3Dtriangulation data structures are meant to maintain the combinatorial information for 3Dgeometric triangulations.
In Cgal, a triangulation data structure is a container of cells (3faces) and vertices (0faces). Following the standard vocabulary of simplicial complexes, an iface f_{i} and a jface f_{j} (0 ≤ j < i ≤ 3) are said to be incident in the triangulation if f_{j} is a (sub)face of f_{i}, and two ifaces (0 ≤ i ≤ 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 (1faces) and facets (2faces) 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 40.1).
Thus, a 3Dtriangulation data structure can store a triangulation of a topological sphere S^{d} of ℝ^{d+1}, for any d ∈ {1,0,1,2,3}.
The second template parameter of the basic triangulation class (see Chapter 39 ) Triangulation_3 is a triangulation data structure class. (See Chapter 40.)
To ensure all the flexibility of the class Triangulation_3, a model of a triangulation data structure must be templated by the base vertex and the base cell classes (see 40.1): TriangulationDataStructure_3<TriangulationVertexBase_3,TriangulationCellBase_3>. The optional functionalities related to geometry are compulsory for this use as a template parameter of Triangulation_3.
A class that satisfies the requirements for a triangulation data structure class must provide the following types and operations.
Vertices and cells are usually manipulated via handles, which support the two dereference operators operator* and operator>.
TriangulationDataStructure_3::Vertex_handle  
TriangulationDataStructure_3::Cell_handle 
Requirements for Vertex and Cell are described in TriangulationDataStructure_3::Vertex and TriangulationDataStructure_3::Cell .
template <typename Vb2>  
TriangulationDataStructure_3:: struct Rebind_vertex  
This nested template class allows to get the type of a triangulation
data structure that only changes the vertex type. It has to define a type
Other which is a rebound triangulation data structure, that is, the
one whose TriangulationDSVertexBase_3 will be Vb2.
 
template <typename Cb2>  
TriangulationDataStructure_3:: struct Rebind_cell  
This nested template class allows to get the type of a triangulation
data structure that only changes the cell type. It has to define a type
Other which is a rebound triangulation data structure, that is, the
one whose TriangulationDSCellBase_3 will be Cb2.

typedef Triple<Cell_handle, int, int>  
Edge;  (c,i,j) is the edge of cell c whose vertices indices are i and j. (See Section 40.1.)  
typedef std::pair<Cell_handle, int>  
Facet;  (c,i) is the facet of c opposite to the vertex of index i. (See Section 40.1.) 
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.
TriangulationDataStructure_3::Cell_iterator  
TriangulationDataStructure_3::Facet_iterator  
TriangulationDataStructure_3::Edge_iterator  
TriangulationDataStructure_3::Vertex_iterator 
The following circulators allow us to visit all the cells and facets incident to a given edge. They are bidirectional and nonmutable.
TriangulationDataStructure_3::Facet_circulator  
TriangulationDataStructure_3::Cell_circulator 
Iterators and circulators are convertible to the corresponding handles, thus the user can pass them directly as arguments to the functions.
TriangulationDataStructure_3 tds;  
Default constructor.
 
TriangulationDataStructure_3 tds ( tds1);  
Copy constructor. All vertices and cells are duplicated.

TriangulationDataStructure_3&  tds = tds1  Assignment operator. All vertices and cells are duplicated, and the former data structure of tds is deleted.  
Vertex_handle  tds.copy_tds ( 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.
 
void  tds.swap ( & 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.  
void  tds.clear ()  Deletes all cells and vertices. tds is reset as a triangulation data structure constructed by the default constructor. 
int  tds.dimension () const  The dimension of the triangulated topological sphere. 
size_type  tds.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. 
size_type  tds.number_of_cells () const  The number of cells. Returns 0 if tds.dimension()<3. 
size_type  tds.number_of_facets () const  The number of facets. Returns 0 if tds.dimension()<2. 
size_type  tds.number_of_edges () const  The number of edges. Returns 0 if tds.dimension()<1. 
void  tds.set_dimension ( int n)  Sets the dimension to n. 
bool  tds.is_vertex ( Vertex_handle v) const  
Tests whether v is a vertex of tds.  
bool  tds.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  tds.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  tds.is_edge ( Vertex_handle u, Vertex_handle v) const  
Tests whether (u,v) is an edge of tds.  
bool  tds.is_facet ( Cell_handle c, int i) const  
Tests whether (c,i) is a facet of tds. Answers false when
dimension() <2 .
 
bool 
 
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  tds.is_cell ( Cell_handle c) const  
Tests whether c is a cell of tds. Answers false when dimension() <3 .  
bool 
 
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. 
There is a method has_vertex in the cell class. The analogous methods for facets are defined here.
bool  tds.has_vertex ( 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.
 
bool  tds.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.  
bool  tds.has_vertex ( Facet f, Vertex_handle v) const  
bool  tds.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. 
The following three methods test whether two facets have the same vertices.
bool  tds.are_equal ( Facet f, Facet g) const  
bool  tds.are_equal ( Cell_handle c, int i, Cell_handle n, int j) const  
bool  tds.are_equal ( Facet f, Cell_handle n, int j) const  
For these three methods:

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 40.7(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.
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  tds.flip ( Edge e)  
bool  tds.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.  
void  tds.flip_flippable ( Edge e)  
void  tds.flip_flippable ( Cell_handle c, int i, int j)  
Should be preferred to the previous methods when the edge is
known to be flippable.
 
bool  tds.flip ( Facet f)  
bool  tds.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  tds.flip_flippable ( Facet f)  
void  tds.flip_flippable ( Cell_handle c, int i)  
Should be preferred to the previous methods when the facet is
known to be flippable.

The following modifier member functions guarantee the combinatorial validity of the resulting triangulation.
Vertex_handle  tds.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.
 
Vertex_handle  tds.insert_in_facet ( 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.
 
Vertex_handle  tds.insert_in_facet ( Cell_handle c, int i)  
Creates a new vertex, inserts it in facet i of c and returns its
handle.
 
Vertex_handle  tds.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.
 
Vertex_handle  tds.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.
 
Vertex_handle  tds.insert_increase_dimension ( Vertex_handle star = Vertex_handle())  
Transforms a triangulation of the sphere S^{d} of ℝ^{d+1} into the
triangulation of the sphere S^{d+1} of 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 40.8. 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.

Figure 40.8: insert_increase_dimension (1dimensional case).
template <class CellIt>  
Vertex_handle  tds.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.
 
template <class CellIt>  
Vertex_handle 
 
Same as above, except that newv will be used as the new vertex, which must have been allocated previously with e.g. create_vertex. 
void  tds.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 ℝ^{d+1} into the
triangulation of the sphere S^{d1} of ℝ^{d} by removing the vertex
v. Delete the cells incident to w, keep the others.
 
Cell_handle  tds.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.

The following operation, decrease_dimension, is necessary when the displacement of a vertex decreases the dimension of the triangulation.
void  tds.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 ℝ^{d+1} onto the
triangulation of the sphere S^{d1} of 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. 40.9).

Figure 40.9: From an S^{d} data structure to an S^{d1} data structure (top: d==2, bottom: d==3).
void  tds.reorient () 
Changes the orientation of all cells of the triangulation data structure.
 
Vertex_handle  tds.create_vertex ( Vertex v = Vertex())  
Adds a copy of the vertex v to the triangulation data structure.  
Vertex_handle  tds.create_vertex ( Vertex_handle v)  
Creates a vertex which is a copy of the one pointed to by v and adds it to the triangulation data structure.  
Cell_handle  tds.create_cell ( Cell c = Cell())  
Adds a copy of the cell c to the triangulation data structure.  
Cell_handle  tds.create_cell ( Cell_handle c)  Creates a cell which is a copy of the one pointed to by c and adds it to the triangulation data structure.  
Cell_handle  tds.create_cell ( Vertex_handle v0, Vertex_handle v1, Vertex_handle v2, Vertex_handle v3)  
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 
 
Creates a cell, initializes its vertices and neighbors, and adds it into the triangulation data structure.  
void  tds.delete_vertex ( Vertex_handle v)  
Removes the vertex from the triangulation data structure.
 
void  tds.delete_cell ( Cell_handle c) 
Removes the cell from the triangulation data structure.
 
template <class VertexIt>  
void  tds.delete_vertices ( VertexIt first, VertexIt last)  
Calls delete_vertex over an iterator range of value type Vertex_handle.  
template <class CellIt>  
void  tds.delete_cells ( CellIt first, CellIt last)  
Calls delete_cell over an iterator range of value type Cell_handle. 
Cell_circulator  tds.incident_cells ( Edge e) const  
Starts at an arbitrary cell incident to e.
 
Cell_circulator  tds.incident_cells ( Cell_handle c, int i, int j) const  
As above for edge (i,j) of c.  
Cell_circulator  tds.incident_cells ( Edge e, Cell_handle start) const  
Starts at cell start.
 
Cell_circulator  tds.incident_cells ( Cell_handle c, int i, int j, Cell_handle start) const  
As above for edge (i,j) of c. 
The following circulators on facets are defined only in dimension 3, though facets are defined also in dimension 2: there are only two facets sharing an edge in dimension 2.
Facet_circulator  tds.incident_facets ( Edge e) const  
Starts at an arbitrary facet incident to e.
 
Facet_circulator  tds.incident_facets ( Cell_handle c, int i, int j) const  
As above for edge (i,j) of c.  
Facet_circulator  tds.incident_facets ( Edge e, Facet start) const  
Starts at facet start.
 
Facet_circulator  tds.incident_facets ( Edge e, Cell_handle start, int f) const  
Starts at facet of index f in start.  
Facet_circulator  tds.incident_facets ( Cell_handle c, int i, int j, Facet start) const  
As above for edge (i,j) of c.  
Facet_circulator  tds.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). 
int  tds.mirror_index ( Cell_handle c, int i) const  
Returns the index of c in its i^{th} neighbor.
 
Vertex_handle  tds.mirror_vertex ( Cell_handle c, int i) const  
Returns the vertex of the i^{th} neighbor of c that is opposite to
c.
 
Facet  tds.mirror_facet ( Facet f) const  Returns the same facet seen from the other adjacent cell. 
bool  tds.is_valid ( bool verbose = false) const  
Checks the combinatorial validity of the triangulation by checking
the local validity of all its cells and vertices (see functions below).
(See Section 40.1.) 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  tds.is_valid ( Vertex_handle v, bool verbose = false) const  
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  tds.is_valid ( Cell_handle c, bool verbose = false) const  
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. 
istream&  istream& is >> & tds  Reads a combinatorial triangulation from is and assigns it to tds 
ostream&  ostream& os << tds  Writes tds into the stream os 
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.
CGAL::Triangulation_data_structure_3
TriangulationDataStructure_3::Vertex
TriangulationDataStructure_3::Cell