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.

Description

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.

Discussion

• Take to be the unit sphere, so that plane is the equatorial plane. The typical point on the sphere has coordinates . The typical point in the equatorial plane, whose coordinates are , will be called .
  1. Show that the South pole is mapped into the origin under stereographic projection.
  2. Show that under stereographic projection the equator is mapped onto the unit circle, that is the circle .
  3. Show that under stereographic projection the lower hemisphere is mapped into the interior of this circle, that is the disk .

  4. Show that under stereographic projection the upper hemisphere is mapped into the exterior of this circle, that is into .

     For this to be true where do we have to think of as lying: interior to or exterior to it?

  5. What projects on to the -axis?

     What projects onto the ? Call the set of points that project onto the prime meridian.

  6. The prime meridian divides the sphere into two hemispheres, the front hemisphere and the back hemisphere. What is the image of the back hemisphere under stereographic projection? The front hemisphere?

  7. Under stereographic projection what is the image of a great circle passing through the north pole? Of any circle (not necessarily a great circle) passing through the north pole?

  8. Under stereographic projection, what projects onto the -axis? onto any vertical line, not necessarily the axis?

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

If

lies on the plane,

Thus

Or

Whence,

Or

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?

• Angles: To see that stereographic projection preserves angles at , we project onto the horizontal plane through . Then by symmetry the tangent planes and at and make the same angle with , as also does , by properties of parallelism (see figure # 1 at the end of this handout).

  So and are images of each other in the (``mirror'') plane through and perpendicular to .

  For a point on the sphere near , the line is nearly parallel to , so that for points near , stereographic projection is approximately the reflection in .

• Circles: To see that stereographic projection takes circles to circles, first note that any circle is where some cone touches the sphere, say the cone of tangent lines to the sphere from a point .

  Now project onto the horizontal plane through .

  In figure # 2 which NEED NOT be a vertical plane, the four angles are equal, for the same reasons as before, so that . The image of is therefore the horizontal circle of the same radius centered at .

• Inversion: Another proof uses the fact that stereographic projection may be regarded as a particular case of inversion in three dimensions. You might like to prove that inversion preserves angles and circularity in two dimensions. The inverse of a point in the circle of radius centered at is the unique point on the ray for which .