ◆ weighted_circumcenter()

template<typename Traits , typename Tds >

Point CGAL::Regular_triangulation_2< Traits, Tds >::weighted_circumcenter | ( | const Face_handle & | f | ) | const

returns the center of the circle orthogonal to the three weighted points corresponding to the vertices of face `f`.

Precondition
  `f` is not infinite.