= 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 is
$$
\boxed{z\longmapsto\overline z.}
$$
This 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.
Back to article page
