Take a sphere sitting on a plane. Draw a line from the top of the sphere to a point . in the plane to intersect the sphere at a point . The stereographic projection of is the point .
The mapping works both ways so you can think of projecting down from the sphere to the plane using the same intersecting line.
Stereographic projection maps the points of a line or a circle in the plane to circles on the sphere. Also, stereographic projection is conformal, which means that angles are preserved.
Although every point in the plane maps up to a point on the sphere, the top point on the sphere has no corresponding point in the plane. Points close to the top map back into the plane far from the sphere, so the top is said to represent the plane's "point at infinity."