The Hyperbolic Derivative in the Poincar´e Ball Model of Hyperbolic Geometry

The generic M¨obius transformation of the complex open unit disc induces a binary operation in the disc, called the M¨obius addition. Following its introduction, the extension of the M¨obius addition to the ball of any real inner product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincar´e ball model of hyperbolic geometry. The M¨obius gyrovector spaces that emerge provide the setting for the Poincar´e ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the M¨obius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry.