Angular variables have an interesting property in that for some applications, e.g. angle at which the light beam of a lighthouse is pointing, we expect some functions of that variable to not depend on how many times the beam has rotated around its axis. In other words $f(x) = f(x+n \ 2 \pi) \ \ \forall n \in \mathbb{Z}$.
It seems to me that in such cases the real line is not a convenient representation of this kind of variables. A closed curve would be more natural. However, in my limited perspective of mathematics, a curve would need to live in a space of at least 2 dimensions even if the angle space is 1-dimensional. It is disconcerting for me to need 2 dimensions to represent something that is clearly 1-dimensional.
I want to learn more about this kind of spaces but I'm a bit at a loss about how to describe the problem and try to undersand better its geometry. What branch of mathematics studies this sort of problems? Is this a case of non-Euclidean geometry?