
> 
> "A *translation surface* is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence."
> 
> 
> 


I take that succinct (and not fully precise) definition from a paper by
Lelievre & Weiss.1


My question is whether the computational complexity is known
for recognizing when
two different finite collection of polygons, together with edge identifications,
represent the same translation surface. Say the total number of edges in the collections is at most $n$.
Is it even known to be decidable?




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1Lelievre, Samuel, and Barak Weiss. "Translation surfaces with no convex presentation." [arXiv:1306.3606](http://arxiv.org/abs/1306.3606) (2013).



Complexity of recognizing equivalent translation surfaces
Decidable? A translation surface is just a (special kind of) a singular Euclidean surface. If you define it as a union of polygons as above, you can start with a triangulation (by triangulating each polygon), then get a canonical (Delaunay) triangulation (with respect to the singularities) by edge flipping, then the two surfaces are the same if and only if the triangulations are the same. This is fast polynomial time.


