Abstract Voronoi diagram in 3-space

摘要

We propose a class of abstract Voronoi diagrams in 3-space that generalizes the planar abstract Voronoi diagram of Klein. Our class of abstract Voronoi diagrams includes the Voronoi diagram of point sites in general position under any convex distance function. To characterize the abstract Voronoi diagram in 3-space, we introduce the notion of intersection characteristic. We determine the intersection characteristic for the simplex, the L∞, and the Lp distance function. We find that the intersection characteristic in case of the simplex distance function is similar to that of the usual Euclidean distance. This enables us to give a randomized incremental algorithm for computing the Voronoi diagram under the simplex distance function in quadratic expected time.