# Chapter 38
3D Triangulation Data Structure

Sylvain Pion and Monique Teillaud

The triangulation data structure is able to represent a triangulation of a topological sphere Sd of d+1, for d {-1,0,1,2,3}. (See .)

The vertex class of a 3D-triangulation data structure must define a number of types and operations. The requirements that are of geometric nature are required only when the triangulation data structure is used as a layer for the geometric triangulation classes. (See Section .)

The cell class of a triangulation data structure stores four handles to its four vertices and four handles to its four neighbors. The vertices are indexed 0, 1, 2, and 3 in a consistent order. The neighbor indexed i lies opposite to vertex i.

In degenerate dimensions, cells are used to store faces of maximal dimension: in dimension 2, each cell represents only one facet of index 3, and 3 edges (0,1), (1,2) and (2,0); in dimension 1, each cell represents one edge (0,1). (See Section .)

### Concepts

TriangulationDataStructure_3

TriangulationDataStructure_3::Cell
TriangulationDataStructure_3::Vertex

TriangulationDSCellBase_3
TriangulationDSVertexBase_3

### Classes

CGAL::Triangulation_data_structure_3<TriangulationDSVertexBase_3,TriangulationDSCellBase_3>

This class is a model for the concept of the 3D-triangulation data structure TriangulationDataStructure_3. It is templated by base classes for vertices and cells.

CGAL provides base vertex classes and base cell classes:

CGAL::Triangulation_ds_cell_base_3<>
CGAL::Triangulation_ds_vertex_base_3<>
CGAL::Triangulation_cell_base_3<TriangulationTraits_3, TriangulationDSCellBase_3>
CGAL::Triangulation_vertex_base_3<TriangulationTraits_3, TriangulationDSVertexBase_3>
CGAL::Triangulation_hierarchy_vertex_base_3<TriangulationVertexBase_3>

#### A class of tools

CGAL::Triangulation_utils_3

It defines operations on the indices of vertices and neighbors within a cell of a triangulation.

### Links to the Reference Sections

- TriangulationDataStructure_3

- TriangulationDataStructure_3::Cell

- TriangulationDataStructure_3::Vertex

- TriangulationDSCellBase_3

- TriangulationDSVertexBase_3

- Triangulation_data_structure_3<TriangulationDSVertexBase_3,TriangulationDSCellBase_3>

- Triangulation_ds_cell_base_3<>

- Triangulation_ds_vertex_base_3<>

- Triangulation_utils_3