Computational Information Geometry Wonderland

Geometries are abstract by essence but we visualize them using embedding into the good old Euclidean 2D/3D spaces: our sheet of paper, or 3D browser. So consider hyperbolic geometry, it has many different realizations: conformal or not. In a conformal representation such as Poincare upper-space or ball, angles are preserved. That is, angles measured in the Euclidean geometry coincide with angles in the hyperbolic geometry. Conformal representations are therefore often wished for mapping because it tends to minimize distortions locally.

The Voronoi diagram on hyperbolic geometry has thus been studied in conformal Poincare upper-plane representation. However, geodesics are visualized by arcs of circles and it makes computation more difficult if not tricky. Another problem is that it requires more numerical precision to carry out predicate evaluations.

Now consider the Klein ball realization. In this non-conformal representation, geodesics are straight line segment (but the mid Euclidean point is not the mid hyperbolic point). Bisectors are also hyperplanes and the diagram is therefore affine, and can be computed from an equivalent power diagram. So hyperbolic Voronoi diagrams are handy and do not require more specific implementation than a weighted Voronoi diagram.

The details are explained in the following report, illustrated with an application on image browsing
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