A conformal map that sends points of a sphere minus one distinguished pole to points of a plane by projecting along lines through the pole; it provides a bijection between the punctured sphere and the plane.

Principle

Project from a chosen pole: each point on the sphere (except the pole) lies on a unique line through the pole which meets the plane in a single point; stereographic projection preserves angles (conformal) and maps circles not passing through the pole to circles in the plane.

Demonstration

For the unit sphere in R^3 with north pole N=(0,0,1), projecting from N to the plane z=0 sends (x,y,z) (z≠1) to (x/(1-z), y/(1-z)). In complex analysis this gives the identification of the Riemann sphere with the extended complex plane by mapping a point to its complex coordinate.

Misapplication

Assuming stereographic projection preserves area or lengths (it does not), or applying it without handling the pole (which maps to infinity) leads to misinterpretation; using it as an isometry is incorrect.

Consequence

Provides a conformal parametrization of the sphere by the plane, turns spherical geometry problems into planar ones, and compactifies the plane by a point at infinity corresponding to the omitted pole.

Reversal

Has a well-defined inverse that lifts a planar point to the sphere; the pole corresponds to the point at infinity in the plane. The inverse recovers spherical coordinates from planar ones except at infinity.

Boundary

Defined for the sphere minus the projection pole; global statements require adding the point at infinity on the plane to obtain a bijection with the whole sphere. It depends on the choice of pole and plane.

Semantic Tension

Sometimes confused with other sphere-to-plane maps (e.g., orthographic, gnomonic, Lambert azimuthal) that trade conformality for area preservation or geodesic properties; stereographic is distinguished by angle preservation and circle-to-circle mapping (excluding circles through the pole).

Synthesis

Stereographic projection is the angle-preserving correspondence between the sphere with one point removed and the plane: it projects along rays from a pole, mapping circles to circles (unless passing through the pole) and identifying the pole with the point at infinity.