Visualizing hyperbolic Voronoi diagrams

ACM Symp. on Computational Geometry (SOCG), video track
[32 points] [300 points] [Isometries] [k-order HVDs]
Klein Poincaré

Illustrating figures

(Implemented in Java with export in SVG, PDF and PNG)

32 points

Klein affine diagram (non-empty cells, non-conformal):
HVD.K.2014-2-10-11-29-34.pdf
Equivalent power diagram (observe some cells may be empty):
HVD.PD2014-2-10-11-29-34.pdf
Conformal Poincare disk:
HVD.P.2014-2-10-11-29-34.pdf
Overlaying the conformal Poincare disk with the non-conformal Beltrami-Klein disk (observe that bisectors match on the boundary circle):
HVD.KP.2014-2-10-11-29-34.pdf
Hyperbolic Voronoi diagram on the conformal upper plane:
HVD.U.2014-2-10-11-29-34.pdf

Isometries in hyperbolic space

Navigate in hyperbolic geometry space (using isometries, Moebius transformations)
KP-beforeisometry.pdf
KP-afterisometry.pdf

300 points

Klein affine diagram (non-empty cells, non-conformal):
HVD.K.2014-2-7-14-6-39.pdf
Equivalent power diagram (observe some cells may be empty):
HVD.PD2014-2-7-14-6-39.pdf
Conformal Poincare disk:
HVD.P.2014-2-7-14-6-39.pdf
Overlaying the conformal Poincare disk with the non-conformal Beltrami-Klein disk (observe that bisectors match on the boundary circle):
HVD.KP.2014-2-7-14-6-39.pdf
Hyperbolic Voronoi diagram on the conformal upper plane:
HVD.U.2014-2-7-14-6-39.pdf

k-Order Hyperbolic Voronoi diagrams

The k-order Voronoi diagram of a set of sites in the Klein hyperbolic disk is affine. (It can be computed from an equivalent power diagram.) The k-order Poincare hyperbolic Voronoi diagrams are obtained by a rescaling conformation around the disk origin. Below is an example for n=16 points.
k Klein disk (non-conformal) Poincare disk (conformal)
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