Further results on the hyperbolic Voronoi diagrams

In Euclidean geometry, it is well-known that the k-order Voronoi diagram in R can be computed from the vertical projection of the k-level of an arrangement of hyperplanes tangent to a convex potential function in R: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a concave potential function: the northern hemisphere. Furthermore, we also show how to build the hyperbolic k-order diagrams as equivalent clipped power diagrams in R. We investigate the hyperbolic Voronoi diagram in the hyperboloid model and show how it reduces to a Kleintype model using central projections. Keywords-Voronoi diagram; hyperbolic geometry; clipping.