Mahurin Honors College Capstone Experience/Thesis Projects
Department
Mathematics
Document Type
Thesis
Abstract
In this thesis, we discuss several properties of Möbius transformations and hyperbolic geometry, a type of non-Euclidean geometry, in the upper half-plane using tools of complex analysis. We begin with preliminaries for our work, comprising the stereographic projection, the representation of circles and lines in the complex plane, conformal maps, and a result on cross-products, which we include for further development. We proceed to Möbius transformations and discuss their properties, cross-ratios, and various mappings. We additionally provide useful calculations. Lastly, we conclude with the hyperbolic metric in the upper half-plane and explore hyperbolic distance, including its invariance under Möbius transformations. We use the Euler-Lagrange equation from the calculus of variations to prove that geodesics are either vertical lines or are parts of circles centered on the 𝑥-axis. Futhermore, we construct a specific mapping (Möbius transformation) and provide important results, which we use to prove the analog of Pythagoras’ Theorem in hyperbolic space.
Advisor(s) or Committee Chair
Tilak Bhattacharya, Ph.D.
Disciplines
Geometry and Topology | Other Mathematics
Recommended Citation
Bunch, Emma, "Möbius Transformations and Hyperbolic Geometry in the Upper Half-Plane" (2025). Mahurin Honors College Capstone Experience/Thesis Projects. Paper 1068.
https://digitalcommons.wku.edu/stu_hon_theses/1068