Gyrotrigonometry

Abstract

Trigonometry is the study of relations between the triangle three sides and three angles. In full analogy, gyrotrigonometry is the study of relations between the gyrotriangle three sides and three gyroangles. In this chapter, gyrotrigonometry is studied in Einstein gyrovector spaces or, equivalently, in the Cartesian–Beltrami–Klein ball model of hyperbolic geometry. Remarkably, the elementary gyrotrigonometric functions coincide with the common elementary trigonometric functions. They, however, stem from a system of two distinct Pythagorean identities that each right-gyroangled gyrotriangle possesses. In the transition from hyperbolic to Euclidean geometry, these two distinct Pythagorean identities degenerate into the single Pythagorean identity that each right-angled triangle possesses in Euclidean geometry.