stereographic projection

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

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

The stereographic projection maps all points of $S^n$ tothe $n$-dimensional Euclidean space ${ℝ}^n$ except one. Let$N:=(0,\mathrm{\dots },0,1)\in S^n$ be this point (it is usually called thenorth 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 projectiondegenerates; in all other cases, however, $\sigma$ is a smoothbijective mapping.

The image $P^{\prime }$ of a point $P$ under $\sigma$ can be geometricallyconstructed as follows. Embed ${ℝ}^n$ into${ℝ}^{n+1}$ as a hyperplane at $x_{n+1}=c$. Unless $c=1$, thestraight line defined by $N$ and $P$ intersects with ${ℝ}^n$ inprecisely 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 unitsphere for $\mathrm{c}\mathrm{=}\mathrm{-}\mathrm{1}$

Figure 2: Stereographic projection of the one dimensional unitsphere 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 andbijectively to ${ℝ}^n$. Together, $\sigma$ and $\tilde{\sigma }$form an atlas of $S^n$, so $S^n$ is an $n$-dimensional smoothmanifold.

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