Hyperbolic Delaunay Complexes and Voronoi Diagrams Made Practical

Mikhail Bogdanov 1 Olivier Devillers 1 Monique Teillaud 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : We study Delaunay complexes and Voronoi diagrams in the Poincaré ball, a conformal model of the hyperbolic space, in any dimension. We elaborate on our earlier work on the space of spheres [CCCG'92], giving a detailed description of algorithms. We also study algebraic and arithmetic issues, observing that only rational computations are needed. All proofs are based on geometric reasoning; they do not resort to any use of the analytic formula of the hyperbolic distance. This allows for an exact and efficient implementation in 2D. All degenerate cases are handled. The implementation will be submitted to the CGAL editorial board for future integration into the CGAL library.
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Mikhail Bogdanov, Olivier Devillers, Monique Teillaud. Hyperbolic Delaunay Complexes and Voronoi Diagrams Made Practical. Journal of Computational Geometry, Carleton University, Computational Geometry Laboratory, 2014, 5 (1), pp.56-85. ⟨http://jocg.org/v5n1p4⟩. ⟨10.20382/jocg.v5i1a4⟩. ⟨hal-00961390⟩

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