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What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by

$$x_1=r \cos \theta$$ $$x_2=r \sin \theta$$

For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by

$$x_1=r \sin \theta \cos \varphi$$ $$x_2=r \sin \theta \sin \varphi$$ $$x_3=r \cos \theta$$

So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n>3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks.

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Hyperspherical coordinates are on Wikipedia. – anon Aug 9 '11 at 19:22
up vote 9 down vote accepted

These are hyperspherical coordinates. You can see an example of them being put to use in this answer.

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That's really funny, because I happen to read that answer just yesterday. And I was going to reference it for this question too. – mixedmath Aug 9 '11 at 19:29

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