Revisiting hyperbolic Voronoi diagrams in two and higher dimensions from theoretical, applied and generalized viewpoints

Toshihiro Tanuma, Hiroshi Imai, Sonoko Moriyama

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Title of host publicationTransactions on Computational Science XIV - Special Issue on Voronoi Diagrams and Delaunay Triangulation
Pages1-30
Number of pages30
DOIs
Publication statusPublished - 2011 Nov 25
Event7th International Symposium on Voronoi Diagrams - Quebec City, QC, Canada
Duration: 2010 Jun 282010 Jun 30

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6970 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other7th International Symposium on Voronoi Diagrams
CountryCanada
CityQuebec City, QC
Period10/6/2810/6/30

Keywords

  • Voronoi diagrams
  • divergences
  • geodesic segments
  • greedy embedding
  • hyperbolic geometry
  • spheres

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Tanuma, T., Imai, H., & Moriyama, S. (2011). Revisiting hyperbolic Voronoi diagrams in two and higher dimensions from theoretical, applied and generalized viewpoints. In Transactions on Computational Science XIV - Special Issue on Voronoi Diagrams and Delaunay Triangulation (pp. 1-30). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6970 LNCS). https://doi.org/10.1007/978-3-642-25249-5_1