Centroidal Voronoi tessellation in universal covering space of manifold surfaces
Computer Aided Geometric Design
Transactions on Computational Science XIV
On approximating the Riemannian 1-center
Computational Geometry: Theory and Applications
Hyperbolic delaunay complexes and voronoi diagrams made practical
Proceedings of the twenty-ninth annual symposium on Computational geometry
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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 consistsin computinganequivalent 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). Wediscuss 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) computingsmallest enclosing balls.