Three-dimensional beta shapes

Abstract

The Voronoi diagram of a point set has been extensively used in various disciplines ever since it was first proposed. Its application realms have been even further extended to estimate the shape of point clouds when Edelsbrunner and Mucke introduced the concept of alpha-shape based on the Delaunay triangulation of a point set. In this paper, we present the theory of beta-shape for a set of three-dimensional spheres as the generalization of the well-known alpha-shape for a set of points. The proposed beta-shape fully accounts for the size differences among spheres and therefore it is more appropriate for the efficient and correct solution for applications in biological systems such as proteins. Once the Voronoi diagram of spheres is given, the corresponding beta-shape can be efficiently constructed and various geometric computations on the sphere complex can be efficiently and correctly performed. It turns out that many important problems in biological systems such as proteins can be easily solved via the Voronoi diagram of atoms in proteins and beta-shapes transformed from the Voronoi diagram.