On the unification of hyperbolic and euclidean geometry

Abstract

The polar decomposition of Möbius transformation of the complex open unit disc gives rise to the Möbius addition in the disc and, more generally, in the ball. Möbius addition and Einstein addition in the ball of a real inner product space are isomorphic gyrogroup operations that play in the hyperbolic geometry of the ball a role analogous to the role that ordinary vector addition plays in the Euclidean geometry of . Möbius (Einstein) addition governs the Poincaré (Beltrami) ball model of hyperbolic geometry just as vector addition governs the standard model of Euclidean geometry. Accordingly, we show in this article that resulting analogies enable Euclidean and hyperbolic geometry to be unified