Mathematics-Online lexicon: Spherical Coordinates

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	Mathematics-Online lexicon:
	Spherical Coordinates

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overview

A point

can be represented by the following 3 values: the point's distance from the origin $r=\vert\overline{OP}\vert$ , the angle $\varphi\in[0,2\pi)$ formed by the

-axis and the projection of $\overline{OP}$ onto the

-plane, and the angle $\vartheta\in[0,\pi]$ formed by $\overline{OP}$ and the

-axis.

$\includegraphics[width=.6\linewidth]{kugelkoordinaten_en}$

We have

$\displaystyle x = r\cos(\varphi)\sin(\vartheta),\quad y = r\sin(\varphi)\sin(\vartheta),\quad z = r\cos(\vartheta)$

and

$\displaystyle r = \sqrt{x^2+y^2+z^2},\quad \varphi = \arctan(y/x),\quad \vartheta = \arctan(\sqrt{x^2+y^2}/z)\,,$

but you have got to select the appropriate branch of $\arctan$ depending on the signs of

and

$\begin{displaymath} \begin{array}{\vert l\vert l\vert} \hline x\ge 0 \wedge y... ... \le 0 & \vartheta \in [\pi/2,\pi] \\ \hline \end{array} \end{displaymath}$

[Examples] [Links]

automatically generated 3/17/2011