I am trying to understand how one can calculate the transition function in general by calculating it from the stereographic projection.
If we have $n=2$ and the two maps
$$\phi_1^{-1}:u\rightarrow \Big(\frac{2r\cos\theta}{1+r^2},\frac{2r\sin\theta}{1+r^2},\frac{1-r^2}{1+r^2}\big)$$ $$\phi_2^{-1}:v\rightarrow \Big(\frac{2r\cos\theta}{1+r^2},\frac{-2r\sin\theta}{1+r^2},\frac{r^2-1}{1+r^2}\big) $$ how are you supposed to get the transition function? I have tried it a couple different ways but none has gotten me any far. I think the best way to progress should be from the knowledge that the transition function is $\phi_2\circ\phi_1^{-1}$ but I don't understand how I am supposed to get $\phi_2$ from just $\phi_2^{-1}$.
Another method I tried is by comparing the coordinates from the inverse functions and trying to see what I need to multiply to get the other. I know that it is possible to go about it this way but I loose myself in the algebra.