PlanetMath: stereographic projection

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stereographic projection

(Definition)

The -dimensional Euclidean unit sphere is defined as a subset of $\mathbb{R}^{n+1}$ :

$\displaystyle S^n=\biggl\{(x_1,\ldots,x_{n+1})\in\mathbb{R}^{n+1}\mid\sum\limits _{k=1}^{n+1}x_k^2=1\biggr\}.$

The stereographic projection maps all points of

to the

-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

$\displaystyle \sigma\colon S^n\setminus N\to\mathbb{R}^n,\quad(x_1,\ldots,x_{n+1})\mapsto\frac{c-1}{x_{n+1}-1}(x_1,\ldots,x_n).$

Here,

is an arbitrary real number. If

, the projection degenerates; in all other cases, however, $\sigma$ is a smooth bijective mapping.

The image of a point 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 , the straight line defined by and intersects with $\mathbb{R}^n$ in precisely one point, . The most common values for are and , see figures and .

**Figure 1:** Stereographic projection of the one dimensional unit sphere for
$\includegraphics{StereographicProjection.1.eps}$

**Figure 2:** Stereographic projection of the one dimensional unit sphere for
$\includegraphics{StereographicProjection.2.eps}$

Let $-{\mathrm{id}}\colon\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ be the map $x\mapsto -x$ , then $\tilde{\sigma}:=\sigma\circ(-{\mathrm{id}})$ (a suitably restricted composition) maps all points of 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 , so is an -dimensional smooth manifold.

"stereographic projection" is owned by GrafZahl.

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Also defines:

north pole, south pole

Keywords:

projection, map

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Cross-references: smooth manifold, atlas, composition, intersects, line, straight, hyperplane, image, mapping, bijective, smooth, projection, real number, Euclidean space, points, maps, subset, Euclidean
There are 6 references to this entry.

This is version 2 of stereographic projection, born on 2005-05-24, modified 2005-05-25.
Object id is 7110, canonical name is StereographicProjection.
Accessed 7178 times total.

Classification:

AMS MSC:	51M15 (Geometry :: Real and complex geometry :: Geometric constructions)
	54C05 (General topology :: Maps and general types of spaces defined by maps :: Continuous maps)
	54C25 (General topology :: Maps and general types of spaces defined by maps :: Embedding)
	54E40 (General topology :: Spaces with richer structures :: Special maps on metric spaces)

Pending Errata and Addenda

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