Mathematics Interdisciplinary Research

Mathematics Interdisciplinary Research

An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach

Document Type : Special Issue: AIMC 51

Authors
Mahfouz Rostamzadeh 1
Sayed-Ghahreman Taherian 2
1 Department of Mathematics, University of Kurdistan, P. O. Box 416 Sanandaj, I. R. Iran
2 Department of Mathematical Sciences, Isfahan University of Technology, 84156 Isfahan, I R Iran
10.22052/mir.2016.13923
Abstract
‎The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]‎. ‎In [1]‎, ‎Ungar and Chen showed that the algebra of the group SL(2, C) naturally leads to the notion of gyrogroups ‎and gyrovector spaces for dealing with the Lorentz group and its ‎underlying hyperbolic geometry‎. ‎They defined the Chen addition and then Chen model of hyperbolic geometry‎. ‎In this paper‎, ‎we directly use the isomorphism properties of gyrovector spaces to recover the Chen's addition and then Chen model of hyperbolic geometry‎. ‎We show that this model is an extension of the Poincaré model of hyperbolic geometry‎. ‎For our purpose we consider ‎the Poincaré plane model of hyperbolic geometry inside the complex open unit disc D. ‎Also we prove that this model is isomorphic to the Poincaré model and then to other models of hyperbolic geometry‎. ‎Finally‎, ‎by gyrovector space approach we verify some properties of this model in details in full analogue with Euclidean geometry‎.
Keywords
Hyperbolic geometry‎
‎Gyrogroup‎
‎Gyrovector space‎
‎Poincar'e model‎
‎Analytic hyperbolic geometry‎
Subjects
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Mathematics Interdisciplinary Research
Volume 1, Issue 1 - Serial Number 1
Winter 2016
Pages 187-198