CGAL 5.3  3D Triangulations

A threedimensional triangulation is a threedimensional simplicial complex, pure connected and without singularities [4]. Its cells (3
faces) are such that two cells either do not intersect or share a common facet (2
face), edge (1
face) or vertex (0
face).
The basic 3Dtriangulation class of CGAL is primarily designed to represent the triangulations of a set of points \( A \) in \( \mathbb{R}^3 \). It can be viewed as a partition of the convex hull of \( A \) into tetrahedra whose vertices are the points of \( A \). Together with the unbounded cell having the convex hull boundary as its frontier, the triangulation forms a partition of \( \mathbb{R}^3 \).
In order to deal only with tetrahedra, which is convenient for many applications, the unbounded cell can be subdivided into tetrahedra by considering that each convex hull facet is incident to an infinite cell having as fourth vertex an auxiliary vertex called the infinite vertex. In that way, each facet is incident to exactly two cells and special cases at the boundary of the convex hull are simple to deal with.
A triangulation is a collection of vertices and cells that are linked together through incidence and adjacency relations. Each cell gives access to its four incident vertices and to its four adjacent cells. Each vertex gives access to one of its incident cells.
The four vertices of a cell are indexed with 0, 1, 2 and 3 in positive orientation, the positive orientation being defined by the orientation of the underlying Euclidean space \( \mathbb{R}^3 \). 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 45.1.
TriangulationTraits_3
DelaunayTriangulationTraits_3
RegularTriangulationTraits_3
TriangulationVertexBase_3
TriangulationVertexBaseWithInfo_3
TriangulationCellBase_3
TriangulationCellBaseWithInfo_3
DelaunayTriangulationCellBase_3
RegularTriangulationVertexBase_3
RegularTriangulationCellBase_3
RegularTriangulationCellBaseWithWeightedCircumcenter_3
TriangulationDataStructure_3
WeightedPoint
CGAL::Triangulation_3<TriangulationTraits_3,TriangulationDataStructure_3,SurjectiveLockDataStructure>
CGAL::Delaunay_triangulation_3<DelaunayTriangulationTraits_3,TriangulationDataStructure_3,LocationPolicy,SurjectiveLockDataStructure>
CGAL::Regular_triangulation_3<RegularTriangulationTraits_3,TriangulationDataStructure_3,SurjectiveLockDataStructure>
CGAL::Triangulation_vertex_base_3<TriangulationTraits_3, TriangulationDSVertexBase_3>
CGAL::Triangulation_vertex_base_with_info_3<Info, TriangulationTraits_3, TriangulationVertexBase_3>
CGAL::Triangulation_cell_base_3<TriangulationTraits_3, TriangulationDSCellBase_3>
CGAL::Triangulation_cell_base_with_info_3<Info, TriangulationTraits_3, TriangulationCellBase_3>
CGAL::Delaunay_triangulation_cell_base_3<DelaunayTriangulationTraits_3,Cb>
CGAL::Delaunay_triangulation_cell_base_with_circumcenter_3<DelaunayTriangulationTraits_3,Cb>
CGAL::Regular_triangulation_vertex_base_3<RegularTriangulationTraits_3,Vb>
CGAL::Regular_triangulation_cell_base_3<RegularTriangulationTraits_3,Cb>
CGAL::Regular_triangulation_cell_base_with_weighted_circumcenter_3<RegularTriangulationTraits_3,Cb>
CGAL::Triangulation_simplex_3<Triangulation_3>
CGAL::Regular_triangulation_euclidean_traits_3<K,Weight>
CGAL::Robust_weighted_circumcenter_filtered_traits_3<K>
Modules  
Concepts  
Triangulation Classes  
Traits Classes  
Vertex and Cell Classes  
Draw a Triangulation 3  
I/O for a Triangulation 3  
Functions  
template<class Triangulation , class TriangleMesh >  
boost::graph_trait< FG >::vertex_descriptor  CGAL::link_to_face_graph (const Triangulation &t, typename Triangulation::Vertex_handle vh, TriangleMesh &tm, bool no_infinite_faces=true) 
fills the face graph tm with the link of triangulation vertex vh . More...  
boost::graph_trait<FG>::vertex_descriptor CGAL::link_to_face_graph  (  const Triangulation &  t, 
typename Triangulation::Vertex_handle  vh,  
TriangleMesh &  tm,  
bool  no_infinite_faces = true 

) 
#include <CGAL/link_to_face_graph.h>
fills the face graph tm
with the link of triangulation vertex vh
.
T.dimension()
==3.Triangulation  must be a CGAL 3D triangulation. 
TriangleMesh  must be a model of the concept MutableFaceGraph . 
t  the 3D triangulation 
vh  the vertex handle of the vertex 
tm  the triangle mesh 
no_infinite_faces  If vh is on the convex hull of the triangulation, no_infinite_faces == true generates a triangle mesh with a border. Otherwise, this parameter is ignored. 
tm
corresponding to the infinite vertex of t
, if vh
is on the convex hull of the triangulation, and if no_infinite_faces == false
. Otherwise, an arbitrary vertex descriptor of the triangle mesh tm
.convex_hull_3_to_face_graph()