# 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?

## Answers and Replies

Mentor
2021 Award
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.