# Stereographic Projection

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Stereographic Projection of a Sphere
A stereographic projection of a sphere onto a plane.

# Basic Description

Cross section of arbitrary points on a sphere being mapped to points on a plane.

Stereographic projection is a method of mapping an object into a lower dimensional space. This page's main image shows a sphere being mapped into a plane. In this context mapping means matching points on the sphere with points on the plane using a specific rule. The rule used in the diagram to the left is as follows: draw a line from the 'north pole' of the sphere and let it pass through a point on the sphere, point A. The point that the line hits on the plane, point B, is the point that A is mapped to.

The main image uses a similar procedure, except the plane is drawn under the sphere instead of cutting through it. The coloring helps give an idea of where regions of the sphere end up on the plane. Note that the projection is still from the top of the sphere, with the coloring not centered around the top creating an interesting formation of ellipses on the plane.

The following applet demonstrates how a sphere is projected onto a plane. A sphere with coaxial bands of color is stereographically projected onto a plane in the background. You can rotate the sphere with the mouse, changing the orientation of the colors on the sphere which changes the projection on the plane. The sphere and projection point remain fixed; only the colors are shifted.

If you can see this message, you do not have the Java software required to view the applet.

# A More Mathematical Explanation

[[Image:Sphereproject.gif|thumb|500px|left|Cutaway view of some points on the sphere, tracked by the [...]

Cutaway view of some points on the sphere, tracked by the green dot when visible, being mapped onto a plane

An example of a mapping from a sphere onto a plane, shown graphically to the left, is

$(X, Y) = \left(\frac{x}{1 - z}, \frac{y}{1 - z}\right),$

where X,Y are coordinates on the plane and x,y,z are coordinates on the sphere.

Since coordinates on the sphere are mapped uniquely to coordinates on the plane, this function is invertible. The explicit inverse is:

$(x, y, z) = \left(\frac{2 X}{1 + X^2 + Y^2}, \frac{2 Y}{1 + X^2 + Y^2}, \frac{-1 + X^2 + Y^2}{1 + X^2 + Y^2}\right).$

An important mathematical application of stereographic projections is the Riemann Sphere.

# Teaching Materials

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# About the Creator of this Image

Thomas F. Banchoff is a geometer, and a professor at Brown University since 1967.

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