Delaunay Tetrahedralizations: Honor Degenerated Cases

The definition of a Delaunay tetrahedralization (DT) of a set S of points is well known: a DT is a tetrahedralization of S in which every simplex (tetrahedron, triangle, or edge) is Delaunay. A simplex is Delaunay if all of its vertices can be connected by a circumsphere that encloses no other vertex. An important remark made in virtually all papers on this topic is that “although any number of vertices is permitted on the sphere”, a Delaunay tetrahedralization is unique “only if the points are in general position, which is the absence of degeneracy (i.e., five or more vertices possible on the circumsphere)”. For applications in which the DT must be unique and invariant invariable under either geometrical rotation or the numbering of the points of set S, degenerated cases are expected to be resolved or at least indicated. This research, however, will direct to a method in which degenerated cases are not solved by a geometrically distorted set of input data using Steiner Points, nor is just one of the possible ‘unique’ DTs generated. To the contrary, an extra type of ‘non-geometrical’, flat, zero-volume ‘tetrahedra’ is purposely introduced in order to indicate the degenerated cases within the applied data structure of the tetrahedralization. The four vertices of these flat tetrahedra lie on the same circumsphere as the vertices of their neighboring tetrahedra. Once the set S of points is tetrahedralized, these flat tetrahedra provide a tool with which to discover other ‘unique’ DTs of the set S of points. From this set of unique DTs, the one having a global optimum can be selected for a given purpose, e.g. to support datadependent applications. Another benefit in creating flat tetrahedra is the ability to indicate nearly degenerated cases, eliminating the need for exact arithmetic. 1. DELAUNAY DIAGRAMS