Definition:Stereographic Projection

Definition
Let $\mathcal P$ be a the plane.

Let $\mathbb S$ be a sphere which is tangent to $\mathcal P$ at the origin $\left({0, 0}\right)$.

Let the diameter of $\mathbb S$ perpendicular to $\mathcal P$ through $\left({0, 0}\right)$ be $NS$ where $S$ is the point $\left({0, 0}\right)$.

Let the point $N$ be referred to as the north pole of $\mathbb S$ and $S$ be referred to as the south pole of $\mathbb S$.

Let $A$ be a point on $P$.

Let the line $NA$ be constructed.


 * Spherical-Represenation-of-Complex-Number.png

Then $NA$ passes through a point of $\mathbb S$.

Thus any point on $P$ can be represented by a point on $\mathbb S$.

With this construction, the point $N$ on $\mathbb S$ maps to no point on $\mathbb S$.

Also see

 * Definition:Spherical Representation of Complex Number, where this technique is used to map the complex plane to the unit sphere.


 * Definition:Riemann Sphere