Möbius transformation, unit disk to real axis

Question

$$\begin{align}i &\to 1 \\1 &\to \infty\\-1& \to 0\end{align}$$

Unusually use the cross product but I can't do this with the infinity. I know it's the unit disk to the real axis.

Answer 1

$W=(-z+iz-1+i)/(z-i+iz-1)$

An use cross product just cancel term with infinity