Hyperbolic trigonometry and its application in the Poincare ball model of hyperbolic geometry

Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigono metric law of cosines and of sines in the Poincare ball model of n-dimensio nal hyperbolic geometry, as well as their application. The Poincare ball mo del of three-dimensional hyperbolic geometry is becoming increasingly impor tant in the construction of hyperbolic browsers in computer graphics. These allow in computer graphics the exploitation of hyperbolic geometry in the development of visualization techniques. It is, therefore, clear that hyper bolic trigonometry in the Poincare ball model of hyperbolic geometry, as pr esented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics. Hyperbolic trigonometry is governed by gyrov ector spaces in the same way that Euclidean trigonometry is governed by vec tor spaces. The capability of gyrovector space theory to capture analogies and its powerful elegance is thus demonstrated once more. (C) 2001 Elsevier Science Ltd. All rights reserved.