The Riemann sphere, denoted $\hat{\mathbb{C}}$ , is the one-point compactification of the complex plane $\mathbb{C}$ , obtained by identifying the limits of all infinitely extending rays from the origin as one single ``point at infinity.'' Heuristically, $\hat{\mathbb{C}}$ can be viewed as a 2-sphere with the top point corresponding to the point at infinity, and the bottom point corresponding the origin. An atlas for the Riemann sphere is given by two charts:

Concretely, the bijective correspondence of the points of the closed complex plane and the Riemann sphere is implemented by the stereographic projection. Think a sphere of radius $R$ being above the complex plane and having it as tangent plane with the origin as the point of tangency. Call this point the South Pole and the opposite point $N$ of the sphere the North Pole. For an arbitrary point $P$ of the complex plane, set the line through it and $N$ . The line intersects the sphere in another point $P'$ . The mapping

(1)

If we equip the sphere with geographic coordinates, the longitude $\lambda$ ($-\pi < \lambda \leqq \pi$ ) and the latitude $\varphi$ ($-\frac{\pi}{2} \leqq \varphi \leqq \frac{\pi}{2}$ ) and fix that the points of the positive real axis are mapped onto the zero meridian $\lambda = 0$ , then the polar coordinates (argument and modulus) $\theta$ and $r$ of $P$ in the mapping (1) are connected with the geographic coordinates of $P'$ by the equations $$\theta \;\equiv\; \lambda \!\pmod{2\pi}, \quad r \;=\; 2R\tan\left(\frac{\varphi}{2}+\frac{\pi}{4}\right),$$ as is easily checked. One can also state that the distance $h$ of $P'$ from the plane is given by $$h \;=\; \frac{2Rr^2}{4R^2\!+\!r^2}.$$