The function is diffeomorphism

Question

Define $$ \phi:\mathbb{R}^2\setminus\{(0,0)\}\to (-1,1)\times(-1,1) $$ defined by, $\phi(u,v)=\left(\frac{u}{u^2+v^2},\frac{v}{u^2+v^2}\right)$.

I am facing problem in finding out the inverse of this function. Please help me.

Answer 1

The function $\phi$ is its own inverse if we consider $\phi:\mathbb R^2\setminus \{(0,0)\}\to \mathbb R^2\setminus \{(0,0)\}$. The resulting map is a diffeomorphism ...

This $\phi$ is "reflection" of a point about the unit circle...