Geometric Insights into Non-Euclidean Spaces: A Study in Hyperbolic Geometry

Abstract

This work evaluates the core principles along with models and usages of hyperbolic geometry which establishes itself as a main category of non-Euclidean geometry because it lacks Euclid's parallel axiom. The analysis of lines angles and triangles in hyperbolic spaces takes place through the study of the Poincaré disk model and the hyperboloid model. This paper examines the separation between Euclidean structures and hyperbolic structures while showing their distinctive properties which affect mathematical and physical applications.