Looking for a coordinate system

  • #1
coolnessitself
35
0
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 [tex](x,y)[/tex] can always be contained in [tex][-a, a], [-b, b][/tex], if the following are true. Along y, the plane wraps up on itself at b, so that [tex](x,-b)=(x,b)[/tex]. For x, if the state travels beyond a, it goes back to -a, but y will also be shifted, so that [tex](a,y) = (-a,y+b)[/tex]. Note that this shift might also cause a jump in y from [tex](x,-b)=(x,b)[/tex].
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 [tex](a,y) = (a,y+b) [/tex], which toroidal coordinates wouldn't allow(?). I'm not really sure what to search for. Suggestions?
 

Answers and Replies

  • #2
fresh_42
Mentor
Insights Author
2021 Award
17,250
17,255
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.
 

Suggested for: Looking for a coordinate system

  • Last Post
Replies
2
Views
314
Replies
4
Views
274
Replies
8
Views
624
Replies
3
Views
1K
Replies
0
Views
1K
Replies
5
Views
1K
Replies
15
Views
724
Replies
3
Views
394
Top