Sylvain Pion and Monique Teillaud
A geometric triangulation has two aspects: the combinatorial structure, which gives the incidence and adjacency relations between faces, and the geometric information related to the position of vertices.
Cgal provides 3D geometric triangulations in which these two aspects are clearly separated. As described in Chapter 38, a geometric triangulation of a set of points in ℝ^{d}, d ≤ 3 is a partition of the whole space ℝ^{d} into cells having d+1 vertices. Some of them are infinite, they are obtained by linking an additional vertex at infinity to each facet of the convex hull of the points (see Section 38.1). The underlying combinatorial graph of such a triangulation without boundary of ℝ^{d} can be seen as a triangulation of the topological sphere S^{d} in ℝ^{d+1}.
This chapter deals with 3D-triangulation data structures, meant to maintain the combinatorial information for 3D-geometric triangulations. The reader interested in geometric triangulations of ℝ^{3} is advised to read Chapter 38.
In Cgal, a 3D 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 ≤ j < i ≤ 3) are said to be incident in the triangulation if f_{j} is a (sub)face of f_{i}, and two i-faces (0 ≤ i ≤ 3) are said to be adjacent if they share a commun 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 39.1).
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).
Degenerate Dimensions 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.
Thus, a 3D-triangulation data structure can store a triangulation of a topological sphere S^{d} of ℝ^{d+1}, for any d ∈ {-1,0,1,2,3}.
Let us give, for each dimension, the example corresponding to the triangulation data structure having a minimal number of vertices, i.e. a simplex. These examples are illustrated by presenting their usual geometric embedding.
The last three cases are defined uniquely:
Note that the notion of infinite vertex has no meaning for the triangulation data structure. The infinite vertex of the geometric embedding is a vertex that cannot be distinguished from the other vertices in the combinatorial triangulation.
The same cell class is used in all cases: triangular faces in 2D can be considered as degenerate cells, having only three vertices (resp. neighbors) numbered (0,1,2); edges in 1D have only two vertices (resp. neighbors) numbered 0 and 1.
The implicit representation of facets (resp. edges) still holds for degenerate (< 3) dimensions : in dimension 2, each cell has only one facet of index 3, and 3 edges (0,1), (1,2) and (2,0); in dimension 1, each cell has one edge (0,1).
Validity A 3D combinatorial triangulation is said to be locally valid iff the following is true:
(a) When a cell c has a neighbor pointer to another cell c', then reciprocally this cell c' has a neighbor pointer to c, and c and c' have three vertices in common. These cells are called adjacent.
(b) The cells have a coherent orientation: if two cells c_{1} and c_{2} are adjacent and share a facet with vertices u,v,w, then the vertices of c_{1} are numbered (v_{0}^{1} = u, v_{1}^{1} = v, v_{2}^{1} = w, v_{3}^{1}), and the vertices of c_{2} are numbered (v_{0}^{2} = v, v_{1}^{2} = u, v_{2}^{2} = w, v_{3}^{2}), up to positive permutations of (0,1,2,3). In other words, if we embed the triangulation in ℝ^{3}, then the fourth vertices v_{3}^{1} and v_{3}^{2} of c_{1} and c_{2} see the common facet in opposite orientations. See Figure 39.5.
The set σ_{4} of permutations of (0,1,2,3) has cardinality 24, and the set of positive permutations A_{4} has cardinality 12. Thus, for a given orientation, there are up to 12 different orderings of the four vertices of a cell. Note that cyclic permutations are negative and so do not preserve the orientation of a cell.
The is_valid() method provided by Triangulation_data_structure_3 checks the local validity of a given triangulation data structure.
The 3D-triangulation data structure class of Cgal, Triangulation_data_structure_3, is designed to be used as a combinatorial layer upon which a geometric layer can be built [Ket98]. This geometric layer is typically one of the 3D-triangulation classes of Cgal: Triangulation_3, Delaunay_triangulation_3 and Regular_triangulation_3. This relation is described in more details in Chapter 38, where the Section 38.4 explains other important parts of the design related to the geometry.
We focus here on the design of the triangulation data structure (TDS) itself, which the Figure 39.6 illustrates.
In order for the user to be able to add his own data in the vertices and cells, the design of the TDS is split into two layers:
The user has several ways to add his own data in the vertex and cell base classes used by the TDS. He can either:
Since adjacency relations are stored in the vertices and cells, it means that the vertex and cell base classes have to be able to store handles (an entity akin to pointers) to their neighbors in the TDS. This in turns means that the vertex and cell base classes have to know the types of these handles, which are provided by the TDS. So in a sense, the base classes are parameterized by the TDS, and the TDS is parameterized by the vertex and cell base classes ! This is a cycle which cannot be resolved easily.
The solution that we have chosen is similar to the mechanism used by the standard class std::allocator: the vertex and cell base classes are initially given a fake or dummy TDS template parameter, whose unique purpose is to provide the types that can be used by the vertex and cell base classes (such as handles). Then, inside the TDS itself, these base classes are rebound to the real TDS type, that is we obtain the same vertex and cell base classes, but parameterized with the real TDS instead of the dummy one. Rebinding is performed by a nested template class of the vertex and cell base classes (see code below), which provides a type which is the rebound vertex or cell base class^{1}.
Here is how it works, schematically:
template < class Vb, class Cb > class TDS { typedef TDS<Vb, Cb> Self; // Rebind the vertex and cell base to the actual TDS (Self). typedef typename Vb::template Rebind_TDS<Self>::Other VertexBase; typedef typename Cb::template Rebind_TDS<Self>::Other CellBase; // ... further internal machinery leads to the final public types: public: typedef ... Vertex; typedef ... Cell; typedef ... Vertex_handle; typedef ... Cell_handle; }; template < class TDS = ... > // The default is some internal type faking a TDS class Triangulation_ds_vertex_base_3 { public: template < class TDS2 > struct Rebind_TDS { typedef Triangulation_ds_vertex_base_3<TDS2> Other; }; ... };
When derivation is used for the vertex or cell base classes, which is the case at the geometric level with Triangulation_vertex_base_3, then it gets slightly more involved because its base class has to be rebound as well:
template < class GT, class Vb = Triangulation_ds_vertex_base_3<> > class Triangulation_vertex_base_3 : public Vb { public: template < class TDS2 > struct Rebind_TDS { typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2; typedef Triangulation_vertex_base_3<GT, Vb2> Other; }; ... };
File: examples/Triangulation_3/tds.cpp
#include <CGAL/Triangulation_data_structure_3.h> #include <iostream> #include <fstream> #include <cassert> #include <vector> typedef CGAL::Triangulation_data_structure_3<> Tds; typedef Tds::size_type size_type; typedef Tds::Cell_handle Cell_handle; typedef Tds::Vertex_handle Vertex_handle; int main() { Tds T; assert( T.number_of_vertices() == 0 ); assert( T.dimension() == -2 ); assert( T.is_valid() ); std::vector<Vertex_handle> PV(7); PV[0] = T.insert_increase_dimension(); assert( T.number_of_vertices() == 1 ); assert( T.dimension() == -1 ); assert( T.is_valid() ); // each of the following insertions of vertices increases the dimension for ( int i=1; i<5; i++ ) { PV[i] = T.insert_increase_dimension(PV[0]); assert( T.number_of_vertices() == (size_type) i+1 ); assert( T.dimension() == i-1 ); assert( T.is_valid() ); } assert( T.number_of_cells() == 5 ); // we now have a simplex in dimension 4 // cell incident to PV[0] Cell_handle c = PV[0]->cell(); int ind; bool check = c->has_vertex( PV[0], ind ); assert( check ); // PV[0] is the vertex of index ind in c // insertion of a new vertex in the facet opposite to PV[0] PV[5] = T.insert_in_facet(c, ind); assert( T.number_of_vertices() == 6 ); assert( T.dimension() == 3 ); assert( T.is_valid() ); // insertion of a new vertex in c PV[6] = T.insert_in_cell(c); assert( T.number_of_vertices() == 7 ); assert( T.dimension() == 3 ); assert( T.is_valid() ); std::ofstream oFileT("output_tds",std::ios::out); // writing file output_tds; oFileT << T; return 0; }
File: examples/Triangulation_3/linking_2d_and_3d.cpp
#include <CGAL/Triangulation_data_structure_2.h> #include <CGAL/Triangulation_data_structure_3.h> #include <cassert> // declare the 2D vertex base type, parametrized by some 3D TDS. template < typename T3, typename Vb = CGAL::Triangulation_ds_vertex_base_2<> > class My_vertex_2; // declare the 3D vertex base type, parametrized by some 2D TDS. template < typename T2, typename Vb = CGAL::Triangulation_ds_vertex_base_3<> > class My_vertex_3; // Then, we have to break the dependency cycle. // we need to refer to a dummy 3D TDS. typedef CGAL::Triangulation_ds_vertex_base_3<>::Triangulation_data_structure Dummy_tds_3; // the 2D TDS, initially plugging a dummy 3D TDS in the vertex type // (to break the dependency cycle). typedef CGAL::Triangulation_data_structure_2<My_vertex_2<Dummy_tds_3> > TDS_2; // the 3D TDS, here we can plug the 2D TDS directly. typedef CGAL::Triangulation_data_structure_3<My_vertex_3<TDS_2> > TDS_3; template < typename T3, typename Vb > class My_vertex_2 : public Vb { public: typedef typename Vb::Face_handle Face_handle; template <typename TDS2> struct Rebind_TDS { typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2; // we also have to break the cycle here by hardcoding TDS_3 instead of T3. typedef My_vertex_2<TDS_3, Vb2> Other; }; My_vertex_2() {} My_vertex_2(Face_handle f) : Vb(f) {} // we store a vertex handle of the 3D TDS. typename T3::Vertex_handle v3; }; template < typename T2, typename Vb > class My_vertex_3 : public Vb { public: typedef typename Vb::Cell_handle Cell_handle; template <typename TDS2> struct Rebind_TDS { typedef typename Vb::template Rebind_TDS<TDS2>::Other Vb2; typedef My_vertex_3<T2, Vb2> Other; }; My_vertex_3() {} My_vertex_3(Cell_handle c) : Vb(c) {} // we store a vertex handle of the 2D TDS. typename T2::Vertex_handle v2; }; int main() { TDS_2 t2; TDS_3 t3; TDS_2::Vertex_handle v2 = t2.insert_dim_up(); TDS_3::Vertex_handle v3 = t3.insert_increase_dimension(); v2->v3 = v3; v3->v2 = v2; assert(t2.is_valid()); assert(t3.is_valid()); return 0; }
Monique Teillaud introduced the triangulation of the topological sphere S^{d} in ℝ^{d+1} to manage the underlying graph of geometric triangulations and handle degenerate dimensions [Tei99].
Sylvain Pion improved the software in several ways, in particular regarding the memory management.
^{1} | It is logically equivalent to a mechanism that does not exist yet in the C++ language: template typedef or template aliasing |