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

‎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 $SL(2,mathbb C)$ 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 Poincar'e model of hyperbolic geometry‎. ‎For our purpose we consider ‎the Poincar'e plane model of hyperbolic geometry inside the complex open unit disc $mathbb{D}$‎. ‎Also we prove that this model is isomorphic to the Poincar'e 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.