Definition:Spherical Representation of Complex Number

Definition

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

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 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.

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

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Spherical Representation of Complex Numbers. Stereographic Projection