stereographic projection

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

$$S^n=\{(x_1,\mathrm{\dots },x_{n+1})\in {ℝ}^{n+1}\mid \sum _{k=1}^{n+1}{x}_k^2=1\}.$$

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

$$\sigma :S^n\setminus N\to {ℝ}^n,(x_1,\mathrm{\dots },x_{n+1})\mapsto \frac{c-1}{x_{n+1}-1}(x_1,\mathrm{\dots },x_n).$$

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 ${ℝ}^n$ into ${ℝ}^{n+1}$ as a hyperplane at $x_{n+1}=c$. Unless $c=1$, the straight line defined by $N$ and $P$ intersects with ${ℝ}^n$ 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 $\mathrm{c}\mathrm{=}\mathrm{-}\mathrm{1}$

Figure 2: Stereographic projection of the one dimensional unit sphere for $\mathrm{c}\mathrm{=}\mathrm{0}$

Let $-\mathrm{id}:{ℝ}^{n+1}\to {ℝ}^{n+1}$ be the map $x\mapsto -x$, then $\tilde{\sigma }:=\sigma \circ (-\mathrm{id})$ (a suitably restricted composition) maps all points of $S^n$ except the south pole $S:=(0,\mathrm{\dots },0,-1)$ smoothly and bijectively to ${ℝ}^n$. 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