Riemann Sphere and Stereographic Projection
Project a finite complex point onto the unit sphere and identify infinity with the missing north pole.
By Riano · Published · Updated · Report a correction
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):
(X, Y, Z) = (2x, 2y, x2 + y2 - 1) / (1 + x2 + y2), with inverse x = X / (1 - Z) and y = Y / (1 - Z) for Z ≠ 1.
Adjoining the point at infinity z = ∞ ↔ N(0, 0, 1) compactifies the complex plane into the Riemann sphere C ∪ {∞}.
Worked Example: Off-Axis Point z = 1 + i and South/North Poles
Compute the stereographic sphere coordinates and inverse recovery for z = 1 + i (x = 1, y = 1):
- Squared modulus and denominator:
r2 = x2 + y2 = 12 + 12 = 2, so1 + r2 = 3. - Unit sphere coordinates:
X = 2(1)/3 = 2/3 ≈ 0.6666666666666666,Y = 2(1)/3 = 2/3 ≈ 0.6666666666666666, andZ = (2 - 1)/3 = 1/3 ≈ 0.3333333333333333. - Unit norm check and inverse:
X2 + Y2 + Z2 = 4/9 + 4/9 + 1/9 = 1, andX / (1 - Z) = (2/3) / (2/3) = 1 = x,Y / (1 - Z) = 1 = y.
Interactive Experiment
Explore equatorial, polar, and north-pole compactification points in the Riemann Sphere and Stereographic Projection interactive workspace:
- With Point mode = Finite z, set Real x = 1 and Imaginary y = 0 (or Real x = 0, Imaginary y = 1). Because
|z| = 1lies on the unit circle, the point maps onto the equatorZ = 0at(1, 0, 0)(or(0, 1, 0)). - Set Real x = 0 and Imaginary y = 0. The origin
z = 0maps to the south pole(0, 0, -1)with Sphere norm squared = 1. - 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. - 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).
Limitations and Assumptions
The forward rational parameterization covers S2 \ {N}; the north pole N(0, 0, 1) requires the explicit compactification extension z = ∞ where 1 - Z = 0 and no finite Cartesian inverse exists. Note also that the canvas renders a fixed orthographic 2D projection of R3: although stereographic projection between C and S2 is conformal in three dimensions, apparent 2D screen angles on the orthographic canvas are foreshortened and are not true 3D spherical angles.
References
The explicit stereographic projection formulas between the complex plane and the unit sphere in R3 and the one-point compactification into the Riemann sphere are derived in the following reference:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Riemann Sphere and Stereographic Projection module.
Try Riemann Sphere and Stereographic Projection on Riano →