2D Triangulation Data Structure

The representation of CGAL 2D triangulations is based on faces and vertices, Edges are only implicitly represented trough the adjacency relations between two faces.

The triangulation data structure can be seen as a container for faces and vertices maintaining incidence and adjacency relations among them.

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

The three vertices of a face are indexed with 0, 1 and 2.
The neighbors of a face are also
indexed with 0,1,2 in such a way that the neighbor indexed by *i*
is opposite to the vertex with the same index.
See Figure 26.1,
the functions *ccw(i)*
and *cw(i)* shown on this figure
compute respectively $$*i+1* and $$*i-1* modulo 3

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

**Figure 26.1: **Vertices and neighbors.

This kind or representation of simplicial complexes extends in any
dimension. More precisely, in dimension $$*d*, the data structure
will explicitly represents cells (i. e. faces of maximal dimension)
and vertices (i. e. faces of dimension 0).
All faces of dimension between $$*1* and $$*d-1*
will have an implicit representation.
The 2D triangulation data structure can represent simplicial complexes
of dimension 2, 1 or 0.

This rules extends to lower dimensional triangulation data structure
arising in degenerate cases or when the triangulations
have less than three vertices.
A one dimensional triangulation structure maintains a set of vertices
and edges which forms a ring
topologically equivalent to a $$*1*-sphere.

A zero dimensional triangulation data structure
only includes two adjacent vertices
that is
topologically equivalent to a $$*0*-sphere.

The triangulation data structure is required to provide:

- the types
*Vertex*and*Face*for the vertices and faces of the triangulations - the type
*Vertex_handle*and*Face_handle*which are models of the concept*Handle*and through which the vertices and faces are accessed. - iterators to visit all the vertices, edges and faces of the triangulation,
- circulators to visit all the vertices, edges and faces incident to a given vertex

The triangulation data structure is responsible for the creation and removal of faces and vertices (memory management). It provides function that gives the number of faces, edges and vertices of the triangulation.

The triangulation data structure provides member functions to perform the following combinatorial transformation of the triangulation:

- flip of two adjacent faces,
- addition of a new vertex splitting a given face see Figure 26.2,
- addition of a new vertex splitting a given edge,
- addition of a new vertex raising by one the dimension of a degenerate - lower dimensional triangulation,
- removal of a vertex incident to three faces,
- removal of a vertex lowering the dimension of the triangulation

**Figure 26.2: **Insertion of a new vertex, splitting a face

CGAL provides the class
*CGAL::Triangulation_data_structure_2<Vb,Fb>*
as a default triangulation data structure.

This design allows the user to plug in the triangulation data structure his own vertex or face classes tuned for his application.

**Figure 26.3: **The cyclic dependency in triangulations software design.

The solution proposed by CGAL to resolve this cyclic dependency
is based on a rebind mechanism similar to the mechanism used in the
standard allocator class std::allocator.
The vertex and face classes plugged in the instantiation of a
triangulation data structure are themselves instantiated with a
fake data structure. The triangulation data structure
will then rebind these classes, plugging itself
at the place of the fake data structure, before using them
to derive the vertex and face classes. The rebinding is performed
through a nested template class *Rebind_TDS* in the vertex and
face class, which provide the rebound class
as a type called *Other*.

Here is how it works schematically. First, here is the rebinding taking place in the triangulation data structure.

template < class Vb, class Fb > class Triangulation_data_structure { typedef Triangulation_data_structure<Vb,Fb> Self; // Rebind the vertex and face base to the actual TDS (Self). typedef typename Vb::template Rebind_TDS<Self>::Other VertexBase; typedef typename Fb::template Rebind_TDS<Self>::Other FaceBase; // ... further internal machinery leads to the final public types: public: typedef ... Vertex; typedef ... Face; typedef ... Vertex_handle; typedef ... Face_handle; };

Then, here is the vertex class with its nested
*Rebind_TDS* template class and its template parameter
set by default to an an internal type faking a triangulation data
structure.

template < class TDS = an internal type faking a triangulation data structure > class Vertex_base { public: template < class TDS2 > struct Rebind_TDS { typedef Vertex_base<TDS2> Other; }; ... };Imagine an analog

typedef Triangulation_data_structure< Vertex_base<>, Face_base<> > TDS;

There is several possibilities to make use of the flexibility offered by the triangulation data structure.

- First, when the user needs to have,
in vertices and faces, additional information
which do not depend on types defined by the
triangulated data structure, predefined classes
*Triangulation_vertex_base_with_info*and*Triangulation_face_base_with_info*can be plugged in. Those classes have a template parameter*Info*to be instantiated by a user defined type. They store a data member of this type and gives access to it. - Second, the user can derive
his own base classes from the default base
classes:
*Triangulation_ds_vertex_base_2*, and*Triangulation_ds_face_base_2*are the default base classes to be plugged in a triangulation data structure used alone. Triangulation classes requires a data structure in which other base classes have been plugged it. The default base classes for most of the triangulation classes are*Triangulation_vertex_base_2*, and*Triangulation_face_base_2*are the default base classes to be used when the triangulation data structure is plugged in a triangulation class.When derivation is used, the rebind mechanism is slightly more involved, because it is necessary to rebind the base class itself. However the user will be able to use in his classes references to types provided by the triangulation data structure. For example,

template < class Gt, class Vb = CGAL::Triangulation_vertex_base_2<Gt> > class My_vertex_base : public Vb { public : template < typename TDS2 > struct Rebind_TDS { typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2; typedef My_vertex_base<Gt,Vb2> Other; }; typedef typename Vb::Triangulation_data_structure Tds; typedef typename Tds::Vertex_handle Vertex_handle; ...... };

- At last the user can write his own base classes.
If the triangulation data structure is used alone,
the requirements for the base classes are described by the concepts
*TriangulationDSVertexBase_2*and*TriangulationDSFaceBase_2*. If the triangulation data structure is plugged into a triangulation class, the concepts for the vertex and base classes depends on the triangulation class. The most basic concepts, valid for basic and Delaunay triangulations are*TriangulationVertexBase_2*and*TriangulationFaceBase_2*.

See section 25.11 for examples of using the triangulation data structure flexibility.