## curvilinear coordinate systems and "periodic" coordinates

Hello,

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

$$\rho = \rho(x,y)$$
$$\tau = \tau(x,y)$$

Sometimes, it can happen that one of the coordinates, say $\tau$, 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.
Thanks.
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 Recognitions: Science Advisor 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.