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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