Is there any 3D coordinate system that is periodic in two angles?

Question

In spherical coordinates the angle $\varphi = \arctan y/x$ (or $\theta$, whichever convention you prefer) is fully periodic, i.e. all values in the range $0$ to $2\pi$ are reached. However, the second angle, $\vartheta=\arctan \sqrt{x^2+y^2}/z$, is not periodic, ranging only from $0$ to $\pi$. Now I am wondering, is there also a coordinate system using two angles both of which have a periodic range? Or is there a proof that this is not possible?

Answer 1

Didn't understand your question to be honest, but how about light with its electric field's plane of oscillation along plane x=y in normal cartesian system. EDIT: On second thoughts that won't even be periodic. EDIT 2:- how about $\frac{x-y}{\sqrt{2}}=asint$

Answer 2

The question uses both notational conventions and defintions in a nonstandard way:

1. The polar angle $\arctan(y/x)$ is usually denoted $\theta$, not $\varphi$.
2. The azimuthal angle $\arctan(\sqrt{x^2+y^2}/z)$ is normally denoted $\phi$, not $\vartheta$.
3. The word "periodic" describes any function or phenomenon that repeats itself at fixed intervals. It is not correct to describe the azimuthal angle as "not periodic" because it ranges "only from $0$ to $\pi$" -- indeed that is fully periodic, and the period is $\pi$.

However, if for some reason you want to describe 3-dimensional space using two coordinates that have the same period, I suppose you could always just define a new coordinate, say $\psi$, by $\psi = 2\phi$. Then both $\theta$ and $\psi$ would range between $0$ and $2\pi$, with points in the $xy$-plane corresponding to $\psi = \pi$ and points on the unit sphere at the "South pole" corresponding to $\psi = 2\pi$. In this case $\psi$ would not really correspond to an angle measurement in a geometric sense, but it would have the mathematical properties that you are asking for.