Index

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