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.