Hyperbolic Voronoi diagrams made easy

Frank Nielsen, Richard Nock

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.

Illustrating figures

@inproceedings{HVD-2010,
 author = {Nielsen, Frank and Nock, Richard},
 title = {Hyperbolic {V}oronoi Diagrams Made Easy},
 booktitle = {Proceedings of the International Conference on Computational Science and Its Applications},
 series = {ICCSA},
 year = {2010},
 isbn = {978-0-7695-3999-7},
 pages = {74--80},
 doi = {10.1109/ICCSA.2010.37},
 publisher = {IEEE Computer Society},
 address = {Washington, DC, USA}
}