Hyperbolic trigonometry in the Einstein relativistic velocity model of hyperbolic geometry

Hyperbolic geometry is a fundamental aspect of modern physics. We explore i n this paper the use of Einstein's velocity addition as a model of vector a ddition in hyperbolic geometry. Guided by analogies with ordinary vector ad dition, we develop hyperbolic vector spaces, called gyrovector spaces, whic h provide the setting for hyperbolic geometry in the same way that vector s paces provide the setting for Euclidean geometry. The resulting gyrovector spaces enable Euclidean trigonometry to be extended to hyperbolic trigonome try. In particular, we present the hyperbolic law of cosines and sines and the Hyperbolic Pythagorean Theorem emerges when the common vector addition is replaced by the Einstein velocity addition. (C) 2000 Elsevier Science Lt d. All rights reserved.