How Similar Are Quasi-, Regular, and Delaunay Triangulations in ℝ3?

Voronoi diagrams and quasi-triangulations are powerful for solving spatial problems among spherical particles with different radii. However, a quasi-triangulation can be a non-simplicial complex due to anomaly conditions. While quasi-triangulation is straightforward to use when it is a simplicial complex, it may not seem so if it is not. In this paper, we report the experimental statistics of showing the phenomena related with two fundamental issues: i) How frequently anomalies occur in the quasi-triangulation of the arrangement of spherical atoms in ℝ3 and ii) how much similar or dissimilar the three related structures (i.e., the quasi-triangulation, the regular triangulation, and the Delaunay triangulation of an atomic arrangements) are. The observations from the experiments are as follows: i) Anomalies occur extremely rarely in molecular structures and occur very rarely even in random sphere sets, and ii) the three dual structures of a given set of spheres are not similar.

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