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 . Show that under stereographic projection the upper hemisphere is mapped into the exterior of this circle, that is into . Call the set of points that project onto the prime meridian. The prime meridian divides the sphere into two hemispheres, the front hemisphere and the back hemisphere.
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.