---
tag: AXDKKVD18N
kind: concept
name: Stereographic projection
msc: [30A99, 51N20]
sources:
  - url: https://en.wikipedia.org/wiki/Stereographic_projection
same_as:
  wikidata: Q866099
  wikipedia: Stereographic projection
---

## Summary

Stereographic projection maps the unit sphere minus one point, its north pole, bijectively onto a plane, see the [[definition|RN8RBM2A8A]].
Sending the north pole to a point at infinity makes the sphere a model of the [[Riemann sphere|0J78KBPFZB]] $\C \cup \set{\infty}$.
The map preserves angles and sends circles on the sphere to circles or lines in the plane.
It is the bridge between the geometry of the sphere and complex analysis on the extended plane.

## Explanations

### L1 human

Put a ball on a flat floor so that it touches the floor at its south pole.
Shine a light from the north pole: each other point of the ball casts a shadow on the floor.
The map from a point to its shadow is the stereographic projection.
The south pole goes to itself, the equator goes to a circle, and points near the north pole go far away.
So the whole floor, together with one extra point at infinity, is a copy of the sphere.