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 . Show that the South pole is mapped into the origin under stereographic projection. Show that under stereographic projection the equator is mapped onto the unit circle, that is the circle . Show that under stereographic projection the lower hemisphere is mapped into the interior of this circle, that is the disk .