stereographic projection

The n-dimensional Euclidean sphereMathworldPlanetmath Sn is defined as a subset of n+1:


The stereographic projection maps all points of Sn to the n-dimensional Euclidean spaceMathworldPlanetmath n except one. Let N:=(0,,0,1)Sn be this point (it is usually called the north pole). Then the stereographic projection is defined by


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

The image P of a point P under σ can be geometrically constructed as follows. Embed n into n+1 as a hyperplaneMathworldPlanetmath at xn+1=c. Unless c=1, the straight line defined by N and P intersects with n in precisely one point, P. 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 -id:n+1n+1 be the map x-x, then σ~:=σ(-id) (a suitably restricted compositionMathworldPlanetmath) maps all points of Sn except the south pole S:=(0,,0,-1) smoothly and bijectively to n. Together, σ and σ~ form an atlas of Sn, so Sn is an n-dimensional smooth manifoldMathworldPlanetmath.

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