From the Hyperboloid to the Disk: Models, Geodesics, and Isometries of the Hyperbolic Plane

Abstract

Hyperbolic geometry is a non-Euclidean geometry where geodesics play the role of straight lines. It is characterised by a constant negative curvature. The idea was conceived when mathematicians tried to understand the parallel postulate. This expository paper develops the foundations of the hyperbolic plane using the classical models, such as the hyperboloid, Beltrami–Klein, Poincaré disk, and upper half-plane models. It examines Gaussian curvature, hyperbolic distance, geodesics, metric transformations, Möbius transformations, and isometry groups, and explains how the different models represent the same geometry. The paper also discusses hyperbolic tessellations and the applications of Hyperbolic Geometry in mathematics, physics, art, and data representation.

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