#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

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) ; 

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, 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`. More...

istream &	operator>> (istream &is, const Triangulation_2< Traits, Tds > &T)
	Reads a triangulation from stream `is` and assigns it to the triangulation. More...

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).

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.
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 the finite vertices of the triangulation.

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.

Creation
	Triangulation_2 (const Traits &gt=Traits())
	Introduces an empty triangulation.

	Triangulation_2 (const Triangulation_2 &tr)
	Copy constructor. More...

Triangulation_2	operator= (const Triangulation_2< Traits, Tds > &tr)
	Assignment. More...

void	swap (Triangulation_2 &tr)
	The triangulations `tr` and `*this` are swapped. More...

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. More...

bool	includes_edge (Vertex_handle va, Vertex_handle vb, Vertex_handle &vbr, Face_handle &fr, int &i)
	`true` if the line segment from `va` to `vb` includes an edge `e` incident to `va`. More...

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. More...

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. More...

Face_handle	inexact_locate (const Point &query, Face_handle start=Face_handle()) const
	Same as `locate()` but uses inexact predicates. More...

Face_handle	locate (const Point &query, Locate_type &lt, int &li, Face_handle h=Face_handle()) const
	Same as above. More...

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`. More...

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`. More...

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)`. More...

Vertex_handle	insert (const Point &p, Face_handle f=Face_handle())
	Inserts point `p` in the triangulation and returns the corresponding vertex. More...

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 InputIterator >
std::ptrdiff_t	insert (InputIterator first, InputIterator last)
	Inserts the points in the range `[first,last)`. More...

void	remove (Vertex_handle v)
	Removes the vertex from the triangulation. More...

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. More...

Vertex_handle	move (Vertex_handle v, const Point &p)
	same as above if there is no collision. More...

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`. More...

Vertex_handle	insert_in_edge (const Point &p, Face_handle f, int i)
	Inserts vertex v in edge `i` of `f`. More...

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. More...

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. More...

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)`. More...

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. More...

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.

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.

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`. More...

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`. More...

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`. More...

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`. More...

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`. More...

int	mirror_index (Face_handle f, int i) const
	returns the index of `f` in its \( i^{th}\) neighbor. More...

Edge	mirror_edge (Edge e) const
	returns the same edge seen from the other adjacent face. More...

Miscellaneous
int	ccw (int i) const
	Returns \( i+1\) modulo 3. More...

int	cw (int i) const
	Returns \( i+2\) modulo 3. More...

Triangle	triangle (Face_handle f) const
	Returns the triangle formed by the three vertices of `f`. More...

Segment	segment (Face_handle f, int i) const
	Returns the line segment formed by the vertices `ccw(i)` and `cw(i)` of face `f`. More...

Segment	segment (const Edge &e) const
	Returns the line segment corresponding to edge `e`. More...

Segment	segment (const Edge_circulator &ec) const
	Returns the line segment corresponding to edge `*ec`. More...

Segment	segment (const Edge_iterator &ei) const
	Returns the line segment corresponding to edge `*ei`. More...

Point	circumcenter (Face_handle f) const
	Compute the circumcenter of the face pointed to by f. More...

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

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. More...

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

◆ 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: \( 0\leq i \leq2\).

◆ 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 &	vbr,
		Face_handle &	fr,
		int &	i
	)

true if the line segment from va to vb includes an edge e incident to va.

If true, vbr 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() [1/2]

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, and Triangulation_2/colored_face.cpp.

◆ insert() [2/2]

template<typename Traits , typename Tds >

template<class InputIterator >

std::ptrdiff_t CGAL::Triangulation_2< Traits, Tds >::insert	(	InputIterator	first,
		InputIterator	last
	)

Inserts the points in the range [first,last).

Returns the number of inserted points.

Precondition: The value_type of InputIterator is Point.

◆ 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 ofp, 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 != NULL, 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 \( (\infty, p, NULL)\) 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 NULL.

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
	)

same as above if there is no collision.

Otherwise, v is deleted and the vertex placed on 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	(	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.

◆ segment() [2/4]

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() [3/4]

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() [4/4]

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.

◆ 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 construtor. 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
	)

Precondition: The extract operator must be defined for Point.

Inherits from

Definition

Related Functions

Types

Handles, Iterators, and Circulators

Creation

Access Functions

Non const access

Predicates

Queries

Modifiers

Specialized Modifiers

Finite Face, Edge and Vertex Iterators

All Face, Edge and Vertex Iterators

Line Face Circulator

Face, Edge and Vertex Circulators

Traversal Between Adjacent Faces

Miscellaneous

Setting

Checking

Additional Inherited Members

Member Enumeration Documentation

◆ Locate_type

Constructor & Destructor Documentation

◆ Triangulation_2()

Member Function Documentation

◆ ccw()

◆ circumcenter()

◆ cw()

◆ flip()

◆ incident_edges()

◆ incident_faces()

◆ incident_vertices()

◆ includes_edge()

◆ inexact_locate()

◆ insert() [1/2]

◆ insert() [2/2]

◆ insert_in_edge()

◆ insert_in_face()

◆ insert_outside_convex_hull()

◆ is_edge()

◆ is_face()

◆ is_valid()

◆ line_walk()

◆ locate() [1/2]

◆ locate() [2/2]

◆ mirror_edge()

◆ mirror_index()

◆ mirror_vertex()

◆ move()

◆ move_if_no_collision()

◆ operator=()

◆ oriented_side()

◆ remove()

◆ remove_degree_3()

◆ segment() [1/4]

◆ segment() [2/4]

◆ segment() [3/4]

◆ segment() [4/4]

◆ set_infinite_vertex()

◆ side_of_oriented_circle()

◆ star_hole() [1/2]

◆ star_hole() [2/2]

◆ swap()

◆ triangle()

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. PreconditionThe insert operator must be defined for Point.

◆ operator>>()

◆ 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`.