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Triangulation of simple arbitrarily shaped polyhedra by cutting off one vertex at a time

We describe a heuristic method of triangulating arbitrarily shaped polyhedra without the addition of Steiner points. The polyhedra are simple, with each vertex connected to at least 3 other vertices (ie, coplanarity and colinearity are not considered). They may, however, be convex or concave and consist of dozens or even hundreds of facets. This makes the treatment universal enough to well meet the requirements of models used to simulate fractured rock masses. Certain concepts are defined in the work, eg, adjacent vertices, polygon of adjacent vertices, and closed cone of a vertex. A polygon of adjacent vertices of an apex can be subdivided into a set of nonoverlapping triangles without adding any vertices. These triangles, together with the apex, form tetrahedra whose union is the closed cone of the apex. The polyhedron is thus the union of the closed cones. Subsequently, we triangulate the polyhedron by gradually removing the closed cones of its vertices. The number of vertices of the polyhedron decreases by one each time a closed cone is removed. A block with n vertices can produce no more than n−3 tetrahedra. We present the analysis procedure and discuss the core issues of the method proposed.