Hyperbolic Geometry and Its Metric in the Upper Half-Plane and Poincare Disk Models

Abstract

Hyperbolic geometry is a non-Euclidean geometry defined by its violation of Euclid’s parallel postulate, which states that for any point not on a given line, exactly one line can be drawn through that point parallel to the given line. In contrast, hyperbolic geometry permits infinitely many lines through a point that do not intersect a given line. This geometry is visualized on surfaces known as the hyperbolic plane, represented through models such as the Poincare Disk Model, the Upper Half-Plane Model, and the Klein Model, each offering a distinct representation of hyperbolic space. The hyperbolic metric, which defines distances between points, differs significantly from the Euclidean distance. Notably, triangles in hyperbolic geometry have sum of interior angles less than 180°, and the concept of parallel lines differ from Euclidean principles. These unique characteristics make hyperbolic geometry essential for understanding spaces with distinct properties. Distances in hyperbolic models are calculated using their respective metrics, and the analytical representation of hyperbolic lines has been established through examples in Poincare Disk Model, the Upper Half-Plane Model.