# Triangulation

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A triangulation of a polyhedron, or rectilinear triangulation, is a representation of the polyhedron (cf. Polyhedron, abstract) as the space of a geometric simplicial complex , that is, a decomposition of it into closed simplices such that any two simplices either do not intersect or intersect along a common face. Rectilinear triangulations of polyhedra serve as the main tool for studying them. Any polyhedron has a triangulation and any two triangulations of it have a common subdivision.
The closed star of a simplex in a triangulation is the union of the simplices of containing . There is a representation of the closed star of a simplex as the union (or join, cf. Union of sets) of and its link: . In particular, the star of a vertex is a cone over its link. If a simplex is represented as the join of two of its faces and , then . The link of a simplex does not depend on : If is a simplex in rectilinear triangulations , of the same polyhedron, then the polyhedra and are PL-homeomorphic. The open star of a simplex is defined as the union of the interiors of those simplices of containing as a face. The open stars of the vertices of a triangulation of a polyhedron form an open covering of . The nerve of this covering (cf. Nerve of a family of sets) is simplicially isomorphic to the triangulation. Two triangulations and of polyhedra and are combinatorially equivalent if certain subdivisions of them are simplicially isomorphic. In order that two triangulations and be combinatorially equivalent it is necessary and sufficient that and be PL-homeomorphic. A triangulation of a manifold is said to be combinatorial if the star of any of its vertices is combinatorially equivalent to a simplex. In this case the star of any simplex of the triangulation is also combinatorially equivalent to a simplex.
If is a closed subpolyhedron of a polyhedron , then any triangulation of can be extended to some triangulation of . In this case one says that the pair of geometric simplicial complexes triangulates the pair . A triangulation of the direct product of two simplices , can be constructed as follows. The vertices of the triangulation are the points , , where are the vertices of and are the vertices of . The vertices , where , span a -dimensional simplex if and only if none of these coincide and . A triangulation of the direct product of two simplicial complexes with ordered vertices can be carried out in the same way.
A triangulation of a topological space, or curvilinear triangulation, is a pair , where is a geometric simplicial complex and is a homeomorphism. Two triangulations and of a space coincide if is a simplicial isomorphism. If is a simplex of a complex and is a triangulation of , then the space endowed with the homeomorphism is called a topological simplex. The star and the link of a topological simplex of a triangulated topological space are defined in the same way as in the case of rectilinear triangulations. If a point is a vertex of triangulations and of , then its links in these triangulations are homotopy equivalent.