Hyperbolic Voronoi diagrams made easy

TL;DR

This paper introduces a straightforward method to compute hyperbolic Voronoi diagrams using affine diagrams and power diagrams, simplifying the process in various hyperbolic models.

Contribution

The authors demonstrate that hyperbolic Voronoi diagrams can be computed as affine diagrams via power diagrams, providing a unified framework for different hyperbolic representations.

Findings

Hyperbolic bisectors are hyperplanes in Klein's model.

Voronoi diagrams can be obtained through clipped power diagrams.

Applications include nearest neighbor search and smallest enclosing ball computations.

Abstract

We present a simple framework to compute hyperbolic Voronoi diagrams of finite point sets as affine diagrams. We prove that bisectors in Klein's non-conformal disk model are hyperplanes that can be interpreted as power bisectors of Euclidean balls. Therefore our method simply consists in computing an equivalent clipped power diagram followed by a mapping transformation depending on the selected representation of the hyperbolic space (e.g., Poincar\'e conformal disk or upper-plane representations). We discuss on extensions of this approach to weighted and $k$-order diagrams, and describe their dual triangulations. Finally, we consider two useful primitives on the hyperbolic Voronoi diagrams for designing tailored user interfaces of an image catalog browsing application in the hyperbolic disk: (1) finding nearest neighbors, and (2) computing smallest enclosing balls.