On the unification of hyperbolic and euclidean geometry

The polar decomposition of Mobius transformation of the complex open unit disc gives rise to the Mobius addition in the disc and, more generally, in the ball. Mobius 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 . Mobius (Einstein) addition governs the Poincare (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