MÖBIUS TRANSFORMATIONS AND THEIR APPLICATIONS: A COMPREHENSIVE REVIEW
Keywords:
Möbius transformation, linear fractional transformation, conformal mapping, complex analysis, cross-ratio, extended complex plane, higher-dimensional geometry, biholomorphic function.Abstract
Möbius transformations constitute a fundamental class of conformal mappings in complex analysis, with significant applications across pure and applied mathematics. Also known as linear fractional transformations, they map the extended complex plane onto itself while preserving angles and the general structure of circles and lines. This review paper presents a comprehensive examination of Möbius transformations, including their algebraic formulation, geometric interpretation, and key properties such as conformality, group structure, and fixed-point behaviour. We further survey diverse applications spanning artificial intelligence, higher-dimensional geometry, cryptography, molecular chemistry, complex dynamics, computer graphics, and machine learning. The unique ability of Möbius transformations to preserve cross-ratios while maintaining analytical tractability underscores their continued relevance as versatile tools in modern mathematical analysis and its interdisciplinary applications.
