Stereographic Projection

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Stereographic Projection

We let be a sphere in Euclidean three space. We want to obtain a picture of the sphere on a flat piece of paper or a plane. Whenever one projects a higher dimensional object onto a lower dimensional object, some type of distortion must occur. There are a number of different ways to project and each projection preserves some things and distorts others. Later we will explain why we choose stereographic projection, but first we describe it.


We shall map the sphere onto the plane containing its equator. Connect a typical point on the surface of the sphere to the north pole by a straight line in three space. This line will intersect the equatorial plane at some point . We call the projection of .

Using this recipe every point of the sphere except the North pole projects to some point on the equatorial plane. Since we want to include the North pole in our picture, we add an extra point , called the point at infinity, to the equatorial plane and we view as the image of under stereographic projection.


What's good about stereographic projection?

Stereographic projection preserves circles and angles. That is, the image of a circle on the sphere is a circle in the plane and the angle between two lines on the sphere is the same as the angle between their images in the plane. A projection that preserves angles is called a conformal projection.

We will outline two proofs of the fact that stereographic projection preserves circles, one algebraic and one geometric. They appear below.

Before you do either proof, you may want to clarify in your own mind what a circle on the surface of a sphere is. A circle lying on the sphere is the intersection of a plane in three space with the sphere. This can be described algebraically. For example, the sphere of radius 1 with center at the origin is given by


An arbitrary plane in three-space is given by


for some arbitrary choice of the constants ,, , and . Thus a circle on the unit sphere is any set of points whose coordinates simultaneously satisfy equations 2 and 3.

The algebraic proof

The fact that the points , and all lie on one line can be expressed by the fact that


for some non-zero real number . (Here .)

The idea of the proof is that one can use equations 2 and 4 to write as a function of and , as a function of and , and as a function of and to simplify equation 3 to an equation in and . Since the equation in and so obtained is clearly the equation of a circle in the plane, the projection of the intersection of 2 and 3 is a circle.

To be more precise:

Equation 4 says that . Set and verify that


lies on the plane,





Recalling that

, we see


Since the coefficients of the

and the

terms are the same, this is the equation of a circle in the plane.

The geometric proofs

The geometric proofs sketched below use the following principle:

It doesn't really make much difference if instead of projecting onto the equatorial plane, we project onto another horizontal plane (not through N), for example the plane that touches the sphere at the South pole, S. Just what difference does this make?

Next: The orbifold shop Up: Geometry and the Imagination Previous: Names for symmetry

Peter Doyle