Marcus Strandman: Möbiustransformationer
Degree Project for teacher
Tid: To 2025-02-06 kl 15.30 - 16.30
Plats: Cramérrummet
Respondent: Marcus Strandman
Handledare: Alan Sola
Abstract.
This paper studies Möbius transformations, which are a type of conformal mapping within the field of complex analysis. Elementary concepts such as complex numbers, analytic functions, and matrix representations are introduced to present the theory and applications of Möbius transformations.
First, important theoretical concepts, such as the Riemann sphere, are introduced, along with their connection to the theory of Möbius transformations. Next, the definition and properties of Möbius transformations are explored, including the inverse, composition, and geometrical interpretation.
The fixed points of Möbius transformations, and their classification, are discussed. The discriminant \(D = \operatorname{tr}(M)^2 − 4 \det(M )\) of a quadratic equation described in terms of elementary operations from linear algebra and our definition of the function \(\tau(M)\) proves to be important tools in the process of classification.
The transformations can be divided into four categories: parabolic, elliptic, loxodromic, and hyperbolic. Finally, the Riemann mapping theorem is introduced, and the connection between Möbius transformations and the Riemann mapping theorem is briefly discussed. Examples of applications in mathematics, physics, computer science, and cartography are provided, highlighting the transformation’s ability to locally preserve angles. This property proves to be a powerful tool for solving complex geometrical problems.
By combining rigorous theory with concrete examples, this paper aims to provide an understanding of Möbius transformations’ mathematical structure and how they are used.