Numerical Aspects of Hyperbolic Geometry

Hyperbolic geometry has recently found applications in social networks, machine learning and computational biology. With the increasing popularity, questions about the best representations of hyperbolic spaces arise, as each representation comes with some numerical instability. This paper compares various 2D and 3D hyperbolic geometry representations. To this end, we conduct an extensive simulational scheme based on six tests of numerical precision errors. Our comparisons include the most popular models and less-known mixed and reduced representations. According to our results, polar representation wins, although the halfplane invariant is also very successful. We complete the comparison with a brief discussion of the non-numerical advantages of various representations.