Curvilinear coordinate systems and periodic coordinates

[mnb96]

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.

 

[Ben Niehoff]

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.

 

[mnb96]

Ok! thanks for your answer Ben.

 

[Bacle2]

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

 

[blue_raver22]

There are several families of curvilinear coordinate systems in 2d that have periodic coordinates, such as polar coordinates, cylindrical coordinates, and spherical coordinates. These coordinate systems are commonly used in physics and engineering to describe the position and motion of objects in space. The periodic nature of the angle coordinates allows for a more intuitive and efficient representation of rotational or circular motion. Additionally, these coordinate systems often have simpler mathematical expressions for certain physical phenomena, making them useful in various applications. Overall, the use of curvilinear coordinate systems with periodic coordinates can greatly aid in the understanding and analysis of complex systems.