This illustrates the stereographic projection homeomorphism
$f\colon S^2\setminus\{N\}\to\mathbb{R}^2$, which is given by
$$ f(x,y,z) = \left(\frac{x}{1-z},\;\frac{y}{1-z}\right)
\hspace{4em}
f^{-1}(u,v) =
\left(
\frac{2u}{u^2+v^2+1},
\frac{2v}{u^2+v^2+1},
\frac{u^2+v^2-1}{u^2+v^2+1},
\right)
$$