# Curvilinear coordinates from orbits

Hello,

I have the following problem where I have two groups of transformations $R_\alpha$ (rotation) and $S_\lambda$ (scaling) acting on the plane, so that the orbits of any arbitrary point x=(x0,y0) under the actions of $S_\lambda$ and $R_\alpha$ are known (in the former case they are straight lines from the origin; in the latter case they are concentric circles with center in 0).

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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quasar987
Homework Helper
Gold Member
The rotation group is diffeomorphic to the circle S^1, while the scaling group is diffeomorphic to the real line. It is clear that if you take the standard coordinates θ on S^1 and r on R^1, then the map

(-pi,pi) x (0,infty)-->R x S --->R^2
(θ,r)------------->(Rθ,Sr)--->(Rθ$\bullet$Sr$\bullet$(1,0)

is a diffeomorphism onto R²\{the x<=0 ray} commonly refered to as "the polar coordinates on R^2"! ("$\bullet$" stands for the group action)

But there is no Log anywhere in this construction so perhaps this is not what you are after?

Hi quasar987,

thanks! your explanation was pretty clear. I simply didn't think of making such observations in terms of diffeomorphisms. And by the way you are right, the log-polar is not important in this case. I was just a bit confused.

One more thing I would be very interested to know. If I replace the rotation and scaling groups with two different and not well-known groups of transformations, can we do the same? What is the procedure to find a system of curvilinear coordinates in the plane, when we know the orbits of the points under the action of the two groups?

quasar987