## Discrete groups and hyperbolic geometry

Institution Of The Thesis: Bursa Uludağ University, Fen Bilimleri Enstitüsü, Fen Bilimleri Enstitüsü, Turkey

Approval Date: 2008

Thesis Language: Turkish

Student: Osman Avcıoğlu

Supervisor: Osman Bizim

Abstract:

In this study, the conics of hyperbolic geometry have been studied and for this study upper half plane has been used. The study consists of two chapters:The first chapter has basic studies for the second chapter.The second chapter has four sections studying conics of hyperbolic geometry:In the first section hyperbolic circle is defined and it is proved that every hyperbolic circle is an Euclidean circle and every Euclidean circle is a hyperbolic circle. In the second section hyperbolic ellipse, its focuses, center, focus distance, major axis and minor axis are defined and the general equation of an ellipse is given. The hyperbolic ellipse with center i, of which focuses are on the imaginary axis, the central ellipse, is examined. The findings obtained for central ellipse are transferred to an ordinary ellipse of U using the elements of Möb(U), so all required knowledge for an ordinary ellipse of U are obtained and related examples are given. In the third section hyperbolic hyperbola, its focuses, center, focus distance and major axis are defined and the general equation of an hyperbola is given. The hyperbolic hyperbola with center i, of which focuses are on the imaginary axis, the central hyperbola, is examined, the points of boundary at infinity and the asymptotes of the central hyperbola are obtained. After that, the findings obtained for the central hyperbola are transferred to an ordinary hyperbola of U using the elements of Möb(U). In the fourth section hyperbolic parabola, its focus, directrix, axis and vertex are defined and the general equation of a parabola is given. The hyperbolic parabola with the focus where and with the hyperbolic line passing through i perpendicular to the imaginary axis as the directrix (so with the imaginary axis as the axis), the central parabola, is examined and the points of boundary at infinity of the central parabola are obtained. After that, the findings obtained for the central parabola are transferred to an ordinary parabola of U using the elements of Möb(U), so all required knowledge for an ordinary parabola of U are obtained and related examples are given.