Notes

Let N=(0,,0,1)N=(0,\dots,0,1) be the "north pole". The formulas for stereographic projection are f ⁣:Sn{N}Rnf(p)=11pn(p0,,pn1)f1 ⁣:RnSn{N}f1(x)=N+21+x2(x0,,xn1,1)\begin{aligned} f\colon &S^n-\{N\} \to \R^n\\ f(p) &= \frac1{1-p_n}(p_0,\dots,p_{n-1})\\[2em] f^{-1} \colon& \R^n \to S^n-\{N\}\\ f^{-1}(x) &= N + \frac2{1+\|x\|^2}(x_0,\dotsc,x_{n-1},-1) \end{aligned}