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I have the following problem where I have two groups of transformations [itex]R_\alpha[/itex] (rotation) and [itex]S_\lambda[/itex] (scaling) acting on the plane, so that the orbits of any arbitrary pointx=(x_{0},y_{0}) under the actions of [itex]S_\lambda[/itex] and [itex]R_\alpha[/itex] are known (in the former case they are straight lines from the origin; in the latter case they are concentric circles with center in0).

From this information, how can I "build" a system of curvilinear coordinates, where the coordinates are exactly the parameters (α,λ) of the transformations?

PS: I know that the answer leads to the log-polar coordinates, but I need a procedure to arrive at it.

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# Curvilinear coordinates from orbits

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