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Curvilinear coordinate systems and periodic coordinates

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Oct13-11, 08:40 AM
P: 625

we can consider a generic system of curvilinear coordinates in the 2d plane:

[tex]\rho = \rho(x,y)[/tex]
[tex]\tau = \tau(x,y)[/tex]

Sometimes, it can happen that one of the coordinates, say [itex]\tau[/itex], represents an angle, and so it is "periodic". This clearly happens for example, in polar coordinates.

What are the families of curvilinear coordinates systems in 2d, that have one or more coordinates that are angles?

I hope the question is not too vague to be answered.
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Ben Niehoff
Oct13-11, 12:22 PM
Sci Advisor
P: 1,591
Any coordinate system whose coordinate lines form closed curves is going to have at least one periodic coordinate. The periodic coordinate doesn't have to represent an "angle", necessarily. For example, in elliptical coordinates, one of the coordinates might represent a parameter for traveling around an ellipse.

One example that can have two periodic coordinates is the bipolar system.
Oct14-11, 12:32 PM
P: 625
Ok! thanks for your answer Ben.

Oct17-11, 09:39 PM
Sci Advisor
P: 1,169
Curvilinear coordinate systems and periodic coordinates

Or spherical coordinates, or coordinates in S^n, or on any subspace that "turns on itself" , or is closed.

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