Definition:Spherical Representation of Complex Number

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

Let $\mathbb S$ be the unit sphere which is tangent to $\mathcal P$ at $\tuple {0, 0}$ (that is, where $z = 0$).

Let the diameter of $\mathbb S$ perpendicular to $\mathcal P$ 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.


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

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

Thus any complex number can be represented by a point on the surface of the unit sphere.

The point $N$ on $\mathbb S$ corresponds to the point at infinity.

Thus any point on the surface of the unit sphere corresponds to a point on the extended complex plane.

Also see

 * Definition:Stereographic Projection, the technique for mapping the plane to the general sphere.


 * Definition:Riemann Sphere