Definition:Stereographic Projection

Definition

Let $\PP$ be a the plane.

Let $\mathbb S$ be a sphere which is tangent to $\PP$ at the origin $\tuple {0, 0}$.

Let the diameter of $\mathbb S$ perpendicular to $\PP$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$.

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.

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 $\mathbb S$.