# stereographic projection

The $n$-dimensional Euclidean http://planetmath.org/node/186unit sphere  $S^{n}$ is defined as a subset of $\mathbb{R}^{n+1}$:

 $S^{n}=\biggl{\{}(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}\mid\sum\limits_{k=1}% ^{n+1}x_{k}^{2}=1\biggr{\}}.$

The stereographic projection maps all points of $S^{n}$ to the $n$-dimensional Euclidean space  $\mathbb{R}^{n}$ except one. Let $N:=(0,\ldots,0,1)\in S^{n}$ be this point (it is usually called the north pole). Then the stereographic projection is defined by

 $\sigma\colon S^{n}\setminus N\to\mathbb{R}^{n},\quad(x_{1},\ldots,x_{n+1})% \mapsto\frac{c-1}{x_{n+1}-1}(x_{1},\ldots,x_{n}).$

Here, $c$ is an arbitrary real number. If $c=1$, the projection degenerates; in all other cases, however, $\sigma$ is a smooth bijective mapping.

The image $P^{\prime}$ of a point $P$ under $\sigma$ can be geometrically constructed as follows. Embed $\mathbb{R}^{n}$ into $\mathbb{R}^{n+1}$ as a hyperplane  at $x_{n+1}=c$. Unless $c=1$, the straight line defined by $N$ and $P$ intersects with $\mathbb{R}^{n}$ in precisely one point, $P^{\prime}$. The most common values for $c$ are $c=-1$ and $c=0$, see figures 1 and 2. Figure 1: Stereographic projection of the one dimensional unit sphere for c=-1 Figure 2: Stereographic projection of the one dimensional unit sphere for c=0

Let $-\operatorname{id}\colon\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ be the map $x\mapsto-x$, then $\tilde{\sigma}:=\sigma\circ(-\operatorname{id})$ (a suitably restricted composition  ) maps all points of $S^{n}$ except the south pole $S:=(0,\ldots,0,-1)$ smoothly and bijectively to $\mathbb{R}^{n}$. Together, $\sigma$ and $\tilde{\sigma}$ form an atlas of $S^{n}$, so $S^{n}$ is an $n$-dimensional smooth manifold  .

 Title stereographic projection Canonical name StereographicProjection Date of creation 2013-03-22 15:18:35 Last modified on 2013-03-22 15:18:35 Owner GrafZahl (9234) Last modified by GrafZahl (9234) Numerical id 5 Author GrafZahl (9234) Entry type Definition Classification msc 54E40 Classification msc 54C25 Classification msc 54C05 Classification msc 51M15 Related topic CoordinateSystems Related topic ClosedComplexPlane Related topic RiemannSphere Defines north pole Defines south pole