CGAL 4.8.2  3D Triangulation Data Structure

3Dtriangulation data structures are meant to maintain the combinatorial information for 3Dgeometric 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 41.1).
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 3Dtriangulation data structure in CGAL can handle the cases when the dimension of the triangulation is lower than 3 (see Section Representation).
Thus, a 3Dtriangulation 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.
Concepts  
concept  Cell 
The concept TriangulationDataStructure_3::Cell stores four Vertex_handle s to its four vertices and four Cell_handle s 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 3Dtriangulation 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, nonmutable 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 nonmutable.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 constanttime 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 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 threedimensional 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 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 halfspheres 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,  
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 
Lowlevel 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_handle s 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 Facet s incident to v to the output iterator facets . More...  
template<class OutputIterator >  
OutputIterator  incident_edges (Vertex_handle v, OutputIterator edges) const 
Copies all Edge s incident to v to the output iterator edges . More...  
template<class OutputIterator >  
OutputIterator  adjacent_vertices (Vertex_handle v, OutputIterator vertices) const 
Copies the Vertex_handle s 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  
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.)
using TriangulationDataStructure_3::Rebind_cell = unspecified_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
.
using TriangulationDataStructure_3::Rebind_vertex = unspecified_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
.
TriangulationDataStructure_3::TriangulationDataStructure_3  (  const TriangulationDataStructure_3 &  tds1) 
Copy constructor.
All vertices and cells are duplicated.
OutputIterator TriangulationDataStructure_3::adjacent_vertices  (  Vertex_handle  v, 
OutputIterator  vertices  
)  const 
Copies the Vertex_handle
s of all vertices adjacent to v
to the output iterator vertices
.
If tds.dimension()
\( <0\), then do nothing. Returns the resulting output iterator.
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:
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.
v
is a vertex of tds1
. 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.
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
.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
.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.
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.
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.
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.
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.
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.
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 \( S^{d1}\) 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).
v
must be equal to the total number of vertices of the triangulation data structure minus 1.size_type TriangulationDataStructure_3::degree  (  Vertex_handle  v)  const 
Returns the degree of a vertex, that is, the number of incident vertices.
v
\( \neq\) Vertex_handle()
, tds.is_vertex(v)
. void TriangulationDataStructure_3::delete_cell  (  Cell_handle  c) 
This is an advanced function.
Removes the cell from the triangulation data structure.
tds
. void TriangulationDataStructure_3::delete_cells  (  CellIt  first, 
CellIt  last  
) 
This is an advanced function.
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.
Removes the vertex from the triangulation data structure.
tds
. void TriangulationDataStructure_3::delete_vertices  (  VertexIt  first, 
VertexIt  last  
) 
This is an advanced function.
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.
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.
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
.
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
.
tds.dimension()
\( =3\) Cell_circulator TriangulationDataStructure_3::incident_cells  (  const Edge &  e, 
Cell_handle  start  
)  const 
Starts at cell start
.
tds.dimension()
\( =3\) and start
is incident to e
. OutputIterator TriangulationDataStructure_3::incident_cells  (  Vertex_handle  v, 
OutputIterator  cells  
)  const 
Copies the Cell_handle
s of all cells incident to v
to the output iterator cells
.
Returns the resulting output iterator.
tds.dimension()
\( =3\), v
\( \neq\) Vertex_handle()
, tds.is_vertex(v)
. OutputIterator TriangulationDataStructure_3::incident_edges  (  Vertex_handle  v, 
OutputIterator  edges  
)  const 
Copies all Edge
s incident to v
to the output iterator edges
.
Returns the resulting output iterator.
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.
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.
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.
OutputIterator TriangulationDataStructure_3::incident_facets  (  Vertex_handle  v, 
OutputIterator  facets  
)  const 
Copies the Facet
s incident to v
to the output iterator facets
.
Returns the resulting output iterator.
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
.
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.
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.
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.
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.
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
. 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.
tds
.dimension()
\( \geq2\), the set of cells (resp. facets) is connected, and its boundary is connected. 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 halfspheres of dimension \( (d+1)\).
Vertex star
is used to triangulate the second halfsphere (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.
tds
.dimension()
\( = d < 3\). When tds
.number_of_vertices()
\( >0\), \( star \neq\) Vertex_handle()
and star
is a vertex of tds
.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()
\( <1\) .
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()
\( <2\) .
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.
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.
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.
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.
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
.
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 \( S^{d1}\) of \( \mathbb{R}^{d}\) by removing the vertex v
. Delete the cells incident to w
, keep the others.
tds
.dimension()
\( = d \geq1\). 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.
tds
.degree(v)
\( =\) tds
.dimension()+1
. void TriangulationDataStructure_3::reorient  (  ) 
This is an advanced function.
Changes the orientation of all cells of the triangulation data structure.
tds
.dimension()
\( \geq1\). void TriangulationDataStructure_3::set_dimension  (  int  n) 
This is an advanced function.
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.