Revisiting Hyperbolic Voronoi Diagrams in Two and Higher Dimensions from Theoretical, Applied and Generalized Viewpoints

This paper revisits hyperbolic Voronoi diagrams, which have been investigated since mid 1990’s by Onishi et al., from three standpoints, background theory, new applications, and geometric extensions.

First, we review two ideas to compute hyperbolic Voronoi diagrams of points. One of them is Onishi’s method to compute a hyperbolic Voronoi diagram from a Euclidean Voronoi diagram. The other one is a linearization of hyperbolic Voronoi diagrams. We show that a hyperbolic Voronoi diagram of points in the upper half-space model becomes an affine diagram, which is part of a power diagram in the Euclidean space. This gives another proof of a result obtained by Nielsen and Nock on the hyperbolic Klein model. Furthermore, we consider this linearization from the view point of information geometry. In the parametric space of normal distributions, the hyperbolic Voronoi diagram is induced by the Fisher metric while the divergence diagram is given by the Kullback-Leibler divergence on a dually flat structure. We show that the linearization of hyperbolic Voronoi diagrams is similar to one of two flat coordinates in the dually flat space, and our result is interesting in view of the linearization having information-geometric interpretations.

Secondly, from the viewpoint of new applications, we discuss the relation between the hyperbolic Voronoi diagram and the greedy embedding in the hyperbolic plane. Kleinberg proved that in the hyperbolic plane the greedy routing is always possible. We point out that results of previous studies about the greedy embedding use a property that any tree is realized as a hyperbolic Delaunay graph easily.

Finally, we generalize hyperbolic Voronoi diagrams. We consider hyperbolic Voronoi diagrams of spheres by two measures and hyperbolic Voronoi diagrams of geodesic segments, and propose algorithms for them, whose ideas are similar to those of computing hyperbolic Voronoi diagrams of points.