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,...

in this paper, we prove that every metric line in the poincare ball model of hyperbolic geometry is exactly a classical line of itself. we also proved nonexistence of periodic lines in the poincare ball model of hyperbolic geometry.

In this paper, we prove that every metric line in the Poincare ball model of hyperbolic geometry is exactly a classical line of itself. We also proved nonexistence of periodic lines in the Poincare ball model of hyperbolic geometry.

‎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 geomet...

the only justification for the einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in einstein addition remained for a long time a mystery to be conquered. accordingly, the aim of this expository article is to present (i) the einstein relativistic vector addition, (ii) the resulting einstein scalar multiplication, (iii) the einstein rel...

the only justification for the einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in einstein addition remained for a long timea mystery to be conquered. accordingly, the aim of this expository article is to present(i) the einstein relativistic vector addition,(ii) the resulting einstein scalar multiplication,(iii) the einstein relativ...

Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the in...

Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincaré ball model of n-dimensional hyperbolic geometry, as well as their application. The Poincaré ball model of 3-dimensional hyperbolic geometry is becoming increasingly important in the construction o...

In 1958, the Dutch artist M.C. Escher became the first person to create artistic patterns in hyperbolic geometry. He used the Poincar é circle model of hyperbolic geometry. Slightly more than 20 years later, my students and I implemented a computer program that could draw repeating hyperbolic patterns in this model. The ...