Stereographic projection

Definition

Setup

Consider ${\mathbb{R}}^3$, three-dimensional Euclidean space, with coordinates $x,y,z$. Denote:

$S^2:=\{(x,y,z)\in {\mathbb{R}}^3\mid x^2+y^2+z^2=1\}$

Identify $\mathbb{C}$ with the $xy$-plane under the map:

$(x,y,0)\mapsto x+iy$

Let $N=(0,0,1)$ denote the north pole in $S^2$. Then, the stereographic projection is a bijective map:

$S^2\setminus \{N\}\to \mathbb{C}$

Definition of the map

Geometrically stereographic projection is defined as follows:

For any point $P\in S^2\setminus \{N\}$, consider the line passing through $N$ and $P$. This line is not parallel to $\mathbb{C}$, hence intersects $\mathbb{C}$ at exactly one point. That point is the image of $P$ under stereographic projection.

The formula for stereographic projection is given as:

$(x,y,z)\mapsto \frac{x+iy}{1-z}$

Stereographic projection is invertible, giving a reverse map:

$\mathbb{C}\to S^2\setminus \{N\}$

The reverse map is given by:

$x+iy\mapsto (\frac{2x}{x^2+y^2+1},\frac{2y}{x^2+y^2+1},\frac{x^2+y^2-1}{x^2+y^2+1})$

In terms of $z$ and $\overline{z}$, this is written as:

$z\mapsto (\frac{z+\overline{z}}{|z{|}^2+1},\frac{z-\overline{z}}{i(|z{|}^2+1)},\frac{|z{|}^2-1}{|z{|}^2+1})$

Facts

• Stereographic projection is conformal: Stereographic projection is a conformal mapping when thought of as a mapping between the Riemannian manifolds $S^2\setminus \{N\}$ and $\mathbb{C}$. In other words, it preserves angles between curves (provided we make a matching choice of orientation for the two manifolds).
• Stereographic projection preserves circles: Under stereographic projection, circles in $S^2\setminus \{N\}$ get mapped to circles in $\mathbb{C}$, while circles passing through $N$ (with $N$ removed), get mapped to straight lines.