Stereographic projection

Remarks

The inverse map is $$\sigma^{-1}(z) = \left( \frac{2 \operatorname{Re} z}{\abs{z}^2 + 1}, \frac{2 \operatorname{Im} z}{\abs{z}^2 + 1}, \frac{\abs{z}^2 - 1}{\abs{z}^2 + 1} \right).$$ Example: the south pole $(0, 0, -1)$ goes to $0/2 = 0$. A point $(x_1, x_2, 0)$ of the equator goes to $x_1 + i x_2$, so the equator goes onto the unit circle. Non-example: the projection from the center of the sphere onto a tangent plane is not stereographic. It sends two antipodal points $P$ and $-P$ to the same point, so it is not injective. The map $\sigma$ is a homeomorphism, it preserves angles, and it sends circles on $S^2$ to circles or lines in $\C$; the circles through $N$ go to lines. Some authors project onto the plane tangent at the south pole, which multiplies $\sigma$ by $2$.