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.
For this to be true where do we have to think of as lying: interior to or exterior to it?
What projects onto the ? Call the set of points that project onto the prime meridian.
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 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 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?
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 .
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 .