Stereographic Projection - Definition

Definition

This section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections.

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

In spherical coordinates (φ, θ) on the sphere (with φ the zenith angle, 0 ≤ φ ≤ π, and θ the azimuth, 0 ≤ θ ≤ 2 π) and polar coordinates (R, Θ) on the plane, the projection and its inverse are

Here, φ is understood to have value π when R = 0. Also, there are many ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates (r, θ, z) on the sphere and polar coordinates (R, Θ) on the plane, the projection and its inverse are