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,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.