Hyperbolic Voronoi diagrams made easy

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., Poincaré 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.

Cited by in corpus (9)

• An elementary introduction to information geometry
• On Approximating the Riemannian 1-Center
• Hilbert geometry of the Siegel disk: The Siegel-Klein disk model
• The hyperbolic Voronoi diagram in arbitrary dimension
• Information measures and geometry of the hyperbolic exponential families of Poincaré and hyperboloid distributions
• On information projections between multivariate elliptical and location-scale families
• Further results on the hyperbolic Voronoi diagrams
• How to pick a random integer matrix? (and other questions)
• On Voronoi diagrams and dual Delaunay complexes on the information-geometric Cauchy manifolds