Stereographic Projection and the Riemann Sphere Explained

Stereographic projection maps the complex plane onto a sphere minus one point: a line from the north pole through any other sphere point hits the plane exactly once. Adding the north pole as the point at infinity gives the Riemann sphere, the extended plane Ĉ = C ∪ {∞}.

Infinity becomes a concrete geometric point rather than a vague "far away." Circles and lines in the plane are one object on the sphere (both are circles), and 1/z, which swaps 0 and ∞, becomes a smooth sphere motion instead of a singularity.

Key Concepts

• Stereographic projection: bijection between the unit sphere minus (0,0,1) and C, projecting from the north pole.
• Point at infinity: ∞ identified with the north pole; the one-point compactification of the plane.
• Riemann sphere: the sphere modelling the extended complex plane.
• Lines as circles: sphere circles project to plane circles, or to lines when they pass through ∞.
• Conformality: the projection preserves angles, carrying the plane's angle geometry to the sphere.

Formulas

• Sphere to plane: (X, Y, Z) ↦ z = X/(1 − Z) + iY/(1 − Z).
• Plane to sphere: z = x + iy ↦ X = 2x/(|z|²+1), Y = 2y/(|z|²+1), Z = (|z|²−1)/(|z|²+1).
• Chordal distance: 2|z − w|/(√(|z|²+1)√(|w|²+1)), continuous even at ∞.
• z ↦ 1/z is reflection of the sphere across the equatorial plane, swapping 0 and ∞.

Worked Example

Project z = 1 + i: |z|² = 2, so X = 2/3, Y = 2/3, Z = (2−1)/(2+1) = 1/3. Check: (2/3)² + (2/3)² + (1/3)² = 4/9 + 4/9 + 1/9 = 1, on the sphere. Project back: X/(1−Z) = (2/3)/(2/3) = 1 and likewise Y/(1−Z) = 1, recovering 1 + i. As |z| → ∞, Z = (|z|²−1)/(|z|²+1) → 1, the north pole: large numbers cluster near ∞ on the sphere.

Common Misconceptions

• "∞ is a number." It is a geometric point; ∞ − ∞ stays meaningless, but "1/z → 0 as z → ∞" becomes continuity at the north pole.
• "The Riemann sphere is a solid ball." It is the 2D surface; each point is a complex number or ∞.
• "Projection preserves distances." It preserves angles and circles but stretches regions near the north pole enormously.

Real-World Applications

• Möbius maps z ↦ (az+b)/(cz+d) become sphere motions, used in graphics texture mapping.
• The quantum Bloch sphere is a Riemann sphere: qubit states are points, gates are rotations.
• Cartographers use stereographic projection for polar charts because it preserves angles.

Stereographic projection closes the plane into a sphere with one point at infinity, unifying lines with circles and domesticating behavior at infinity: the natural stage for complex analysis.