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

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 three-dimensional hyperbolic geometry is becoming increasingly important 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 hyperbolic trigonometry in the Poincaré ball model of hyperbolic geometry, as presented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics. Hyperbolic trigonometry is governed by gyrovector spaces in the same way that Euclidean trigonometry is governed by vector spaces. The capability of gyrovector space theory to capture analogies and its powerful elegance is thus demonstrated once more.