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Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call it $r$, and its distance from $x$, call it $L$, and a binary choice, say $+$ or $-$. How does one transform this co-ordinate system with the Cartesian coordinate system, and what is the integrating factor (i.e. area element)?

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en.wikipedia.org/wiki/Two-center_bipolar_coordinates This is what you are looking for. –  progressiveforest Aug 12 '12 at 18:41
    
That's great, thanks! Is there anything like that for higher dimensions? (Again parametrized by the distance to two centers, with the other parameter being an element $\xi \in S^{d-2} $ ? –  Robert Aug 12 '12 at 18:53
    
@Robert: why not? You've pretty much defined it yourself. –  tomasz Aug 12 '12 at 18:55
    
True, but I'd like to know if the volume element (a.k.a. integrating factor) is worked out anywhere of arbitrary dimensions. –  Robert Aug 12 '12 at 19:24
    
I suppose you'd want $n$ distances from $n$ points in $\mathbb{R}^n$ to pin down location of a point to two possible points. –  progressiveforest Aug 13 '12 at 8:35

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