The hyperbolic derivative in the Poincare ball model of hyperbolic geometry

The generic Mobius transformation of the complex open unit disc induces a b inary operation in the disc, called the Mobius addition. Following its intr oduction, the extension of the Mobius addition to the ball of any real inne r product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincare ball model of hyperboli c geometry. The Mobius gyrovector spaces that emerge provide the setting fo r the Poincare ball model of hyperbolic geometry in the same way that vecto r spaces provide the setting for Euclidean geometry. Our summary of the pre sentation of the Mobius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Su bsequently, the hyperbolic derivative and its application to geodesics unco ver novel analogies that hyperbolic geometry shares with Euclidean geometry . (C) 2001 Academic Press.