TriangulationDataStructure_3 Concept Reference

Definition

3D-triangulation data structures are meant to maintain the combinatorial information for 3D-geometric triangulations.

In CGAL, a triangulation data structure is a container of cells ( \( 3\)-faces) and vertices ( \( 0\)-faces). Following the standard vocabulary of simplicial complexes, an \( i\)-face \( f_i\) and a \( j\)-face \( f_j\) \( (0 \leq j < i \leq 3)\) are said to be incident in the triangulation if \( f_j\) is a (sub)face of \( f_i\), and two \( i\)-faces \( (0 \leq i \leq 3)\) are said to be adjacent if they share a common incident (sub)face.

Each cell gives access to its four incident vertices and to its four adjacent cells. Each vertex gives direct access to one of its incident cells, which is sufficient to retrieve all the incident cells when needed.

The four vertices of a cell are indexed with 0, 1, 2 and 3. The neighbors of a cell are also indexed with 0, 1, 2, 3 in such a way that the neighbor indexed by \( i\) is opposite to the vertex with the same index (see Figure 40.1).

Edges ( \( 1\)-faces) and facets ( \( 2\)-faces) are not explicitly represented: a facet is given by a cell and an index (the facet i of a cell c is the facet of c that is opposite to the vertex of index i) and an edge is given by a cell and two indices (the edge (i,j) of a cell c is the edge whose endpoints are the vertices of indices i and j of c).

As CGAL explicitly deals with all degenerate cases, a 3D-triangulation data structure in CGAL can handle the cases when the dimension of the triangulation is lower than 3 (see Section Representation).

Thus, a 3D-triangulation data structure can store a triangulation of a topological sphere \( S^d\) of \( \mathbb{R}^{d+1}\), for any \( d \in \{-1,0,1,2,3\}\).

The second template parameter of the basic triangulation class (see Chapter 3D Triangulations) CGAL::Triangulation_3 is a triangulation data structure class. (See Chapter chapterTDS3.)

To ensure all the flexibility of the class CGAL::Triangulation_3, a model of a triangulation data structure must be templated by the base vertex and the base cell classes (see Representation): TriangulationDataStructure_3<TriangulationVertexBase_3,TriangulationCellBase_3>. The optional functionalities related to geometry are compulsory for this use as a template parameter of CGAL::Triangulation_3.

A class that satisfies the requirements for a triangulation data structure class must provide the following types and operations.

I/O

The information stored in the iostream is: the dimension, the number of vertices, the number of cells, the indices of the vertices of each cell, then the indices of the neighbors of each cell, where the index corresponds to the preceding list of cells. When dimension < 3, the same information is stored for faces of maximal dimension instead of cells.

Has Models:

CGAL::Triangulation_data_structure_3

See Also

TriangulationDataStructure_3::Vertex

TriangulationDataStructure_3::Cell

Concepts
concept	Cell
	The concept TriangulationDataStructure_3::Cell stores four Vertex_handles to its four vertices and four Cell_handles to its four neighbors. The vertices are indexed 0, 1, 2, and 3 in consistent order. The neighbor indexed \( i\) lies opposite to vertex i. More...
concept	Vertex
	The concept TriangulationDataStructure_3::Vertex represents the vertex class of a 3D-triangulation data structure. It must define the types and operations listed in this section. Some of these requirements are of geometric nature, they are optional when using the triangulation data structure class alone. They become compulsory when the triangulation data structure is used as a layer for the geometric triangulation class. (See Section Software Design.) More...

Types
typedef unspecified_type	Vertex
	Vertex type, requirements are described in TriangulationDataStructure_3::Vertex.
typedef unspecified_type	Cell
	Cell type, requirements are described in TriangulationDataStructure_3::Cell.
typedef unspecified_type	size_type
	Size type (unsigned integral type)
typedef unspecified_type	difference_type
	Difference type (signed integral type)
typedef unspecified_type	Vertex_handle
typedef unspecified_type	Cell_handle
typedef unspecified_type	Concurrency_tag
	Can be CGAL::Sequential_tag or CGAL::Parallel_tag. More...
template<typename Vb2 >
using	Rebind_vertex = unspecified_type
	Advanced This template class allows to get the type of a triangulation data structure that only changes the vertex type. More...
template<typename Cb2 >
using	Rebind_cell = unspecified_type
	Advanced This template class allows to get the type of a triangulation data structure that only changes the cell type. More...
typedef Triple< Cell_handle, int, int >	Edge
	(c,i,j) is the edge of cell c whose vertices indices are i and j. More...
typedef std::pair< Cell_handle, int >	Facet
	(c,i) is the facet of c opposite to the vertex of index i. More...

Iterators
The following iterators allow one to visit all the vertices, edges, facets and cells of the triangulation data structure. They are all bidirectional, non-mutable iterators. Iterators are convertible to the corresponding handles, thus the user can pass them directly as arguments to the functions.
typedef unspecified_type	Cell_iterator
typedef unspecified_type	Facet_iterator
typedef unspecified_type	Edge_iterator
typedef unspecified_type	Vertex_iterator

Circulators
The following circulators allow us to visit all the cells and facets incident to a given edge. They are bidirectional and non-mutable.Circulators are convertible to the corresponding handles, thus the user can pass them directly as arguments to the functions.
typedef unspecified_type	Facet_circulator
typedef unspecified_type	Cell_circulator

Creation
	TriangulationDataStructure_3 ()
	Default constructor.
	TriangulationDataStructure_3 (const TriangulationDataStructure_3 &tds1)
	Copy constructor. More...
TriangulationDataStructure_3 &	operator= (const TriangulationDataStructure_3 &tds1)
	Assignment operator. More...
Vertex_handle	copy_tds (const TriangulationDataStructure_3 &tds1, Vertex_handle v=Vertex_handle())
	tds1 is copied into tds. More...
template<class TDS_src , class ConvertVertex , class ConvertCell >
Vertex_handle tds	copy_tds (const TDS_src &tds_src, typename TDS_src::Vertex_handle v, const ConvertVertex &convert_vertex, const ConvertCell &convert_cell)
	tds_src is copied into this. More...
void	swap (TriangulationDataStructure_3 &tds1)
	Swaps tds and tds1. More...
void	clear ()
	Deletes all cells and vertices. More...

Access Functions
int	dimension () const
	The dimension of the triangulated topological sphere.
size_type	number_of_vertices () const
	The number of vertices. More...
size_type	number_of_cells () const
	The number of cells. More...

Non constant-time access functions
size_type	number_of_facets () const
	The number of facets. More...
size_type	number_of_edges () const
	The number of edges. More...

Setting
void	set_dimension (int n)
	This is an advanced function. More...

Queries
bool	is_vertex (Vertex_handle v) const
	Tests whether v is a vertex of tds.
bool	is_edge (Cell_handle c, int i, int j) const
	Tests whether (c,i,j) is an edge of tds. More...
bool	is_edge (Vertex_handle u, Vertex_handle v, Cell_handle &c, int &i, int &j) const
	Tests whether (u,v) is an edge of tds. More...
bool	is_edge (Vertex_handle u, Vertex_handle v) const
	Tests whether (u,v) is an edge of tds.
bool	is_facet (Cell_handle c, int i) const
	Tests whether (c,i) is a facet of tds. More...
bool	is_facet (Vertex_handle u, Vertex_handle v, Vertex_handle w, Cell_handle &c, int &i, int &j, int &k) const
	Tests whether (u,v,w) is a facet of tds. More...
bool	is_cell (Cell_handle c) const
	Tests whether c is a cell of tds. More...
bool	is_cell (Vertex_handle u, Vertex_handle v, Vertex_handle w, Vertex_handle t, Cell_handle &c, int &i, int &j, int &k, int &l) const
	Tests whether (u,v,w,t) is a cell of tds. More...

has_vertex
There is a method has_vertex in the cell class. The analogous methods for facets are defined here.
bool	has_vertex (const Facet &f, Vertex_handle v, int &j) const
	If v is a vertex of f, then j is the index of v in the cell f.first, and the method returns true. More...
bool	has_vertex (Cell_handle c, int i, Vertex_handle v, int &j) const
	Same for facet (c,i). More...
bool	has_vertex (const Facet &f, Vertex_handle v) const
bool	has_vertex (Cell_handle c, int i, Vertex_handle v) const
	Same as the first two methods, but these two methods do not return the index of the vertex.

Equality Tests
The following three methods test whether two facets have the same vertices.
bool	are_equal (const Facet &f, const Facet &g) const
bool	are_equal (Cell_handle c, int i, Cell_handle n, int j) const
bool	are_equal (const Facet &f, Cell_handle n, int j) const
	For these three methods: More...

Flips
Two kinds of flips exist for a three-dimensional triangulation. They are reciprocal. To be flipped, an edge must be incident to three tetrahedra. During the flip, these three tetrahedra disappear and two tetrahedra appear. Figure TDS3figflips (left) shows the edge that is flipped as bold dashed, and one of its three incident facets is shaded. On the right, the facet shared by the two new tetrahedra is shaded. Flips. The following methods guarantee the validity of the resulting 3D combinatorial triangulation. Moreover the flip operations do not invalidate the vertex handles, and only invalidate the cell handles of the affected cells. Flips for a 2d triangulation are not implemented yet
bool	flip (Edge e)
bool	flip (Cell_handle c, int i, int j)
	Before flipping, these methods check that edge e=(c,i,j) is flippable (which is quite expensive). More...
void	flip_flippable (Edge e)
void	flip_flippable (Cell_handle c, int i, int j)
	Should be preferred to the previous methods when the edge is known to be flippable. More...
bool	flip (Facet f)
bool	flip (Cell_handle c, int i)
	Before flipping, these methods check that facet f=(c,i) is flippable (which is quite expensive). More...
void	flip_flippable (Facet f)
void	flip_flippable (Cell_handle c, int i)
	Should be preferred to the previous methods when the facet is known to be flippable. More...

Insertions
The following modifier member functions guarantee the combinatorial validity of the resulting triangulation.
Vertex_handle	insert_in_cell (Cell_handle c)
	Creates a new vertex, inserts it in cell c and returns its handle. More...
Vertex_handle	insert_in_facet (const Facet &f)
	Creates a new vertex, inserts it in facet f and returns its handle. More...
Vertex_handle	insert_in_facet (Cell_handle c, int i)
	Creates a new vertex, inserts it in facet i of c and returns its handle. More...
Vertex_handle	insert_in_edge (Edge e)
	Creates a new vertex, inserts it in edge e and returns its handle. More...
Vertex_handle	insert_in_edge (Cell_handle c, int i, int j)
	Creates a new vertex, inserts it in edge \( (i,j)\) of c and returns its handle. More...
Vertex_handle	insert_increase_dimension (Vertex_handle star=Vertex_handle())
	Transforms a triangulation of the sphere \( S^d\) of \( \mathbb{R}^{d+1}\) into the triangulation of the sphere \( S^{d+1}\) of \( \mathbb{R}^{d+2}\) by adding a new vertex v: v is linked to all the vertices to triangulate one of the two half-spheres of dimension \( (d+1)\). More...
template<class CellIt >
Vertex_handle	insert_in_hole (CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i)
	Creates a new vertex by starring a hole. More...
template<class CellIt >
Vertex_handle	insert_in_hole (CellIt cell_begin, CellIt cell_end, Cell_handle begin, int i, Vertex_handle newv)
	Same as above, except that newv will be used as the new vertex, which must have been allocated previously with e.g. More...

Removal
void	remove_decrease_dimension (Vertex_handle v, Vertex_handle w=v)
	This operation is the reciprocal of insert_increase_dimension(). More...
Cell_handle	remove_from_maximal_dimension_simplex (Vertex_handle v)
	Removes v. More...

Dimension Manipulation
The following operation, decrease_dimension, is necessary when the displacement of a vertex decreases the dimension of the triangulation.
void	decrease_dimension (Cell_handle c, int i)
	The link of a vertex \( v\) is formed by the facets disjoint from \( v\) that are included in the cells incident to \( v\). More...

Other modifiers
The following modifiers can affect the validity of the triangulation data structure.
void	reorient ()
	This is an advanced function. More...
Vertex_handle	create_vertex (const Vertex &v=Vertex())
	This is an advanced function. More...
Vertex_handle	create_vertex (Vertex_handle v)
	This is an advanced function. More...
Cell_handle	create_cell (const Cell &c=Cell())
	This is an advanced function. More...
Cell_handle	create_cell (Cell_handle c)
	This is an advanced function. More...
Cell_handle	create_cell (Vertex_handle v0, Vertex_handle v1, Vertex_handle v2, Vertex_handle v3)
	This is an advanced function. More...
Cell_handle	create_cell (Vertex_handle v0, Vertex_handle v1, Vertex_handle v2, Vertex_handle v3, Cell_handle n0, Cell_handle n1, Cell_handle n2, Cell_handle n3)
	This is an advanced function. More...
void	delete_vertex (Vertex_handle v)
	This is an advanced function. More...
void	delete_cell (Cell_handle c)
	This is an advanced function. More...
template<class VertexIt >
void	delete_vertices (VertexIt first, VertexIt last)
	This is an advanced function. More...
template<class CellIt >
void	delete_cells (CellIt first, CellIt last)
	This is an advanced function. More...

Traversing the triangulation
Cell_iterator	cells_begin () const
	Returns cells_end() when tds.dimension() \( <3\).
Cell_iterator	cells_end () const
Cell_iterator	raw_cells_begin () const
	Low-level access to the cells, does not return cells_end() when tds.dimension() \( <3\).
Cell_iterator	raw_cells_end () const
Facet_iterator	facets_begin () const
	Returns facets_end() when tds.dimension() \( <2\).
Facet_iterator	facets_end () const
Edge_iterator	edges_begin () const
	Returns edges_end() when tds.dimension() \( <1\).
Edge_iterator	edges_end () const
Vertex_iterator	vertices_begin () const
Vertex_iterator	vertices_end () const

Cell and facet circulators
Cell_circulator	incident_cells (const Edge &e) const
	Starts at an arbitrary cell incident to e. More...
Cell_circulator	incident_cells (Cell_handle c, int i, int j) const
	As above for edge (i,j) of c.
Cell_circulator	incident_cells (const Edge &e, Cell_handle start) const
	Starts at cell start. More...
Cell_circulator	incident_cells (Cell_handle c, int i, int j, Cell_handle start) const
	As above for edge (i,j) of c.
Facet_circulator	incident_facets (Edge e) const
	Starts at an arbitrary facet incident to e. More...
Facet_circulator	incident_facets (Cell_handle c, int i, int j) const
	As above for edge (i,j) of c. More...
Facet_circulator	incident_facets (Edge e, Facet start) const
	Starts at facet start. More...
Facet_circulator	incident_facets (Edge e, Cell_handle start, int f) const
	Starts at facet of index f in start. More...
Facet_circulator	incident_facets (Cell_handle c, int i, int j, Facet start) const
	As above for edge (i,j) of c. More...
Facet_circulator	incident_facets (Cell_handle c, int i, int j, Cell_handle start, int f) const
	As above for edge (i,j) of c and facet (start,f). More...

Traversal of the incident cells, facets and edges, and the adjacent vertices of a given vertex
template<class OutputIterator >
OutputIterator	incident_cells (Vertex_handle v, OutputIterator cells) const
	Copies the Cell_handles of all cells incident to v to the output iterator cells. More...
template<class OutputIterator >
OutputIterator	incident_facets (Vertex_handle v, OutputIterator facets) const
	Copies the Facets incident to v to the output iterator facets. More...
template<class OutputIterator >
OutputIterator	incident_edges (Vertex_handle v, OutputIterator edges) const
	Copies all Edges incident to v to the output iterator edges. More...
template<class OutputIterator >
OutputIterator	adjacent_vertices (Vertex_handle v, OutputIterator vertices) const
	Copies the Vertex_handles of all vertices adjacent to v to the output iterator vertices. More...
size_type	degree (Vertex_handle v) const
	Returns the degree of a vertex, that is, the number of incident vertices. More...

Traversal between adjacent cells
int	mirror_index (Cell_handle c, int i) const
	Returns the index of c in its \( i^{th}\) neighbor. More...
Vertex_handle	mirror_vertex (Cell_handle c, int i) const
	Returns the vertex of the \( i^{th}\) neighbor of c that is opposite to c. More...
Facet	mirror_facet (Facet f) const
	Returns the same facet seen from the other adjacent cell.

Checking
bool	is_valid (bool verbose=false) const
	This is a function for debugging purpose. More...
bool	is_valid (Vertex_handle v, bool verbose=false) const
	This is a function for debugging purpose. More...
bool	is_valid (Cell_handle c, bool verbose=false) const
	This is a function for debugging purpose. More...
istream &	operator>> (istream &is, TriangulationDataStructure_3 &tds)
	Reads a combinatorial triangulation from is and assigns it to tds
ostream &	operator<< (ostream &os, const TriangulationDataStructure_3 &tds)
	Writes tds into the stream os

Member Typedef Documentation

typedef unspecified_type TriangulationDataStructure_3::Concurrency_tag

Can be CGAL::Sequential_tag or CGAL::Parallel_tag.

If it is CGAL::Parallel_tag, the following functions can be called concurrently: create_vertex, create_cell, delete_vertex, delete_cell.

typedef Triple<Cell_handle, int, int> TriangulationDataStructure_3::Edge

(c,i,j) is the edge of cell c whose vertices indices are i and j.

(See Section Representation.)

typedef std::pair<Cell_handle, int> TriangulationDataStructure_3::Facet

(c,i) is the facet of c opposite to the vertex of index i.

(See Section Representation.)

template<typename Cb2 >

using TriangulationDataStructure_3::Rebind_cell = unspecified_type

Advanced

This template class allows to get the type of a triangulation data structure that only changes the cell type.

It has to define a type Rebind_cell<Cb2>::Other which is a rebound triangulation data structure, that is, the one whose TriangulationDSCellBase_3 will be Cb2.

Note

It can be implemented using a nested template class.

template<typename Vb2 >

using TriangulationDataStructure_3::Rebind_vertex = unspecified_type

Advanced

This template class allows to get the type of a triangulation data structure that only changes the vertex type.

It has to define a type Rebind_vertex<Vb2>::Other which is a rebound triangulation data structure, that is, the one whose TriangulationDSVertexBase_3 will be Vb2.

Note

It can be implemented using a nested template class.

Constructor & Destructor Documentation

TriangulationDataStructure_3::TriangulationDataStructure_3

(

const TriangulationDataStructure_3 &

tds1)

Copy constructor.

All vertices and cells are duplicated.

Member Function Documentation

template<class OutputIterator >

OutputIterator TriangulationDataStructure_3::adjacent_vertices	(	Vertex_handle	v,
		OutputIterator	vertices
	)		const

Copies the Vertex_handles of all vertices adjacent to v to the output iterator vertices.

If tds.dimension() \( <0\), then do nothing. Returns the resulting output iterator.

Precondition

v \( \neq\) Vertex_handle(), tds.is_vertex(v).

bool TriangulationDataStructure_3::are_equal	(	const Facet &	f,
		Cell_handle	n,
		int	j
	)		const

For these three methods:

Precondition

tds.dimension()=3.

void TriangulationDataStructure_3::clear

(

)

Deletes all cells and vertices.

tds is reset as a triangulation data structure constructed by the default constructor.

Vertex_handle TriangulationDataStructure_3::copy_tds	(	const TriangulationDataStructure_3 &	tds1,
		Vertex_handle	v = Vertex_handle()
	)

tds1 is copied into tds.

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

Precondition

The optional argument v is a vertex of tds1.

template<class TDS_src , class ConvertVertex , class ConvertCell >

Vertex_handle tds TriangulationDataStructure_3::copy_tds	(	const TDS_src &	tds_src,
		typename TDS_src::Vertex_handle	v,
		const ConvertVertex &	convert_vertex,
		const ConvertCell &	convert_cell
	)

tds_src is copied into this.

As the vertex and cell types might be different and incompatible, the creation of new cells and vertices is made thanks to the functors convert_vertex and convert_cell, that convert vertex and cell 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 cells with the functor convert_cell. 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 ConvertCell must provide two operator()'s that are responsible for converting the source cell c_src into the target cell:
  • Cell operator()(const TDS_src::Cell& c_src); This operator is used to create the cell from c_src.
  • void operator()(const TDS_src::Cell& c_src, Cell& c_tgt); This operator is meant to be used in case heavy data should transferred to c_tgt.

Precondition

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

Cell_handle TriangulationDataStructure_3::create_cell

(

const Cell &

c = Cell())

This is an advanced function.

Advanced

Adds a copy of the cell c to the triangulation data structure.

Cell_handle TriangulationDataStructure_3::create_cell

(

Cell_handle

c)

This is an advanced function.

Advanced

Creates a cell which is a copy of the one pointed to by c and adds it to the triangulation data structure.

Cell_handle TriangulationDataStructure_3::create_cell	(	Vertex_handle	v0,
		Vertex_handle	v1,
		Vertex_handle	v2,
		Vertex_handle	v3
	)

This is an advanced function.

Advanced

Creates a cell and adds it into the triangulation data structure. Initializes the vertices of the cell, its neighbor handles being initialized with the default constructed handle.

Cell_handle TriangulationDataStructure_3::create_cell	(	Vertex_handle	v0,
		Vertex_handle	v1,
		Vertex_handle	v2,
		Vertex_handle	v3,
		Cell_handle	n0,
		Cell_handle	n1,
		Cell_handle	n2,
		Cell_handle	n3
	)

This is an advanced function.

Advanced

Creates a cell, initializes its vertices and neighbors, and adds it into the triangulation data structure.

Vertex_handle TriangulationDataStructure_3::create_vertex

(

const Vertex &

v = Vertex())

This is an advanced function.

Advanced

Adds a copy of the vertex v to the triangulation data structure.

Vertex_handle TriangulationDataStructure_3::create_vertex

(

Vertex_handle

v)

This is an advanced function.

Advanced

Creates a vertex which is a copy of the one pointed to by v and adds it to the triangulation data structure.

void TriangulationDataStructure_3::decrease_dimension	(	Cell_handle	c,
		int	i
	)

The link of a vertex \( v\) is formed by the facets disjoint from \( v\) that are included in the cells incident to \( v\).

When the link of v = c->vertex(i) contains all the other vertices, decrease_dimension crushes the triangulation of the sphere \( S^d\) of \( \mathbb{R}^{d+1}\) onto the triangulation of the sphere \( S^{d-1}\) of \( \mathbb{R}^{d}\) formed by the link of v augmented with the vertex v itself, for \( d\)==2,3; this one is placed on the facet (c, i) (see Fig. TDS3dim_down).

Precondition

The dimension must be 2 or 3. The degree of v must be equal to the total number of vertices of the triangulation data structure minus 1.

From an \( S^d\) data structure to an \( S^{d-1}\) data structure (top: \( d==2\), bottom: \( d==3\)).

size_type TriangulationDataStructure_3::degree

(

Vertex_handle

v)

const

Returns the degree of a vertex, that is, the number of incident vertices.

Precondition

v \( \neq\) Vertex_handle(), tds.is_vertex(v).

void TriangulationDataStructure_3::delete_cell

(

Cell_handle

c)

This is an advanced function.

Advanced

Removes the cell from the triangulation data structure.

Precondition

The cell is a cell of tds.

template<class CellIt >

void TriangulationDataStructure_3::delete_cells	(	CellIt	first,
		CellIt	last
	)

This is an advanced function.

Advanced

Calls delete_cell over an iterator range of value type Cell_handle.

void TriangulationDataStructure_3::delete_vertex

(

Vertex_handle

v)

This is an advanced function.

Advanced

Removes the vertex from the triangulation data structure.

Precondition

The vertex is a vertex of tds.

template<class VertexIt >

void TriangulationDataStructure_3::delete_vertices	(	VertexIt	first,
		VertexIt	last
	)

This is an advanced function.

Advanced

Calls delete_vertex over an iterator range of value type Vertex_handle.

bool TriangulationDataStructure_3::flip	(	Cell_handle	c,
		int	i,
		int	j
	)

Before flipping, these methods check that edge e=(c,i,j) is flippable (which is quite expensive).

They return false or true according to this test.

bool TriangulationDataStructure_3::flip	(	Cell_handle	c,
		int	i
	)

Before flipping, these methods check that facet f=(c,i) is flippable (which is quite expensive).

They return false or true according to this test.

void TriangulationDataStructure_3::flip_flippable	(	Cell_handle	c,
		int	i,
		int	j
	)

Should be preferred to the previous methods when the edge is known to be flippable.

Precondition

The edge is flippable.

void TriangulationDataStructure_3::flip_flippable	(	Cell_handle	c,
		int	i
	)

Should be preferred to the previous methods when the facet is known to be flippable.

Precondition

The facet is flippable.

bool TriangulationDataStructure_3::has_vertex	(	const Facet &	f,
		Vertex_handle	v,
		int &	j
	)		const

If v is a vertex of f, then j is the index of v in the cell f.first, and the method returns true.

Precondition

tds.dimension()=3

bool TriangulationDataStructure_3::has_vertex	(	Cell_handle	c,
		int	i,
		Vertex_handle	v,
		int &	j
	)		const

Same for facet (c,i).

Computes the index j of v in c.

Cell_circulator TriangulationDataStructure_3::incident_cells

(

const Edge &

e)

const

Starts at an arbitrary cell incident to e.

Precondition

tds.dimension() \( =3\)

Cell_circulator TriangulationDataStructure_3::incident_cells	(	const Edge &	e,
		Cell_handle	start
	)		const

Starts at cell start.

Precondition

tds.dimension() \( =3\) and start is incident to e.

template<class OutputIterator >

OutputIterator TriangulationDataStructure_3::incident_cells	(	Vertex_handle	v,
		OutputIterator	cells
	)		const

Copies the Cell_handles of all cells incident to v to the output iterator cells.

Returns the resulting output iterator.

Precondition

tds.dimension() \( =3\), v \( \neq\) Vertex_handle(), tds.is_vertex(v).

template<class OutputIterator >

OutputIterator TriangulationDataStructure_3::incident_edges	(	Vertex_handle	v,
		OutputIterator	edges
	)		const

Copies all Edges incident to v to the output iterator edges.

Returns the resulting output iterator.

Precondition

tds.dimension() \( >0\), v \( \neq\) Vertex_handle(), tds.is_vertex(v).

Facet_circulator TriangulationDataStructure_3::incident_facets

(

Edge

e)

const

Starts at an arbitrary facet incident to e.

Only defined in dimension 3, though are defined also in dimension 2: there are only two facets sahring an edge in dimension 2.

Precondition

tds.dimension() \( =3\)

Facet_circulator TriangulationDataStructure_3::incident_facets	(	Cell_handle	c,
		int	i,
		int	j
	)		const

As above for edge (i,j) of c.

Only defined in dimension 3, though are defined also in dimension 2: there are only two facets sahring an edge in dimension 2.

Facet_circulator TriangulationDataStructure_3::incident_facets	(	Edge	e,
		Facet	start
	)		const

Starts at facet start.

Only defined in dimension 3, though are defined also in dimension 2: there are only two facets sahring an edge in dimension 2.

Precondition

start is incident to e.

Facet_circulator TriangulationDataStructure_3::incident_facets	(	Edge	e,
		Cell_handle	start,
		int	f
	)		const

Starts at facet of index f in start.

Only defined in dimension 3, though are defined also in dimension 2: there are only two facets sahring an edge in dimension 2.

Facet_circulator TriangulationDataStructure_3::incident_facets	(	Cell_handle	c,
		int	i,
		int	j,
		Facet	start
	)		const

As above for edge (i,j) of c.

Only defined in dimension 3, though are defined also in dimension 2: there are only two facets sahring an edge in dimension 2.

Facet_circulator TriangulationDataStructure_3::incident_facets	(	Cell_handle	c,
		int	i,
		int	j,
		Cell_handle	start,
		int	f
	)		const

As above for edge (i,j) of c and facet (start,f).

Only defined in dimension 3, though are defined also in dimension 2: there are only two facets sahring an edge in dimension 2.

template<class OutputIterator >

OutputIterator TriangulationDataStructure_3::incident_facets	(	Vertex_handle	v,
		OutputIterator	facets
	)		const

Copies the Facets incident to v to the output iterator facets.

Returns the resulting output iterator.

Precondition

tds.dimension() \( >1\), v \( \neq\) Vertex_handle(), tds.is_vertex(v).

Vertex_handle TriangulationDataStructure_3::insert_in_cell

(

Cell_handle

c)

Creates a new vertex, inserts it in cell c and returns its handle.

The cell c is split into four new cells, each of these cells being formed by the new vertex and a facet of c.

Precondition

tds.dimension() \( = 3\) and c is a cell of tds.

Vertex_handle TriangulationDataStructure_3::insert_in_edge

(

Edge

e)

Creates a new vertex, inserts it in edge e and returns its handle.

In dimension 3, all the incident cells are split into 2 new cells; in dimension 2, the 2 incident facets are split into 2 new facets; in dimension 1, the edge is split into 2 new edges.

Precondition

tds.dimension() \( \geq1\) and e is an edge of tds.

Vertex_handle TriangulationDataStructure_3::insert_in_edge	(	Cell_handle	c,
		int	i,
		int	j
	)

Creates a new vertex, inserts it in edge \( (i,j)\) of c and returns its handle.

Precondition

tds.dimension() \( \geq1\). \( i\neq j\), \( i,j \in\{0,1,2,3\}\) in dimension 3, \( i,j \in\{0,1,2\}\) in dimension 2, \( i,j \in\{0,1\}\) in dimension 1 and (c,i,j) is an edge of tds.

Vertex_handle TriangulationDataStructure_3::insert_in_facet

(

const Facet &

f)

Creates a new vertex, inserts it in facet f and returns its handle.

In dimension 3, the two incident cells are split into 3 new cells; in dimension 2, the facet is split into 3 facets.

Precondition

tds.dimension() \( \geq2\) and f is a facet of tds.

Vertex_handle TriangulationDataStructure_3::insert_in_facet	(	Cell_handle	c,
		int	i
	)

Creates a new vertex, inserts it in facet i of c and returns its handle.

Precondition

tds.dimension() \( \geq2\), \( i \in\{0,1,2,3\}\) in dimension 3, \( i=3\) in dimension 2 and (c,i) is a facet of tds.

template<class CellIt >

Vertex_handle TriangulationDataStructure_3::insert_in_hole	(	CellIt	cell_begin,
		CellIt	cell_end,
		Cell_handle	begin,
		int	i
	)

Creates a new vertex by starring a hole.

It takes an iterator range [cell_begin; cell_end[ of Cell_handles which specifies a set of connected cells (resp. facets in dimension 2) describing a hole. (begin, i) is a facet (resp. an edge) on the boundary of the hole, that is, begin belongs to the set of cells (resp. facets) previously described, and begin->neighbor(i) does not. Then this function deletes all the cells (resp. facets) describing the hole, creates a new vertex v, and for each facet (resp. edge) on the boundary of the hole, creates a new cell (resp. facet) with v as vertex. v is returned.

Precondition

tds.dimension() \( \geq2\), the set of cells (resp. facets) is connected, and its boundary is connected.

template<class CellIt >

Vertex_handle TriangulationDataStructure_3::insert_in_hole	(	CellIt	cell_begin,
		CellIt	cell_end,
		Cell_handle	begin,
		int	i,
		Vertex_handle	newv
	)

Same as above, except that newv will be used as the new vertex, which must have been allocated previously with e.g.

create_vertex.

Vertex_handle TriangulationDataStructure_3::insert_increase_dimension

(

Vertex_handle

star = Vertex_handle())

Transforms a triangulation of the sphere \( S^d\) of \( \mathbb{R}^{d+1}\) into the triangulation of the sphere \( S^{d+1}\) of \( \mathbb{R}^{d+2}\) by adding a new vertex v: v is linked to all the vertices to triangulate one of the two half-spheres of dimension \( (d+1)\).

Vertex star is used to triangulate the second half-sphere (when there is an associated geometric triangulation, star is in fact the vertex associated with its infinite vertex). See Figure TDS3figtopoinsert_outside_affine_hull.

The numbering of the cells is such that, if f was a face of maximal dimension in the initial triangulation, then (f,v) (in this order) is the corresponding face in the new triangulation. This method can be used to insert the first two vertices in an empty triangulation.

A handle to v is returned.

Precondition

tds.dimension() \( = d < 3\). When tds.number_of_vertices() \( >0\), \( star \neq\) Vertex_handle() and star is a vertex of tds.

insert_increase_dimension (1-dimensional case).

bool TriangulationDataStructure_3::is_cell

(

Cell_handle

c)

const

Tests whether c is a cell of tds.

Answers false when dimension() \( <3\) .

bool TriangulationDataStructure_3::is_cell	(	Vertex_handle	u,
		Vertex_handle	v,
		Vertex_handle	w,
		Vertex_handle	t,
		Cell_handle &	c,
		int &	i,
		int &	j,
		int &	k,
		int &	l
	)		const

Tests whether (u,v,w,t) is a cell of tds.

If the cell c is found, it computes the indices i, j, k and l of the vertices u, v, w and t in c, in this order.

bool TriangulationDataStructure_3::is_edge	(	Cell_handle	c,
		int	i,
		int	j
	)		const

Tests whether (c,i,j) is an edge of tds.

Answers false when dimension() \( <1\) .

Precondition

\( i,j \in\{0,1,2,3\}\)

bool TriangulationDataStructure_3::is_edge	(	Vertex_handle	u,
		Vertex_handle	v,
		Cell_handle &	c,
		int &	i,
		int &	j
	)		const

Tests whether (u,v) is an edge of tds.

If the edge is found, it computes a cell c having this edge and the indices i and j of the vertices u and v, in this order.

bool TriangulationDataStructure_3::is_facet	(	Cell_handle	c,
		int	i
	)		const

Tests whether (c,i) is a facet of tds.

Answers false when dimension() \( <2\) .

Precondition

\( i \in\{0,1,2,3\}\)

bool TriangulationDataStructure_3::is_facet	(	Vertex_handle	u,
		Vertex_handle	v,
		Vertex_handle	w,
		Cell_handle &	c,
		int &	i,
		int &	j,
		int &	k
	)		const

Tests whether (u,v,w) is a facet of tds.

If the facet is found, it computes a cell c having this facet and the indices i, j and k of the vertices u, v and w, in this order.

bool TriangulationDataStructure_3::is_valid

(

bool

verbose = false)

const

This is a function for debugging purpose.

Debugging Support

Checks the combinatorial validity of the triangulation by checking the local validity of all its cells and vertices (see functions below). (See Section Representation.) Moreover, the Euler relation is tested.

When verbose is set to true, messages are printed to give a precise indication on the kind of invalidity encountered.

bool TriangulationDataStructure_3::is_valid	(	Vertex_handle	v,
		bool	verbose = false
	)		const

This is a function for debugging purpose.

Debugging Support

Checks the local validity of the adjacency relations of the triangulation. It also calls the is_valid member function of the vertex. When verbose is set to true, messages are printed to give a precise indication on the kind of invalidity encountered.

bool TriangulationDataStructure_3::is_valid	(	Cell_handle	c,
		bool	verbose = false
	)		const

This is a function for debugging purpose.

Debugging Support

Checks the local validity of the adjacency relations of the triangulation. It also calls the is_valid member function of the cell. When verbose is set to true, messages are printed to give a precise indication on the kind of invalidity encountered.

int TriangulationDataStructure_3::mirror_index	(	Cell_handle	c,
		int	i
	)		const

Returns the index of c in its \( i^{th}\) neighbor.

Precondition

\( i \in\{0, 1, 2, 3\}\).

Vertex_handle TriangulationDataStructure_3::mirror_vertex	(	Cell_handle	c,
		int	i
	)		const

Returns the vertex of the \( i^{th}\) neighbor of c that is opposite to c.

Precondition

\( i \in\{0, 1, 2, 3\}\).

size_type TriangulationDataStructure_3::number_of_cells

(

)

const

The number of cells.

Returns 0 if tds.dimension() \( <3\).

size_type TriangulationDataStructure_3::number_of_edges

(

)

const

The number of edges.

Returns 0 if tds.dimension() \( <1\).

size_type TriangulationDataStructure_3::number_of_facets

(

)

const

The number of facets.

Returns 0 if tds.dimension() \( <2\).

size_type TriangulationDataStructure_3::number_of_vertices

(

)

const

The number of vertices.

Note that the triangulation data structure has one more vertex than an associated geometric triangulation, if there is one, since the infinite vertex is a standard vertex and is thus also counted.

TriangulationDataStructure_3& TriangulationDataStructure_3::operator=

(

const TriangulationDataStructure_3 &

tds1)

Assignment operator.

All vertices and cells are duplicated, and the former data structure of tds is deleted.

void TriangulationDataStructure_3::remove_decrease_dimension	(	Vertex_handle	v,
		Vertex_handle	w = v
	)

This operation is the reciprocal of insert_increase_dimension().

It transforms a triangulation of the sphere \( S^d\) of \( \mathbb{R}^{d+1}\) into the triangulation of the sphere \( S^{d-1}\) of \( \mathbb{R}^{d}\) by removing the vertex v. Delete the cells incident to w, keep the others.

Precondition

tds.dimension() \( = d \geq-1\). tds.degree(v) \( =\) degree(w) \( =\) tds.number_of_vertices() \( -1\).

Cell_handle TriangulationDataStructure_3::remove_from_maximal_dimension_simplex

(

Vertex_handle

v)

Removes v.

The incident simplices of maximal dimension incident to v are replaced by a single simplex of the same dimension. This operation is exactly the reciprocal to tds.insert_in_cell(v) in dimension 3, tds.insert_in_facet(v) in dimension 2, and tds.insert_in_edge(v) in dimension 1.

Precondition

tds.degree(v) \( =\) tds.dimension()+1.

void TriangulationDataStructure_3::reorient

(

)

This is an advanced function.

Advanced

Changes the orientation of all cells of the triangulation data structure.

Precondition

tds.dimension() \( \geq1\).

void TriangulationDataStructure_3::set_dimension

(

int

n)

This is an advanced function.

Advanced

Sets the dimension to n.

void TriangulationDataStructure_3::swap

(

TriangulationDataStructure_3 &

tds1)

Swaps tds and tds1.

There is no copy of cells and vertices, thus this method runs in constant time. This method should be preferred to tds=tds1 or tds(tds1) when tds1 is deleted after that.