##  [Stereographic Projection](/stereographic-projection-0) 

 Definition

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

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

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

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

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

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

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

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

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.