Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 2 / 12G / ii / Solution

Reflection of the sphere in the plane $Y=0$ sends $(X,Y,Z)$ to $(X,-Y,Z)$. It preserves dot products, hence the spherical distance between every pair of points. Its stereographic expression isThis is an isometry of the sphere but is not a Möbius transformation: at any finite point, its complex difference quotient is one along real increments and minus one along imaginary increments, so it is not holomorphic. A Möbius transformation is holomorphic away from its pole. The geometric distinction is that this reflection reverses orientation, whereas a Möbius transformation preserves it.