Curvilinear coordinates from orbits

[mnb96]

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.

 

[quasar987]

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)------------->(R_θ,S_r)--->(R_θ$\bullet$S_r$\bullet$(1,0)

is a diffeomorphism onto R²\{the x<=0 ray} commonly referred 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?

 

[mnb96]

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]

Well, in general, no, the analogue of the map we created for general group is not going to be a diffeomorphism. For instance, if instead of S^1 and R^1, you take two R^1's (both acting in the same way on R²), then we see fairly easily that the differential will be at every point non surjective. Similarly if we take two S^1's. The key for the above example to work was that the orbits are transverse.

 

[blue_raver22]

Hello,

Thank you for sharing your problem with me. It seems like you are trying to construct a system of curvilinear coordinates based on the orbits of points under rotation and scaling transformations. This is a very interesting problem and I will do my best to provide a helpful response.

To begin, let's consider the orbits of points under the rotation transformation R_\alpha. As you mentioned, these orbits are concentric circles with center at the origin. This suggests that we can use the distance from the origin, r, as one of our coordinates. The other coordinate, θ, can be the angle that the point makes with the positive x-axis. This is because as the point moves along the circle, the angle θ changes.

Next, let's look at the orbits of points under the scaling transformation S_\lambda. These orbits are straight lines from the origin. This suggests that we can use the distance from the origin, r, as one of our coordinates again. However, the other coordinate, φ, should be the angle between the line and the positive x-axis. This is because as the point moves along the line, the angle φ changes.

Now, if we combine these two sets of coordinates, we get (r,θ,φ). This is known as the log-polar coordinates, as you mentioned. To arrive at this system, we used the orbits of points under the transformations R_\alpha and S_\lambda.

I hope this helps you understand the procedure for constructing curvilinear coordinates from orbits. Let me know if you have any further questions or if I can be of any more assistance. Best of luck with your research!