Geometric Transformations and Their Applications in Non-Euclidean Spaces

Abstract

This paper investigates the nature of geometric transformations in non-Euclidean spaces, focusing on hyperbolic and spherical geometries. These transformations—such as translations, rotations, reflections, and conformal mappings—differ significantly from their Euclidean counterparts due to the curvature inherent in non-Euclidean spaces. We explore the theoretical foundations of these transformations and their practical applications in fields such as general relativity, where space-time is modeled as a curved manifold, as well as in computer graphics, network visualization, and navigation. Despite the challenges associated with visualization and complex calculations, non-Euclidean transformations provide essential tools for understanding and modeling curved spaces in both two and three dimensions. The paper also addresses current limitations and suggests future directions for expanding the use of these transformations in advanced scientific and technological applications.