A common "trick" used in the study of polygonal billiards is to unfold the trajectory.
I'm interested in billiards which may be bounded in one direction, and unbounded in another. For instance, consider billiards in a square with two opposite edges identified. Aside from the particles departing at an angle equal to $\pm \frac{\pi}{2}$ (or $0, \pi$ depending on which sides are identified), all trajectories will eventually reach one of the identified sides. Does it make any sense to unfold the trajectory in situations like this?
Doing some sketches on paper, this seems perfectly possible (up until the trajectory reaches an identified edge). But I don't know what it means to tile the plane with these objects. For comparison, if we unfold billiards in a square, we end up with a trajectory on the flat torus $\mathbb{T}^2$.