Unique Representation of Complex Number in Spherical Form

Theorem

Let $\PP$ be the complex plane.

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 exactly one point $A'$ on the surface of $\mathbb S$ apart from $N$.

Similarly, let $A'$ be a point on the surface of $\mathbb S$ apart from $N$.

Let the line $NA'$ be constructed.

Then $NA'$ passes through exactly one point $A$ on $P$.

Proof

This theorem requires a proof. In particular: Should be standard Euclidean 3-d stuff but I haven't got that far in Euclid yet. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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