On the Poinacré ball model of hyperbolic geometry

This thesis consists of five chapters. The first chapter is devoted to the introduction section and provides a general knowledge of literature. In the second chapter, some required concepts are mentioned. In the third chapter, non-existence of periodic lines in the Möbius gyrovector space (Poinacré ball model) is proved and it is showed that all gyrolines and cogyrolines in this space are metric lines. In the fourth chapter, hyperbolic Breusch lemma, the hyperbolic Urquhart theorem and the hyperbolic Steiner-Lehmus theorem in the Poincaré ball model of hyperbolic geometry are examined. In the fifth chapter, a characterization of Möbius transformations by use of hyperbolic isosceles triangles and hyperbolic regular polygons is given.