On the generalized Poincaré distance

In this study, the Poincaré distance defined on the Euclidean upper half plane has been generalized. Generalized Hyperbolic plane models and hyperbolic space in the Euclidean upper half plane are studied. The thesis consists of seven chapters. The first three chapters of the thesis consist of the introduction and purpose of the thesis, the literature research related to the thesis and the basic concepts of the hyperbolic plane. In the fourth part of the thesis, the Poincare upper half plane model and geodesics of this plane are mentioned. In the fifth chapter, Generalized Hk upper half-plane models have been examined. In the sixth chapter, the concept of Generalized Poincaré distance is introduced and the distance between two points on vertical and horizontal lines in the upper half-plane, as well as on horizontal and vertical ellipse arcs with the center on the x-axis, is investigated. Additionally, it is demonstrated that translations parallel to the x-axis and reflections with respect to vertical lines preserve the generalized Poincaré distance. In the final chapter, hyperbolic space defined on the upper half-space is studied.