Further results on the hyperbolic Voronoi diagrams

Abstract

In Euclidean geometry, it is well-known that the $k$-order Voronoi diagram in ${\mathbb{R}}^d$ can be computed from the vertical projection of the $k$-level of an arrangement of hyperplanes tangent to a convex potential function in ${\mathbb{R}}^{d+1}$: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a {\em concave} potential function: the northern hemisphere. Furthermore, we also show how to build the hyperbolic $k$-order diagrams as equivalent clipped power diagrams in ${\mathbb{R}}^d$. We investigate the hyperbolic Voronoi diagram in the hyperboloid model and show how it reduces to a Klein-type model using central projections.

6 pages, 2 figures (ISVD 2014)

References in corpus (2)

• Hyperbolic Voronoi diagrams made easy
• The hyperbolic Voronoi diagram in arbitrary dimension