Tags : hyperbolic geometry

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Jan 27, 2011

Videos of hyperbolic Voronoi diagrams

I have posted a few videos of interactive hyperbolic Voronoi diagrams. These were rendered brute force using GPU: see here for details.

A more elegant computational geometry proves that those diagrams are affine in the Klein ball (arbitrary dimension, not only 2D) and can thus be computed from equivalent power diagrams. Here are the details.
Frank.

Sep 23, 2010

Hyperbolic centroid and other centers

Post @ 23:37:17 | hyperbolic geometry

We are all familiar with some Euclidean center points like the centroid, the Fermat-Weber median point and the circumcenter. Given 3 points defining a triangle, we can express those centers using the barycentric coordinates. What is an appropriate center point ? Well, that depends (once again, same question/answer). In fact, there is a listing of other 3000+ Euclidean center points defined by their barycentric coordinates.

Now, let us consider hyperbolic geometry. Even the simplest point can be difficult to compute: Define the centroid as the minimizer of the average squared (Riemannian) distance. In hyperbolic geometry, there is unfortunately no closed form solutions for it. (In Euclidean geometry, it is merely the center of gravity, the center of mass). So, one way to bypass this -:) is to give another definition for centroids: Like the model centroid, or relativistic centroid. Then, we have a simple way to compute it, and write it in closed-form.

There is a recent book on that topic:

I also recommend : A concept of the mass center of a system of material points in the constant curvature spaces, Galperin 1993.

So what is the link with information geometry. Well, consider normal distributions with the Fisher-Rao geometry: you get hyperbolic geometry.

Frank.