I've been asked to find $(x,y,z)$ in $\mathcal{S}$. I'm stuck on the question attached because although it gives the formula of how to find $\pi_\mathcal{s}$ (stereographic projection), I'm not sure it can be used to find the reverse i.e. $(x,y,z)$. Can anyone offer some guidance?
Use $\pi_s^{-1}(u, v) = \left(\frac{2u}{1 + u^2 + v^2}, \frac{2 v}{1 + u^2 + v^2}, \frac{-1 + u^2 + v^2}{1 + u^2 + v^2}\right)$
$$\pi_s^{-1}(1, 4) = \left(\frac{2}{18}, \frac{8}{18}, \frac{16}{18}\right)=\left(\frac{1}{9}, \frac{4}{9}, \frac{8}{9}\right)$$