2013 13th International Conference on Computational Science and Its Applications (2010)

Fukuoka, Japan

Mar. 23, 2010 to Mar. 26, 2010

ISBN: 978-0-7695-3999-7

pp: 74-80

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/ICCSA.2010.37

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., Poincare 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.

INDEX TERMS

Voronoi diagrams, conformal/non-conformal geometries, Klein disk, Poincare disk, Poincare upper-plane

CITATION

Frank Nielsen,
Richard Nock,
"Hyperbolic Voronoi Diagrams Made Easy",

*2013 13th International Conference on Computational Science and Its Applications*, vol. 00, no. , pp. 74-80, 2010, doi:10.1109/ICCSA.2010.37