The triangulation data structure is able to represent a triangulation of a topological sphere $S^d$ of $\mathbb{R}^{d+1}$, for $d \in \{-1,0,1,2,3\}$. (See Representation.)

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 Software Design.)

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 Representation.)

Classified Reference Pages

Concepts

• TriangulationDataStructure_3
• TriangulationDataStructure_3::Cell
• TriangulationDataStructure_3::Vertex
• TriangulationDSCellBase_3
• TriangulationDSVertexBase_3

Classes

• CGAL::Triangulation_data_structure_3<TriangulationDSVertexBase_3,TriangulationDSCellBase_3,Vertex_container_strategy,Cell_container_strategy,Concurrency_tag> 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<TDS>
• CGAL::Triangulation_ds_vertex_base_3<TDS>
• CGAL::Triangulation_cell_base_3<TriangulationTraits_3, TriangulationDSCellBase_3>
• CGAL::Triangulation_vertex_base_3<TriangulationTraits_3, TriangulationDSVertexBase_3>
• CGAL::Triangulation_cell_base_with_circumcenter_3<DelaunayTriangulationTraits_3, TriangulationCellBase_3>

Helper Classes

• CGAL::Triangulation_utils_3 defines operations on the indices of vertices and neighbors within a cell of a triangulation.