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Mathematics-Online lexicon:

Spherical Coordinates

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A point $ P=(x,y,z)$ 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 $ x$ -axis and the projection of $ \overline{OP}$ onto the $ xy$ -plane, and the angle $ \vartheta\in[0,\pi]$ formed by $ \overline{OP}$ and the $ z$ -axis.


We have

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


$\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 $ x$ and $ y$ :

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

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  automatically generated 3/17/2011