Hyperbolic Geometry and Mobius Transformations

Emmalee Stevens, John Carroll University


Imagine a world in which there are infinitely many lines through a single point that are all parallel to the same line; this world is referred to as the hyperbolic plane. It starts with a brief history of the development of hyperbolic geometry and the mathematicians who contributed their explorations of the hyperbolic plane. The reader will be reminded of some of the important aspects of Euclidean geometry and some of the basic properties that will later be adjusted to develop hyperbolic geometry. The hyperbolic plane will be constructed from the pseudosphere and the properties of the Poincare Upper Half plane will be explored. The theory of Mobius transformations and their properties will be developed leading to an examination of hyperbolic isometries, which will be used to construct other models of the hyperbolic plane.