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"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?

1Lelievre, Samuel, and Barak Weiss. "Translation surfaces with no convex presentation." arXiv:1306.3606 (2013).

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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.

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  • $\begingroup$ Well, perhaps I should have given more background concerning the phrase "up to cut and paste equivalence," which I think makes it more complicated. But I admit to being uncertain. Perhaps your view is accurate... $\endgroup$
    – Joseph O'Rourke
    Commented Dec 22, 2015 at 20:59
  • $\begingroup$ @JosephO'Rourke I know what a translation surface is, and it is that which I describe. I have no idea why Lelievre and Weiss would say "up to cut and paste equivalence" vs "up to isometry". $\endgroup$
    – Igor Rivin
    Commented Dec 22, 2015 at 21:19
  • $\begingroup$ Thanks, Igor! The "cut and paste equivalence" threw me off. $\endgroup$
    – Joseph O'Rourke
    Commented Dec 22, 2015 at 21:22

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