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 $$P'. The mapping $$ P \mapsto P' is a bijection between the closed complex plane and the sphere. Especially, the origin is mapped onto the South Pole and $$\infty onto the North Pole.

If we equip the sphere with geographic coordinates, the longitude $$\lambda ($$-\pi < \lambda \leq \pi) and the latitude $$\varphi ($$-\frac{\pi}{2} \leq \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 \PMlinkescapetext{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 $$h \;=\; \frac{2Rr^2}{4R^2\!+\!r^2}.$$