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arXiv · 1210.8234

The hyperbolic Voronoi diagram in arbitrary dimension

Frank Nielsen·Richard Nock
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Abstract

We show that in the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and amounts to clip a corresponding power diagram, requiring however algebraic arithmetic. By considering the lesser-known Beltrami hemisphere model of hyperbolic geometry, we overcome the arithmetic limitations of Klein construction. Finally, we characterize the bisectors and geodesics in the other Poincar\' e upper half-space, the Poincar\'e ball, and the Lorentz hyperboloid models, and discusses on degenerate cases for which the dual hyperbolic Delaunay complex is not a triangulation.

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Frank Nielsen, Richard Nock. 2012-10-31. The hyperbolic Voronoi diagram in arbitrary dimension. https://arxiv.org/abs/1210.8234

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