@article{2009-HyperbolicVoronoi-0903.3287
, author={Frank Nielsen and Richard Nock}
, title={Hyperbolic Voronoi diagrams made easy}
, journal={Computing Research Repository (CoRR)}
, month={March}
, year={2009}
, volume={abs/0903.3287}
, 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.
}
}