Definition

The concept TriangulationDataStructure_2 describes the requirements for the second template parameter of the basic triangulation class Triangulation_2<Traits,Tds> and of all other 2D triangulation classes.

The concept can be seen as a container for the faces and vertices of the triangulation. The concept TriangulationDataStructure_2 includes two sub-concepts TriangulationDataStructure_2::Vertex and TriangulationDataStructure_2::Face.

The TriangulationDataStructure_2 maintains incidence and adjacency relations among vertices and faces.

Each triangular face gives access to its three incident vertices and to its three adjacent faces. Each vertex gives access to one of its incident faces and through that face to the circular list of its incident faces.

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

Each edge has two implicit representations : the edge of a face f which is opposed to the vertex indexed i, can be represented as well as an edge of the neighbor(i) of f. See Figure 40.1.

The triangulation data structure is responsible for the combinatorial integrity of the triangulation. This means that the triangulation data structure allows to perform some combinatorial operations on the triangulation and guarantees the maintenance on proper incidence and adjacency relations among the vertices and faces. The term combinatorial operations means that those operations are purely topological and do not depend on the geometric embedding. Insertion of a new vertex in a given face, or in a given edge, suppression of a vertex of degree three, flip of two edges are examples of combinatorial operations.

I/O

The information output in the iostream is: the dimension, the number of (finite) vertices, the number of (finite) faces. Then comes for each vertex, the non combinatorial information stored in that vertex if any. Then comes for each faces, the indices of its vertices and the non combinatorial information (if any) stored in this face. Then comes for each face again the indices of the neighboring faces. The index of an item (vertex of face) 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.

Has Models:: CGAL::Triangulation_data_structure_2<Vb,Fb>

See also: TriangulationDataStructure_2::Face; TriangulationDataStructure_2::Vertex; CGAL::Triangulation_2

Concepts
concept	Face
	The concept `TriangulationDataStructure_2::Face` describes the types used to store the faces face class of a `TriangulationDataStructure_2`. A `TriangulationDataStructure_2::Face` stores three handles to its three vertices and three handles to its three neighbors. The vertices are indexed 0,1, and 2 in counterclockwise order. The neighbor indexed `i` lies opposite to vertex `i`. More...

concept	Face_data
	Various algorithms using a triangulation data structure, such as Delaunay triangulations or Alpha Shapes, must be able to associate a state to a face elemental. For efficiency, this information must be stored directly within the face. More...

concept	Vertex
	The concept `TriangulationDataStructure_2::Vertex` describes the type used by a `TriangulationDataStructure_2` to store the vertices. More...

Types
typedef unspecified_type	size_type
	Size type (unsigned integral type)

typedef unspecified_type	difference_type
	Difference type (signed integral type)

typedef unspecified_type	Vertex
	The vertex type, requirements for this type are described in concept `TriangulationDataStructure_2::Vertex`.

typedef unspecified_type	Face
	The face type, requirements for this type are described in concept `TriangulationDataStructure_2::Face`.

typedef unspecified_type	Face_data
	Face data type, requirements are described in `TriangulationDataStructure_2::Face_data`.

typedef unspecified_type	Vertex_handle
	Handle to a vertex. More...

typedef unspecified_type	Face_handle
	Handle to a face. More...

typedef std::pair< Face_handle, int >	Edge
	The edge type. More...

Iterators and Circulators
The iterators allow one to visit all the vertices, edges and faces of a triangulation data structure. The circulators allow to visit all the edges or faces incident to a given vertex and all the vertices adjacent to a given vertex. The iterators and circulators are bidirectional and non mutable, and they are convertible to the corresponding handles, thus they can be passed directly as argument to the functions expecting a handle. A face circulator is invalidated by any modification of the face it points to. An edge circulator is invalidated by any modification of any of the two faces incident to the edge pointed to. A vertex circulator that turns around vertex `v` and that has as value a handle to vertex `w`, is invalidated by any modification of anyone of the two faces incident to `v` and `w`.
typedef unspecified_type	Face_iterator

typedef unspecified_type	Edge_iterator

typedef unspecified_type	Vertex_iterator

typedef unspecified_type	Face_circulator

typedef unspecified_type	Edge_circulator

typedef unspecified_type	Vertex_circulator

Creation
	TriangulationDataStructure_2 ()
	Default constructor.

	TriangulationDataStructure_2 (const TriangulationDataStructure_2 &tds1)
	Copy constructor, performing a deep copy, that is all vertices and faces are duplicated.

TriangulationDataStructure_2 &	operator= (const TriangulationDataStructure_2 &tds1)
	Assignment. More...

Vertex_handle	copy_tds (const TriangulationDataStructure_2 &tds1, Vertex_handle v=Vertex_handle())
	`tds1` is copied into the triangulation data structure. More...

template<class TDS_src , class ConvertVertex , class ConvertFace >
Vertex_handle	copy_tds (const TDS_src &tds_src, typename TDS_src::Vertex_handle v, const ConvertVertex &convert_vertex, const ConvertFace &convert_face)
	`tds_src` is copied into `this`. More...

void	swap (TriangulationDataStructure_2 &tds1)
	Swaps the triangulation data structure and `tds1`. More...

void	clear ()
	Deletes all faces and all finite vertices.

Access Functions
int	dimension () const
	This is an advanced function. More...

size_type	number_of_vertices () const
	returns the number of vertices in the triangulation data structure.

size_type	number_of_faces () const
	returns the number of two dimensional faces in the triangulation data structure.

size_type	number_of_edges () const
	returns the number of edges in the triangulation data structure.

size_type	number_of_full_dim_faces () const
	returns the number of full dimensional faces, i.e. faces of dimension equal to the dimension of the triangulation data structure. More...

Setting
void	set_dimension (int n)
	sets the dimension.

Queries
bool	is_vertex (Vertex_handle v) const
	returns `true` if `v` is a vertex of the triangulation data structure.

bool	is_edge (Face_handle fh, int i) const
	returns `true` if `(fh,i)` is an edge of the triangulation data structure. More...

bool	is_edge (Vertex_handle va, Vertex_handle vb) const
	returns `true` if `(va, vb)` is an edge of the triangulation data structure.

bool	is_edge (Vertex_handle va, Vertex_handle vb, Face_handle &fr, int &i) const
	returns `true` if `(va, vb)` is an edge of the triangulation data structure. More...

bool	is_face (Face_handle fh) const
	returns `true` if `fh` is a face of the triangulation data structure. More...

bool	is_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3) const
	`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) const
	`true` if there is a face having `v1`, `v2`, and `v3` as vertices. More...

Traversing the Triangulation
Face_iterator	faces_begin () const
	visits all faces.

Face_iterator	faces_end () const

Vertex_iterator	vertices_begin () const
	visits all vertices.

Vertex_iterator	vertices_end () const

Edge_iterator	edges_begin () const
	visits all edges.

Edge_iterator	edges_end () const

Vertex_circulator	incident_vertices (Vertex_handle v, Face_handle f=Face_handle()) const

Edge_circulator	incident_edges (Vertex_handle v, Face_handle f=Face_handle()) const

Face_circulator	incident_faces (Vertex_handle v, Face_handle f=Face_handle()) const

Vertex_handle	mirror_vertex (Face_handle f, int i) const
	returns vertex of `f->neighbor(i)`.

int	mirror_index (Face_handle f, int i) const
	returns the index of `f` as a neighbor of `f->neighbor(i)`.

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

Modifiers
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_first ()
	creates the first vertex and returns a handle to it.

Vertex_handle	insert_second ()
	creates the second vertex and returns a handle to it.

Vertex_handle	insert_in_edge (Face_handle f, int i)
	adds a vertex `v` splitting edge `i` of face `f`. More...

Vertex_handle	insert_in_face (Face_handle f)
	adds a vertex `v` splitting face `f` in three. More...

Vertex_handle	insert_dim_up (Vertex_handle w, bool orient=true)
	adds a vertex `v`, increasing by one the dimension of the triangulation data structure. More...

void	remove_degree_3 (Vertex_handle v, Face_handle f=Face_handle())
	removes a vertex of degree 3. More...

void	remove_second (Vertex_handle v)
	removes the before last vertex.

void	remove_first (Vertex_handle v)
	removes the last vertex.

void	remove_dim_down (Vertex_handle v)
	removes vertex `v` incident to all other vertices and decreases by one the dimension of the triangulation data structure. More...

void	dim_down (Face_handle f, int i)
	must be called when the displacement of a vertex decreases the dimension of the triangulation data structure. More...

Advanced Modifiers
The following modifiers are required for convenience of the advanced user. They do not guarantee the combinatorial validity of the resulting triangulation data structure.
template<class FaceIt >
Vertex_handle	insert_in_hole (FaceIt face_begin, FaceIt face_end)
	creates a new vertex `v` and uses it to star a hole. More...

template<class FaceIt >
void	insert_in_hole (Vertex_handle new_v, FaceIt face_begin, FaceIt face_end)
	same as above, except that `new_v` will be used as the new vertex, which must have been allocated previously, for example with `create_vertex`.

template<class EdgeIt >
Vertex_handle	star_hole (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 (EdgeIt edge_begin, EdgeIt edge_end, FaceIt face_begin, FaceIt face_end)
	same as above, except that, to build the new faces, the algorithm first recycles faces in the sequence `[face_begin, face_end)` and create new ones when the sequence is exhausted.

template<class EdgeIt >
void	star_hole (Vertex_handle v, EdgeIt edge_begin, EdgeIt edge_end)
	uses vertex v to star the hole whose boundary is described by the sequence of edges `[edge_begin, edge_end)`.

template<class EdgeIt , class FaceIt >
void	star_hole (Vertex_handle v, EdgeIt edge_begin, EdgeIt edge_end, FaceIt face_begin, FaceIt face_end)
	same as above, recycling faces in the sequence `[face_begin, face_end)`.

void	make_hole (Vertex_handle v, List_edges &hole)
	removes the vertex v, and store in `hole` the list of edges on the boundary of the hole.

Vertex_handle	create_vertex ()
	adds a new vertex.

Face_handle	create_face (Face_handle f1, int i1, Face_handle f2, int i2, Face_handle f3, int i3)
	adds a face which is the neighbor `i1` of `f1`, `i2` of `f2` and `i3` of `f3`.

Face_handle	create_face (Face_handle f1, int i1, Face_handle f2, int i2)
	adds a face which is the neighbor `i1` of `f1`, and the neighbor `i2` of `f2`.

Face_handle	create_face (Face_handle f1, int i1, Vertex_handle v)
	adds a face which is the neighbor `i1` of `f1`, and has `v` as vertex.

Face_handle	create_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3)
	adds a face with vertices `v1`, `v2` and `v3`.

Face_handle	create_face (Vertex_handle v1, Vertex_handle v2, Vertex_handle v3, Face_handle f1, Face_handle f2, Face_handle f3)
	adds a face with vertices `v1`, `v2` and `v3`, and neighbors `f1`, `f2`, `f3`.

Face_handle	create_face ()
	adds a face whose vertices and neighbors are set to `Vertex_handle()` and `Face_handle()`.

void	delete_face (Face_handle)
	deletes a face.

void	delete_vertex (Vertex_handle)
	deletes a vertex.

Miscellaneous
int	ccw (int i) const
	returns \( i+1\) modulo 3, with \( 0\leq i \leq2\).

int	cw (int i) const
	returns \( i+2\) modulo 3, with \( 0\leq i \leq2\).

bool	is_valid ()
	checks the combinatorial validity of the triangulation data structure: call the `is_valid()` member function for each vertex and each face, checks the number of vertices and the Euler relation between numbers of vertices, faces and edges.

size_type	degree (Vertex_handle v) const
	Returns the degree of `v` in the triangulation data structure.

void	file_output (ostream &os, Vertex_handle v=Vertex_handle(), bool skip_first=false)
	writes the triangulation data structure into the stream `os`. More...

Vertex_handle	file_input (istream &is, bool skip_first=false)
	inputs the triangulation data structure from file and returns a handle to the first input vertex. More...

istream &	operator>> (istream &is, TriangulationDataStructure_2 &tds)
	reads a combinatorial triangulation data structure from `is` and assigns it to tthe triangulation data structure.

ostream &	operator<< (ostream &os, const TriangulationDataStructure_2 &tds)
	writes `tds` into the stream `os`.

Member Typedef Documentation

◆ Edge

typedef std::pair<Face_handle,int> TriangulationDataStructure_2::Edge

The edge type.

The Edge(f,i) is edge common to faces f and f.neighbor(i). It is also the edge joining the vertices vertex(cw(i)) and vertex(ccw(i)) of f.

◆ Face_handle

typedef unspecified_type TriangulationDataStructure_2::Face_handle

Handle to a face.

Is Model Of:: Handle

◆ Vertex_handle

typedef unspecified_type TriangulationDataStructure_2::Vertex_handle

Handle to a vertex.

Is Model Of:: Handle

Member Function Documentation

◆ copy_tds() [1/2]

Vertex_handle TriangulationDataStructure_2::copy_tds	(	const TriangulationDataStructure_2 &	tds1,
		Vertex_handle	v = `Vertex_handle()`
	)

tds1 is copied into the triangulation data structure.

If v != Vertex_handle(), the vertex of the triangulation data structure corresponding to v is returned, otherwise Vertex_handle() is returned.

Precondition: The optional argument v is a vertex of tds1.

◆ copy_tds() [2/2]

template<class TDS_src , class ConvertVertex , class ConvertFace >

Vertex_handle TriangulationDataStructure_2::copy_tds	(	const TDS_src &	tds_src,
		typename TDS_src::Vertex_handle	v,
		const ConvertVertex &	convert_vertex,
		const ConvertFace &	convert_face
	)

tds_src is copied into this.

As the vertex and face types might be different and incompatible, the creation of new faces and vertices is made thanks to the functors convert_vertex and convert_face, that convert vertex and face types. For each vertex v_src in tds_src, the corresponding vertex v_tgt in this is a copy of the vertex returned by convert_vertex(v_src). The same operations are done for faces with the functor convert_face. If v != TDS_src::Vertex_handle(), a handle to the vertex created in this that is the copy of v is returned, otherwise Vertex_handle() is returned.

A model of ConvertVertex must provide two operator()'s that are responsible for converting the source vertex v_src into the target vertex:
- Vertex operator()(const TDS_src::Vertex& v_src); This operator is used to create the vertex from v_src.
- void operator()(const TDS_src::Vertex& v_src, Vertex& v_tgt); This operator is meant to be used in case heavy data should transferred to v_tgt.
A model of ConvertFace must provide two operator()'s that are responsible for converting the source face f_src into the target face:
- Face operator()(const TDS_src::Face& f_src); This operator is used to create the face from f_src.
- void operator()(const TDS_src::Face& f_src, Face& f_tgt); This operator is meant to be used in case heavy data should transferred to f_tgt.

Precondition: The optional argument v is a vertex of tds_src or is Vertex_handle().

◆ dim_down()

void TriangulationDataStructure_2::dim_down	(	Face_handle	f,
		int	i
	)

must be called when the displacement of a vertex decreases the dimension of the triangulation data structure.

The link of a vertex v is formed by the edges disjoint from v that are included in the faces incident to v. When the link of v = f->vertex(i) contains all the other vertices of the two-dimensional triangulation data structure ( \( \mathbb{S}^2\)), dim_down() crushes the two-dimensional data-structure ( \( \mathbb{S}^2\)) onto the one-dimensional data structure ( \( \mathbb{S}^1\)) formed by the link of v augmented with the vertex v itself; this one is placed on the edge (f, i) (see Figure figtdsdim_down_2).

Precondition: dimension() must be equal to 2, the degree of f->vertex(i) must be equal to the total number of vertices minus 1.

From a two-dimensional data structure to a one-dimensional data structure.

◆ dimension()

int TriangulationDataStructure_2::dimension ( ) const

This is an advanced function.

Advanced

returns the dimension of the triangulation data structure.

◆ file_input()

Vertex_handle TriangulationDataStructure_2::file_input	(	istream &	is,
		bool	skip_first = `false`
	)

inputs the triangulation data structure from file and returns a handle to the first input vertex.

If skip_first is true, it is assumed that the first vertex has been omitted when output.

◆ file_output()

void TriangulationDataStructure_2::file_output	(	ostream &	os,
		Vertex_handle	v = `Vertex_handle()`,
		bool	skip_first = `false`
	)

writes the triangulation data structure into the stream os.

If v is not Vertex_handle(), vertex v is output first or skipped if skip_first is true.

◆ flip()

void TriangulationDataStructure_2::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).

Flip

◆ incident_edges()

Edge_circulator TriangulationDataStructure_2::incident_edges	(	Vertex_handle	v,
		Face_handle	f = `Face_handle()`
	)		const

Precondition: If the face f is given, it has to be a face of incident to v and the circulator begins with the edge (f,cw(i)) of f if i is the index of v in f.

◆ incident_faces()

Face_circulator TriangulationDataStructure_2::incident_faces	(	Vertex_handle	v,
		Face_handle	f = `Face_handle()`
	)		const

Precondition: If the face f is given, it has to be a face of incident to v and the circulator begins with the face f.

◆ incident_vertices()

Vertex_circulator TriangulationDataStructure_2::incident_vertices	(	Vertex_handle	v,
		Face_handle	f = `Face_handle()`
	)		const

Precondition: If the face f is given, it has to be incident to be a face incident to v and the circulator begins with the vertex f->vertex(ccw(i)) if i is the index of v in f.

◆ insert_dim_up()

Vertex_handle TriangulationDataStructure_2::insert_dim_up	(	Vertex_handle	w,
		bool	orient = `true`
	)

adds a vertex v, increasing by one the dimension of the triangulation data structure.

Vertex v and the existing vertex w are linked to all the vertices of the triangulation data structure. The Boolean orient decides the final orientation of all faces. A handle to vertex v is returned.

◆ insert_in_edge()

Vertex_handle TriangulationDataStructure_2::insert_in_edge	(	Face_handle	f,
		int	i
	)

adds a vertex v splitting edge i of face f.

Return a handle to v.

◆ insert_in_face()

Vertex_handle TriangulationDataStructure_2::insert_in_face ( Face_handle f )

adds a vertex v splitting face f in three.

Face f is modified, two new faces are created. Return a handle to v

◆ insert_in_hole()

template<class FaceIt >

Vertex_handle TriangulationDataStructure_2::insert_in_hole	(	FaceIt	face_begin,
		FaceIt	face_end
	)

creates a new vertex v and uses it to star a hole.

Given a set of faces 'F' describing a simply connected hole (i.e., a topological disk), the function deletes all the faces in F, creates a new vertex v and for each edge on the boundary of the hole creates a new face with v as a vertex. The input is an iterator range [face_begin, face_end[ of Face_handles over the connected faces in F. The handle to the new vertex v is returned.

Precondition: tds.dimension() = 2 and the set of faces has the topology of a disk.

◆ is_edge() [1/2]

bool TriangulationDataStructure_2::is_edge	(	Face_handle	fh,
		int	i
	)		const

returns true if (fh,i) is an edge of the triangulation data structure.

Returns false when dimension() < 1.

◆ is_edge() [2/2]

bool TriangulationDataStructure_2::is_edge	(	Vertex_handle	va,
		Vertex_handle	vb,
		Face_handle &	fr,
		int &	i
	)		const

returns true if (va, vb) is an edge of the triangulation data structure.

In addition, if true is returned fr and i are set such that the pair (fr,i) is the description of the ordered edge (va, vb).

◆ is_face() [1/2]

bool TriangulationDataStructure_2::is_face ( Face_handle fh ) const

returns true if fh is a face of the triangulation data structure.

Returns false when dimension() < 2.

◆ is_face() [2/2]

bool TriangulationDataStructure_2::is_face	(	Vertex_handle	v1,
		Vertex_handle	v2,
		Vertex_handle	v3,
		Face_handle &	fr
	)		const

true if there is a face having v1, v2, and v3 as vertices.

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

◆ number_of_full_dim_faces()

size_type TriangulationDataStructure_2::number_of_full_dim_faces ( ) const

returns the number of full dimensional faces, i.e. faces of dimension equal to the dimension of the triangulation data structure.

This is the actual number of faces stored in the triangulation data structure.

◆ operator=()

TriangulationDataStructure_2& TriangulationDataStructure_2::operator= ( const TriangulationDataStructure_2 & tds1 )

Assignment.

All the vertices and faces of tds1 are duplicated in the triangulation data structure. Former faces and vertices of the triangulation data structure , if any, are deleted.

◆ remove_degree_3()

void TriangulationDataStructure_2::remove_degree_3	(	Vertex_handle	v,
		Face_handle	f = `Face_handle()`
	)

removes a vertex of degree 3.

Two of the incident faces are destroyed, the third one is modified. If parameter f is specified, it has to be a face incident to v and will be the modified face.

Precondition: Vertex v is a finite vertex with degree 3 and, if specified, face f is incident to v.

Insertion

◆ remove_dim_down()

void TriangulationDataStructure_2::remove_dim_down ( Vertex_handle v )

removes vertex v incident to all other vertices and decreases by one the dimension of the triangulation data structure.

Precondition: If the dimension is 2, the number of vertices is more than 3, if the dimension is 1, the number of vertices is 2.

◆ star_hole()

template<class EdgeIt >

Vertex_handle TriangulationDataStructure_2::star_hole	(	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 vertex.

◆ swap()

void TriangulationDataStructure_2::swap ( TriangulationDataStructure_2 & tds1 )

Swaps the triangulation data structure and tds1.

Should be preferred to an assignment or copy constructor when tds1 is deleted after that.