Stereographic projection maps points on a sphere S² to a plane through the equator,
using a pole as the projection point.
σ(x, y, z) = (x/(1-z), y/(1-z))
Key Properties
•Conformal: Preserves angles between curves
•Circles → Circles: Maps circles to circles or lines
•Bijective: One-to-one except at pole (maps to ∞)
Riemann Sphere
The sphere S² with stereographic projection gives the Riemann sphere,
representing the extended complex plane ℂ ∪ {∞}. The pole maps to the
"point at infinity," compactifying ℂ.
Complex Analysis
Identifying the plane with ℂ, the projection gives a natural identification
of S² with the compactified complex plane. Möbius transformations on ℂ
correspond to rotations of S².
Click a point to highlight its correspondence between sphere and plane.
Latitude Circles
z = 0.8 (near N pole)
z = 0.5
z = 0.25
z = 0 (Equator)
z = −0.25
z = −0.5
z = −0.8 (near S pole)
Projection pole
Circles near the projection pole map to larger circles on the plane.