Further results on the hyperbolic Voronoi diagrams

TL;DR

This paper explores the computation of hyperbolic Voronoi diagrams using various models, extending Euclidean techniques to hyperbolic geometry and demonstrating their equivalences and construction methods.

Contribution

It introduces methods to compute hyperbolic Voronoi diagrams via Klein and hyperboloid models, connecting them to Euclidean power diagrams and providing new geometric insights.

Findings

Hyperbolic Voronoi diagrams can be derived from concave potential functions.

Construction of hyperbolic diagrams as clipped power diagrams in Euclidean space.

Reduction of hyperboloid model diagrams to Klein-type models via projections.

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.