Question

a) Prove stereographic projection maps circles not containing (0,0,1) from the upper half sphere’s surface into circles on the z = 0 projection plane. b) What happens under stereographic projection to circles containing (0,0,1), or otherwise the North Pole? Draw a sketch showing this.

Answer 1

b)

Stereographic projection of the unit sphere from the north pole onto the plane z = 0, shown here in cross section

The unit sphere in three-dimensional space R³ is the set of points (x, y, z) such that x² + y² + z² = 1. Let N = (0, 0, 1) be

The unit sphere in three-dimensional space R³ is the set of points (x, y, z) such that x² + y² + z² = 1. Let N = (0, 0, 1) be the "north pole", and let M be the rest of the sphere. The plane z = 0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane.

For any point P on M, there is a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane.

In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are given by the formulas