Riemann sphereRiemannSphere2003-07-17 12:38:362009-02-02 12:55:24Definitiongeographic coordinateslongitudelatitudecompactification% this is the default PlanetMath preamble. as your knowledge
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\newtheorem{cor}{Corollary}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:
\begin{align*}
\hat{\mathbb{C}}\backslash\{\infty\}\rightarrow\mathbb{C}:z\mapsto z
\end{align*}
and
\begin{align*}
\hat{\mathbb{C}}\backslash\{0\}\rightarrow\mathbb{C}:z\mapsto \frac{1}{z}
\end{align*}
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 $P'$.\, The mapping
\begin{align}
P \mapsto P'
\end{align}
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.
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If we equip the sphere with {\em geographic coordinates}, the {\em longitude} $\lambda$ ($-\pi < \lambda \leqq \pi$) and the {\em 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 \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
$$h \;=\; \frac{2Rr^2}{4R^2\!+\!r^2}.$$