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# Riemann sphere

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:

$\displaystyle\hat{\mathbb{C}}\backslash\{\infty\}\rightarrow\mathbb{C}:z\mapsto z$ |

and

$\displaystyle\hat{\mathbb{C}}\backslash\{0\}\rightarrow\mathbb{C}:z\mapsto% \frac{1}{z}$ |

Any rational function on $\hat{\mathbb{C}}$ has a unique smooth extension to a map $\hat{p}:\hat{\mathbb{C}}\rightarrow\hat{\mathbb{C}}$.

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 $P^{{\prime}}$. The mapping

$\displaystyle P\mapsto P^{{\prime}}$ | (1) |

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 $\lambda$ ($-\pi<\lambda\leqq\pi$) 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 $P^{{\prime}}$ by the equations

$\theta\;\equiv\;\lambda\!\;\;(\mathop{{\rm mod}}2\pi),\quad r\;=\;2R\tan\left(% \frac{\varphi}{2}+\frac{\pi}{4}\right),$ |

as is easily checked. One can also state that the distance $h$ of $P^{{\prime}}$ from the plane is given by

$h\;=\;\frac{2Rr^{2}}{4R^{2}\!+\!r^{2}}.$ |

## Mathematics Subject Classification

32C15*no label found*

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