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:

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

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

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

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.