Riemann Sphere and Stereographic Projection

Stereographic projection establishes a one-to-one rational correspondence between the complex plane z = x + iy and the unit sphere S2 = {(X, Y, Z) ∈ R3 : X2 + Y2 + Z2 = 1} punctured at the north pole N(0, 0, 1):

Interactive Experiment

Explore equatorial, polar, and north-pole compactification points in the Riemann Sphere and Stereographic Projection interactive workspace:

1. With Point mode = Finite z, set Real x = 1 and Imaginary y = 0 (or Real x = 0, Imaginary y = 1). Because |z| = 1 lies on the unit circle, the point maps onto the equator Z = 0 at (1, 0, 0) (or (0, 1, 0)).
2. Set Real x = 0 and Imaginary y = 0. The origin z = 0 maps to the south pole (0, 0, -1) with Sphere norm squared = 1.
3. Switch Point mode = Infinity. Notice that the finite Real x and Imaginary y selectors are disabled while the sphere point jumps to the north pole (0, 0, 1) with Inverse real = Unavailable and Status = North pole: infinity.
4. Return Point mode to Finite z and set Real x = 2, Imaginary y = -2 (r2 = 8) to observe a high northern-hemisphere point (4/9, -4/9, 7/9).