Given the \(n\)-sphere \(\mathbb{S}^{n}\), the north pole \(N=(0,\ldots,0, 1)\in \mathbb{S}^n\), and the south pole \(S=(0,\ldots ,0,-1\).

The stereographic projection from the north pole \(\sigma\colon \mathbb{S}^n\setminus \{N\}\to \mathbb{R}^n\) is defined by

\begin{equation*} \sigma(x^1,\ldots ,x^{n+1})=\frac{(x^1,\ldots ,x^n}{1-x^{n+1}}. \end{equation*}

The stereographic projection from the south pole \(\tilde{\sigma}\colon \mathbb{S}^n\setminus \{S\}\to \mathbb{R}^n\) is given by \(\tilde{\sigma}(x)=-\sigma(-x)\).

Remarks
  • For any \(x\in \mathbb{S}^n\setminus \{N\}\), \(\sigma(x)=u\), and \((u,0)\) is the point where the line through \(N\) and \(x\) intersects with \(\mathbb{R}^d\times \{0\}\). The stereographic projection from the south pole acts similarly.
  • The inverse of \(\sigma\) is given by \[ \sigma^{-1}(u^1,\ldots ,u^n)=\frac{(2u^1,\ldots ,2u^n, \lvert u\rvert^2-1}{\lvert u\rvert^2+1}. \]
  • The collection consisting of \((\mathbb{S}^n\setminus \{N\},\sigma)\) and \((\mathbb{S}^n \setminus \{S\}, \tilde{\sigma} )\) is an atlas on \(\mathbb{S}^n\), and the coordinates are called stereographic coordinates.
  • Stereographic coordinates determine the standard smooth structure, this can be verified by showing that they are compatible with graph coordinate charts (see (0x68f0e23e) ).