Looking for a coordinate system

[coolnessitself]

I'm working with a cartestian system that has certain periodic properties I'd like to exploit with a new coordinate system, but I don't know one that would work. The trajectory of the state of the system is symmetric across non-adjacent squares (ie a checkerboard of sorts), so that $$(x,y)$$ can always be contained in $$[-a, a], [-b, b]$$, if the following are true. Along y, the plane wraps up on itself at b, so that $$(x,-b)=(x,b)$$. For x, if the state travels beyond a, it goes back to -a, but y will also be shifted, so that $$(a,y) = (-a,y+b)$$. Note that this shift might also cause a jump in y from $$(x,-b)=(x,b)$$.
So wrapping in y means I curl my cartesian into a cylinder, and the wrapping in x might change the cylinder into a torus, but it would have to be twisted somehow so that $$(a,y) = (a,y+b)$$, which toroidal coordinates wouldn't allow(?). I'm not really sure what to search for. Suggestions?

 

[fresh_42]

Coordinates with periodicity aren't unique anymore and so no longer coordinates. Mathematicians solve this problem by using local coordinates and patch them, i.e. consider your surface as a manifold with an atlas.