CGAL 5.6.3 - 2D Triangulations

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CGAL::Triangulation_2< Traits, Tds > Class Template Reference
2D Triangulations Reference » Triangulation Classes

`#include <CGAL/Triangulation_2.h>`

## Inherits from

CGAL::Triangulation_cw_ccw_2.

Inherited by CGAL::Constrained_triangulation_2< Traits, Tds, Itag >, CGAL::Delaunay_triangulation_2< Traits, Tds >, and CGAL::Regular_triangulation_2< Traits, Tds >.

## Definition

template<typename Traits, typename Tds>
class CGAL::Triangulation_2< Traits, Tds >

The class `Triangulation_2` is the basic class designed to handle triangulations of set of points \( { A}\) in the plane.

Such a triangulation has vertices at the points of \( { A}\) and its domain covers the convex hull of \( { A}\). It can be viewed as a planar partition of the plane whose bounded faces are triangular and cover the convex hull of \( { A}\). The single unbounded face of this partition is the complementary of the convex hull of \( { A}\).

In many applications, it is convenient to deal only with triangular faces. Therefore, we add to the triangulation a fictitious vertex, called the `infinite vertex` and we make each convex hull edge incident to an `infinite` face having as third vertex the `infinite vertex`. In that way, each edge is incident to exactly two faces and special cases at the boundary of the convex hull are simpler to deal with.

The infinite vertex.

The class `Triangulation_2` implements this point of view and therefore considers the triangulation of the set of points as a set of triangular, finite and infinite faces. Although it is convenient to draw a triangulation as in figure Triangulation_ref_Fig_infinite_vertex, note that the `infinite vertex` has no significant coordinates and that no geometric predicate can be applied on it or on an infinite face.

A triangulation is a collection of vertices and faces that are linked together through incidence and adjacency relations. Each face give access to its three incident vertices and to its three adjacent faces. Each vertex give access to one of its incident faces.

The three vertices of a face are indexed with 0, 1 and 2 in counterclockwise order. The neighbor of a face are also indexed with 0,1,2 in such a way that the neighbor indexed by \( i\) is opposite to the vertex with the same index.

The triangulation class offers two functions `int cw(int i)` and `int ccw(int i)` which, given the index of a vertex in a face, compute the index of the next vertex of the same face in clockwise or counterclockwise order. Thus, for example the neighbor `neighbor(cw(i))` is the neighbor of `f` which is next to `neighbor(i)` turning clockwise around `f`. The face `neighbor(cw(i))` is also the first face encountered after `f` when turning clockwise around vertex `i` of `f` (see Figure Triangulation_ref_Fig_neighbors).

Vertices and neighbors.

Template Parameters

  Traits | is the geometric traits which must be a model of the concept `TriangulationTraits_2`.
  Tds | is the triangulation data structure which must be a model of the concept `TriangulationDataStructure_2`. By default, the triangulation data structure is instantiated by `Triangulation_data_structure_2 < Triangulation_vertex_base_2<Gt>, Triangulation_face_base_2<Gt> >`.

Traversal of the Triangulation

A triangulation can be seen as a container of faces and vertices. Therefore the triangulation provides several iterators and circulators that allow to traverse it completely or partially.

Traversal of the Convex Hull

Applied on the infinite vertex the above functions allow to visit the vertices on the convex hull and the infinite edges and faces. Note that a counterclockwise traversal of the vertices adjacent to the infinite vertex is a clockwise traversal of the convex hull.

typedef CGAL::Triangulation_2<Traits, Tds> Triangulation_2;
Triangulation_2 t;
Triangulation_2::Face_handle f;
Face_circulator incident_faces(t.infinite_vertex()) const;
Face_circulator incident_faces(t.infinite_vertex(), f) const;
Edge_circulator incident_edges(t.infinite_vertex()) const;
Edge_circulator incident_edges(t.infinite_vertex(), f);
Vertex_circulator incident_vertices(t.infinite_vertex() v) ;
Vertex_circulator incident_vertices(t.infinite_vertex(), f) ;
CGAL::Triangulation_2
The class Triangulation_2 is the basic class designed to handle triangulations of set of points in t...
Definition: Triangulation_2.h:170
CGAL::Triangulation_2::incident_edges
Edge_circulator incident_edges(Vertex_handle v) const
Starts at an arbitrary edge incident to v.
CGAL::Triangulation_2::incident_faces
Face_circulator incident_faces(Vertex_handle v) const
Starts at an arbitrary face incident to v.
CGAL::Triangulation_2::Face_handle
Tds::Face_handle Face_handle
handle to a face.
Definition: Triangulation_2.h:266
CGAL::Triangulation_2::infinite_vertex
Vertex_handle infinite_vertex() const
the infinite vertex.
CGAL::Triangulation_2::incident_vertices
Vertex_circulator incident_vertices(Vertex_handle v) const
Starts at an arbitrary vertex incident to v.
unspecified_type

I/O

The I/O operators are defined for `iostream`. The format for the iostream is an internal format.

The information output in the `iostream` is:

- the number of vertices (including the infinite one), the number of faces (including infinite ones), and the dimension.
- for each vertex (except the infinite vertex), the non combinatorial information stored in that vertex (point, etc.).
- for each faces, the indices of its vertices and the non combinatorial information (if any) in this face.
- for each face again the indices of the neighboring faces.

The index of an item (vertex of face) is the rank of this item in the output order. When dimension \( <\) 2, the same information is output for faces of maximal dimension instead of faces.

Implementation

Locate is implemented by a line walk from a vertex of the face given as optional parameter (or from a finite vertex of `infinite_face()` if no optional parameter is given). It takes time \( O(n)\) in the worst case, but only \( O(\sqrt{n})\) on average if the vertices are distributed uniformly at random.

Insertion of a point is done by locating a face that contains the point, and then splitting this face. If the point falls outside the convex hull, the triangulation is restored by flips. Apart from the location, insertion takes a time time \( O(1)\). This bound is only an amortized bound for points located outside the convex hull.

Removal of a vertex is done by removing all adjacent triangles, and re-triangulating the hole. Removal takes time \( O(d^2)\) in the worst case, if \( d\) is the degree of the removed vertex, which is \( O(1)\) for a random vertex.

The face, edge, and vertex iterators on finite features are derived from their counterparts visiting all (finite and infinite) features which are themselves derived from the corresponding iterators of the triangulation data structure.

See also
  `TriangulationTraits_2`
  `TriangulationDataStructure_2`
  `TriangulationDataStructure_2::Face`
  `TriangulationDataStructure_2::Vertex`
  `CGAL::Triangulation_data_structure_2<Vb,Fb>`
  `CGAL::Triangulation_vertex_base_2<Traits>`
  `CGAL::Triangulation_face_base_2<Traits>`

Examples
  Triangulation_2/adding_handles.cpp, Triangulation_2/colored_face.cpp, Triangulation_2/draw_triangulation_2.cpp, Triangulation_2/for_loop_2.cpp, Triangulation_2/low_dimensional.cpp, and Triangulation_2/triangulation_prog1.cpp.

## Related Functions
(Note that these are not member functions.)
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.

## Handles, Iterators, and Circulators
The vertices and faces of the triangulations are accessed through handles, iterators and circulators. The handles are models of the concept `Handle` which basically offers the two dereference operators and `->`. The handles are also model of the concepts `LessThanComparable` and `Hashable`, that is they can be used as keys in containers such as `std::map` and `boost::unordered_map`. The iterators and circulators are all bidirectional and non-mutable. The circulators and iterators are convertible to handles with the same value type, so that whenever a handle appear in the parameter list of a function, an appropriate iterator or circulator can be passed as well. The edges of the triangulation can also be visited through iterators and circulators, the edge circulators and iterators are also bidirectional and non mutable. In the following, we called infinite any face or edge incident to the infinite vertex and the infinite vertex itself. Any other feature (face, edge or vertex) of the triangulation is said to be finite. Some iterators (the `All` iterators ) allows to visit finite or infinite feature while others (the `Finite` iterators) visit only finite features. Circulators visit infinite features as well as finite ones. The triangulation class also defines the following enum type to specify which case occurs when locating a point in the triangulation. In order to write C++ 11 `for`-loops we provide range types.
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.

## Types
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).

## Creation
| 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.

## Access Functions
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.

## Non const access
Attention The responsibility of keeping a valid triangulation belongs to the user when using advanced operations allowing a direct manipulation of the `tds`. This method is mainly a help for users implementing their own triangulation algorithms.
TriangulationDataStructure_2 & | tds ()
| Returns a reference to the triangulation data structure.

## Predicates
The class `Triangulation_2` provides methods to test the finite or infinite character of any feature, and also methods to test the presence in the triangulation of a particular feature (edge or face).
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.

## Queries
The class `Triangulation_2` provides methods to locate a given point with respect to a triangulation. It also provides methods to locate a point with respect to a given finite face of the triangulation.
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`.

## Modifiers
The following operations are guaranteed to lead to a valid triangulation when they are applied on a valid triangulation. Insertion of a point on an edge. Insertion in a face. Insertion outside the convex hull. Removal.
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` .

## Specialized Modifiers
The following member functions offer more specialized versions of the insertion or removal operations to be used when one knows to be in the corresponding case.
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 Face, Edge and Vertex Iterators
The following iterators allow respectively to visit finite faces, finite edges and finite vertices of the triangulation. These iterators are non mutable, bidirectional and their value types are respectively `Face`, `Edge` and `Vertex`. They are all invalidated by any change in the triangulation.
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 Face, Edge and Vertex Iterators
The following iterators allow respectively to visit all (finite or infinite) faces, edges and vertices of the triangulation. These iterators are non mutable, bidirectional and their value types are respectively `Face`, `Edge` and `Vertex`. They are all invalidated by any change in the triangulation.
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
The triangulation defines a circulator that allows to visit all faces that are intersected by a line. A face `f` is considered has being intersected by the oriented line `l` if either: `f` is a finite face whose interior intersects `l`, or `f` is a finite face with an edge collinear with `l` and lies to the left of `l`, or `f` is an infinite face incident to a convex hull edge whose interior is intersected by `l`, or `f` is an infinite face incident to a convex hull vertex lying on `l` and the finite edge of `f` lies to the left of `l`. The circulator has a singular value if the line `l` intersect no finite face of the triangulation. This circulator is non-mutable and bidirectional. Its value type is `Face`. Figure Triangulation_ref_Fig_Line_face_circulator illustrates which finite faces are enumerated. Lines \( l_1\) and \( l_2\) have no face to their left. Lines \( l_3\) and \( l_4\) have faces to their left. Note that the finite faces that are only vertex incident to lines \( l_3\) and \( l_4\) are not enumerated. The line face circulator. A line face circulator is invalidated if the face the circulator refers to is changed.
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, Edge and Vertex Circulators
The triangulation also provides circulators that allows to visit respectively all faces or edges incident to a given vertex or all vertices adjacent to a given vertex. These circulators are non-mutable and bidirectional. The `operator++` moves the circulator counterclockwise around the vertex while the `operator-` moves clockwise. A face circulator is invalidated by any modification of the face pointed to. An edge or a vertex circulator are invalidated by any modification of one of the two faces incident to the edge pointed to.
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`.

## Traversal Between Adjacent Faces
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.

## Miscellaneous
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.

## Setting
void | set_infinite_vertex (const Vertex_handle &v)
| This is an advanced function.

## Checking
Advanced The responsibility of keeping a valid triangulation belongs to the users if advanced operations are used. Obviously the advanced user, who implements higher levels operations may have to make a triangulation invalid at some times. The following method is provided to help the debugging.
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.

## Additional Inherited Members
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.

## Member Enumeration Documentation

## ◆ Locate_type

template<typename Traits , typename Tds >

enum CGAL::Triangulation_2::Locate_type

specifies which case occurs when locating a point in the triangulation.

See also
  `CGAL::Triangulation_2<Traits,Tds>`

Enumerator
VERTEX | when the located point coincides with a vertex of the triangulation
EDGE | when the point is in the relative interior of an edge
FACE | when the point is in the interior of a facet
OUTSIDE_CONVEX_HULL | when the point is outside the convex hull but in the affine hull of the current triangulation
OUTSIDE_AFFINE_HULL | when the point is outside the affine hull of the current triangulation.

## Constructor & Destructor Documentation

## ◆ Triangulation_2()

template<typename Traits , typename Tds >

CGAL::Triangulation_2< Traits, Tds >::Triangulation_2 | ( | const Triangulation_2< Traits, Tds > & | tr | ) |

Copy constructor.

All the vertices and faces are duplicated. After the copy, `*this` and `tr` refer to different triangulations: if `tr` is modified, `*this` is not.

## Member Function Documentation

## ◆ all_face_handles()

template<typename Traits , typename Tds >

All_face_handles CGAL::Triangulation_2< Traits, Tds >::all_face_handles | ( |  | ) | const

returns a range of iterators over all faces.

Note
  While the value type of `All_faces_iterator` is `Face`, the value type of `All_face_handles::iterator` is `Face_handle`.

## ◆ all_vertex_handles()

template<typename Traits , typename Tds >

All_vertex_handles CGAL::Triangulation_2< Traits, Tds >::all_vertex_handles | ( |  | ) | const

returns a range of iterators over all vertices.

Note
  While the value type of `All_vertices_iterator` is `Vertex`, the value type of `All_vertex_handles::iterator` is `Vertex_handle`.

## ◆ ccw()

template<typename Traits , typename Tds >

int CGAL::Triangulation_2< Traits, Tds >::ccw | ( | int | i | ) | const

Returns \( i+1\) modulo 3.

Precondition
  \( 0\leq i \leq2\).

## ◆ circumcenter()

template<typename Traits , typename Tds >

Point CGAL::Triangulation_2< Traits, Tds >::circumcenter | ( | Face_handle | f | ) | const

Compute the circumcenter of the face pointed to by f.

This function is available only if the corresponding function is provided in the geometric traits.

## ◆ cw()

template<typename Traits , typename Tds >

int CGAL::Triangulation_2< Traits, Tds >::cw | ( | int | i | ) | const

Returns \( i+2\) modulo 3.

Precondition
  \( 0\leq i \leq2\).

## ◆ finite_face_handles()

template<typename Traits , typename Tds >

Finite_face_handles CGAL::Triangulation_2< Traits, Tds >::finite_face_handles | ( |  | ) | const

returns a range of iterators over finite faces.

Note
  While the value type of `Finite_faces_iterator` is `Face`, the value type of `Finite_face_handles::iterator` is `Face_handle`.

## ◆ finite_vertex_handles()

template<typename Traits , typename Tds >

Finite_vertex_handles CGAL::Triangulation_2< Traits, Tds >::finite_vertex_handles | ( |  | ) | const

returns a range of iterators over finite vertices.

Note
  While the value type of `Finite_vertices_iterator` is `Vertex`, the value type of `Finite_vertex_handles::iterator` is `Vertex_handle`.

## ◆ flip()

template<typename Traits , typename Tds >

void CGAL::Triangulation_2< Traits, Tds >::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)`.

Precondition
  The faces `f` and `f->neighbor(i)` are finite faces and their union form a convex quadrilateral.

## ◆ incident_edges()

template<typename Traits , typename Tds >

Edge_circulator CGAL::Triangulation_2< Traits, Tds >::incident_edges | ( | Vertex_handle | v,
|  | Face_handle | f
| ) |  | const

Starts at the first edge of `f` incident to `v`, in counterclockwise order around `v`.

Precondition
  Face `f` is incident to vertex `v`.

## ◆ incident_faces()

template<typename Traits , typename Tds >

Face_circulator CGAL::Triangulation_2< Traits, Tds >::incident_faces | ( | Vertex_handle | v,
|  | Face_handle | f
| ) |  | const

Starts at face `f`.

Precondition
  Face `f` is incident to vertex `v`.

## ◆ incident_vertices()

template<typename Traits , typename Tds >

Vertex_circulator CGAL::Triangulation_2< Traits, Tds >::incident_vertices | ( | Vertex_handle | v,
|  | Face_handle | f
| ) |  |

Starts at the first vertex of `f` adjacent to `v` in counterclockwise order around `v`.

Precondition
  Face `f` is incident to vertex `v`.

## ◆ includes_edge()

template<typename Traits , typename Tds >

bool CGAL::Triangulation_2< Traits, Tds >::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`.

If `true`, `vbb` becomes the other vertex of `e`, `e` is the edge `(fr,i)` where `fr` is a handle to the face incident to `e` and on the right side `e` oriented from `va` to `vb`.

## ◆ inexact_locate()

template<typename Traits , typename Tds >

Face_handle CGAL::Triangulation_2< Traits, Tds >::inexact_locate | ( | const Point & | query,
|  | Face_handle | start = `Face_handle()`
| ) |  | const

Same as `locate()` but uses inexact predicates.

This function returns a handle on a face that is a good approximation of the exact location of `query`, while being faster. Note that it may return a handle on a face whose interior does not contain `query`. When the triangulation has dimension smaller than 2, `start` is returned.

## ◆ insert() [1/3]

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert | ( | const Point & | p,
|  | Face_handle | f = `Face_handle()`
| ) |  |

Inserts point `p` in the triangulation and returns the corresponding vertex.

If point `p` coincides with an already existing vertex, this vertex is returned and the triangulation remains unchanged.

If point `p` is on an edge, the two incident faces are split in two.

If point `p` is strictly inside a face of the triangulation, the face is split in three.

If point `p` is strictly outside the convex hull, `p` is linked to all visible points on the convex hull to form the new triangulation.

At last, if `p` is outside the affine hull (in case of degenerate 1-dimensional or 0-dimensional triangulations), `p` is linked all the other vertices to form a triangulation whose dimension is increased by one. The last argument `f` is an indication to the underlying locate algorithm of where to start.

Examples
  Triangulation_2/adding_handles.cpp, Triangulation_2/colored_face.cpp, and Triangulation_2/for_loop_2.cpp.

## ◆ insert() [2/3]

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

std::ptrdiff_t CGAL::Triangulation_2< Traits, Tds >::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 Parameters

  PointInputIterator | must be an input iterator with value type `Point`.

## ◆ insert() [3/3]

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

std::ptrdiff_t CGAL::Triangulation_2< Traits, Tds >::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.

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, and 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

  PointWithInfoInputIterator | must be an input iterator with the value type `std::pair<Point,Vertex::Info>`.

## ◆ insert_in_edge()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert_in_edge | ( | const Point & | p,
|  | Face_handle | f,
|  | int | i
| ) |  |

Inserts vertex v in edge `i` of `f`.

Precondition
  The point in vertex `v` lies on the edge opposite to the vertex `i` of face `f`.

## ◆ insert_in_face()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert_in_face | ( | const Point & | p,
|  | Face_handle | f
| ) |  |

Inserts vertex `v` in face `f`.

Face `f` is modified, two new faces are created.

Precondition
  The point in vertex `v` lies inside face `f`.

## ◆ insert_outside_convex_hull()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::insert_outside_convex_hull | ( | const Point & | p,
|  | Face_handle | f
| ) |  |

Inserts a point which is outside the convex hull but in the affine hull.

Precondition
  The handle `f` points to a face which is a proof of the location of`p`, see the description of the `locate` method above.

## ◆ is_edge()

template<typename Traits , typename Tds >

bool CGAL::Triangulation_2< Traits, Tds >::is_edge | ( | Vertex_handle | va,
|  | Vertex_handle | vb,
|  | Face_handle & | fr,
|  | int & | i
| ) |  |

as above.

In addition, if `true` is returned, the edge with vertices `va` and `vb` is the edge `e=(fr,i)` where `fr` is a handle to the face incident to `e` and on the right side of `e` oriented from `va` to `vb`.

## ◆ is_face()

template<typename Traits , typename Tds >

bool CGAL::Triangulation_2< Traits, Tds >::is_face | ( | Vertex_handle | v1,
|  | Vertex_handle | v2,
|  | Vertex_handle | v3,
|  | Face_handle & | fr
| ) |  |

as above.

In addition, if `true` is returned, fr is a handle to the face with `v1`, `v2` and `v3` as vertices.

## ◆ is_valid()

template<typename Traits , typename Tds >

bool CGAL::Triangulation_2< Traits, Tds >::is_valid | ( | bool | verbose = `false`,
|  | int | level = `0`
| ) |  | const

Checks the combinatorial validity of the triangulation and also the validity of its geometric embedding.

This method is mainly a debugging help for the users of advanced features.

## ◆ line_walk()

template<typename Traits , typename Tds >

Line_face_circulator CGAL::Triangulation_2< Traits, Tds >::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`.

If there is no such face the circulator has a singular value.

The starting point of the circulator is the face `f`, or the first finite face traversed by `l` , if `f` is omitted.

The circulator wraps around the infinite vertex: after the last traversed finite face, it steps through the infinite face adjacent to this face then through the infinite face adjacent to the first traversed finite face then through the first finite traversed face again.

Precondition
  The triangulation must have dimension 2.
  Points `p` and `q` must be different points.
  If `f != nullptr`, it must point to a finite face and the point `p` must be inside or on the boundary of `f`.

## ◆ locate() [1/2]

template<typename Traits , typename Tds >

Face_handle CGAL::Triangulation_2< Traits, Tds >::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.

If the point `query` lies outside the convex hull of the triangulation but in the affine hull, the returned face is an infinite face which is a proof of the point's location:

- for a two dimensional triangulation, it is a face \( (\infty, p, q)\) such that `query` lies to the left of the oriented line \( pq\) (the rest of the triangulation lying to the right of this line).
- for a degenerate one dimensional triangulation it is the (degenerate one dimensional) face \( (\infty, p, nullptr)\) such that `query` and the triangulation lie on either side of `p`.

If the point `query` lies outside the affine hull, the returned `Face_handle` is `nullptr`.

The optional `Face_handle` argument, if provided, is used as a hint of where the locate process has to start its search.

## ◆ locate() [2/2]

template<typename Traits , typename Tds >

Face_handle CGAL::Triangulation_2< Traits, Tds >::locate | ( | const Point & | query,
|  | Locate_type & | lt,
|  | int & | li,
|  | Face_handle | h = `Face_handle()`
| ) |  | const

Same as above.

Additionally, the parameters `lt` and `li` describe where the query point is located. The variable `lt` is set to the locate type of the query. If `lt==VERTEX` the variable `li` is set to the index of the vertex, and if `lt==EDGE` `li` is set to the index of the vertex opposite to the edge. Be careful that `li` has no meaning when the query type is `FACE`, `OUTSIDE_CONVEX_HULL`, or `OUTSIDE_AFFINE_HULL` or when the triangulation is \( 0\)-dimensional.

## ◆ mirror_edge()

template<typename Traits , typename Tds >

Edge CGAL::Triangulation_2< Traits, Tds >::mirror_edge | ( | Edge | e | ) | const

returns the same edge seen from the other adjacent face.

Precondition
  \( 0\leq i \leq2\).

## ◆ mirror_index()

template<typename Traits , typename Tds >

int CGAL::Triangulation_2< Traits, Tds >::mirror_index | ( | Face_handle | f,
|  | int | i
| ) |  | const

returns the index of `f` in its \( i^{th}\) neighbor.

Precondition
  \( 0\leq i \leq2\).

## ◆ mirror_vertex()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::mirror_vertex | ( | Face_handle | f,
|  | int | i
| ) |  | const

returns the vertex of the \( i^{th}\) neighbor of `f` that is opposite to `f`.

Precondition
  \( 0\leq i \leq2\).

## ◆ move()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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` .

Otherwise, `v` is removed and the vertex at point `p` is returned.

Precondition
  Vertex `v` must be finite.

## ◆ move_if_no_collision()

template<typename Traits , typename Tds >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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.

Otherwise, the triangulation is not modified and the vertex at point `p` is returned.

Precondition
  Vertex `v` must be finite.

## ◆ operator=()

template<typename Traits , typename Tds >

Triangulation_2 CGAL::Triangulation_2< Traits, Tds >::operator= | ( | const Triangulation_2< Traits, Tds > & | tr | ) |

Assignment.

All the vertices and faces are duplicated. After the assignment, `*this` and `tr` refer to different triangulations: if `tr` is modified, `*this` is not.

## ◆ oriented_side()

template<typename Traits , typename Tds >

Oriented_side CGAL::Triangulation_2< Traits, Tds >::oriented_side | ( | Face_handle | f,
|  | const Point & | p
| ) |  | const

Returns on which side of the oriented boundary of `f` lies the point `p`.

Precondition
  `f` is finite.

## ◆ remove()

template<typename Traits , typename Tds >

void CGAL::Triangulation_2< Traits, Tds >::remove | ( | Vertex_handle | v | ) |

Removes the vertex from the triangulation.

The created hole is re-triangulated.

Precondition
  Vertex `v` must be finite.

## ◆ remove_degree_3()

template<typename Traits , typename Tds >

void CGAL::Triangulation_2< Traits, Tds >::remove_degree_3 | ( | Vertex_handle | v | ) |

Removes a vertex of degree three.

Two of the incident faces are destroyed, the third one is modified.

Precondition
  Vertex `v` is a finite vertex with degree three.

## ◆ segment() [1/4]

template<typename Traits , typename Tds >

Segment CGAL::Triangulation_2< Traits, Tds >::segment | ( | const Edge & | e | ) | const

Returns the line segment corresponding to edge `e`.

Precondition
  `e` is a finite edge.

## ◆ segment() [2/4]

template<typename Traits , typename Tds >

Segment CGAL::Triangulation_2< Traits, Tds >::segment | ( | const Edge_circulator & | ec | ) | const

Returns the line segment corresponding to edge `*ec`.

Precondition
  `*ec` is a finite edge.

## ◆ segment() [3/4]

template<typename Traits , typename Tds >

Segment CGAL::Triangulation_2< Traits, Tds >::segment | ( | const Edge_iterator & | ei | ) | const

Returns the line segment corresponding to edge `*ei`.

Precondition
  `*ei` is a finite edge.

## ◆ segment() [4/4]

template<typename Traits , typename Tds >

Segment CGAL::Triangulation_2< Traits, Tds >::segment | ( | Face_handle | f,
|  | int | i
| ) |  | const

Returns the line segment formed by the vertices `ccw(i)` and `cw(i)` of face `f`.

Precondition
  \( 0\leq i \leq2\). The vertices `ccw(i)` and `cw(i)` of `f` are finite.

## ◆ set_infinite_vertex()

template<typename Traits , typename Tds >

void CGAL::Triangulation_2< Traits, Tds >::set_infinite_vertex | ( | const Vertex_handle & | v | ) |

This is an advanced function.

Advanced

This method is meant to be used only if you have done a low-level operation on the underlying tds that invalidated the infinite vertex. Sets the infinite vertex.

## ◆ side_of_oriented_circle()

template<typename Traits , typename Tds >

Oriented_side CGAL::Triangulation_2< Traits, Tds >::side_of_oriented_circle | ( | Face_handle | f,
|  | const Point & | p
| ) |  |

Returns on which side of the circumcircle of face `f` lies the point `p`.

The circle is assumed to be counterclockwise oriented, so its positive side correspond to its bounded side. This predicate is available only if the corresponding predicates on points is provided in the geometric traits class.

## ◆ star_hole() [1/2]

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

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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)`.

Returns a handle to the new vertex.

This function is intended to be used in conjunction with the `find_conflicts()` member functions of Delaunay and constrained Delaunay triangulations to perform insertions.

## ◆ star_hole() [2/2]

template<typename Traits , typename Tds >
template<class EdgeIt , class FaceIt >

Vertex_handle CGAL::Triangulation_2< Traits, Tds >::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.

This function is intended to be used in conjunction with the `find_conflicts()` member functions of Delaunay and constrained Delaunay triangulations to perform insertions.

## ◆ swap()

template<typename Traits , typename Tds >

void CGAL::Triangulation_2< Traits, Tds >::swap | ( | Triangulation_2< Traits, Tds > & | tr | ) |

The triangulations `tr` and `*this` are swapped.

This method should be used instead of assignment of copy constructor. if `tr` is deleted after that.

## ◆ triangle()

template<typename Traits , typename Tds >

Triangle CGAL::Triangulation_2< Traits, Tds >::triangle | ( | Face_handle | f | ) | const

Returns the triangle formed by the three vertices of `f`.

Precondition
  The face is finite.

## Friends And Related Function Documentation

## ◆ operator<<()

template<typename Traits , typename Tds >

ostream & operator<< | ( | ostream & | os,
|  | const Triangulation_2< Traits, Tds > & | T
| ) |  |
| related

Inserts the triangulation into the stream `os`.

Precondition
  The insert operator must be defined for `Point`.

## ◆ operator>>()

template<typename Traits , typename Tds >

istream & operator>> | ( | istream & | is,
|  | const Triangulation_2< Traits, Tds > & | T
| ) |  |
| related

Reads a triangulation from stream `is` and assigns it to the triangulation.

Precondition
  The extract operator must be defined for `Point`.