This article provides a simple pictorial introduction to universal hyperbolic geometry. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. This provides a completely algebraic framework for hyperbolic geometry, valid over the rational numbers (and indeed any field not of characteristic two), and gives us many new and beautifu...

In order to obtain a global principle for modeling closed surfaces of arbitrary genus, rst hyperbolic geometry and then discrete groups of motions in planar geometries of constant curvature are studied. The representation of a closed surface as an orbifold leads to a natural parametrization of the surfaces as a subset of one of the classical geome-tries S 2 , E 2 and H 2. This well known connec...

Formulas about the side lengths, diagonal lengths or radius of the circumcircle of a cyclic polygon in Euclidean geometry, hyperbolic geometry or spherical geometry can be unified.

Abstract Based on data from the European banking stress tests of 2014, 2016 and transparency exercise 2018 we construct networks banks demonstrate that latent geometry these financial can be well-represented by negative curvature, i.e., hyperbolic geometry. Using two different embedding methods, hydra+ Mercator, this allows us to connect network structure popularity-vs-similarity model Papdopou...