Stereographic Projection

Why

We want to make physical two dimensional images of parts of the earth's surface. In other words, we want to make maps.

Definition

Denote the unit sphere in $\R ^3$ by $S$ and the northpole $(0, 0, 1) \in S$ by $p$. Denote the sphere without the nortpole by $S^\circ$.¹ In other words, $S^\circ$ is defined by

\[ S^\circ = S \setminus \set{p} \]

The stereopgraphic projection $\pi : S^\circ \to \R ^2$ of $S^\circ$ is defined as follows. The point $\pi (x)$ is the $(x, y)$ value for the point where the line through $N$ and $x$ intersects the plane $P = \Set{(x, y, z) \in \R ^3}{z = 0}$ (the “xy-plane”).²

The stereographic projection is a diffeomorphism.

Future editions will resolve the fact that this conflicts with the notion of polar sets (seeReal Polar Sets ↩︎
Future editions will improve and continue the development as well as include the requisite figures. ↩︎