I would say that the most general version of flat geometry consists of a bunch of polygons …

I just want to point out that a "bunch" could be infinite. I don't want to support infinite surfaces in libflatsurf.

As a general comment, I don't think we loose any generality if we assume that all the polygons are triangles. (We need to store the triangulation map in sage-flatsurf to recover the polygons but I think that can be arranged.)

If we keep everything triangulated, then supporting more general types of surfaces comes down to having some optional gluing information that records how half edge e is glued to half edge -e.

@wphooper wrote at flatsurf/sage-flatsurf#160:

Perhaps slope coordinates would be useful for libflatsurf?

Currently, we store a vector for each half edge in a triangulated surface. I don't understand how storing slopes makes much of a difference in practice but I think I did not fully understand what you were proposing.

I imagine that the only thing we really need to do here is to drop the requirement that the vector associated to a half edge v(e) satisfies that v(-e)=-v(e) for the corresponding half edge -e.

Indeed, a flat structure would associate to each edge a (linear, projective, ...) transformation m(e) such that m(e) v(e) = -v(-e). For a translation structure we have m(e) = identity. In sage-flatsurf we simplified the possible structures as m(e) is always the unique similarity m(e) mapping v(e) to -v(-e).

As discussed today, we probably just want to replace the underlying OddHalfEdgeMap with something more general. A structure that does caching of - but (virtually?) resolves what vector(-halfEdge) when there is a cache miss.

For the data structure it might be important to keep in mind that for each kind of surface there is a hidden subgroup of scalar multiplicaion z -> a z which is

• the trivial group for TranslationSurface
• {+/-1} for HalfTranslationSurface
• R_{> 0} for DilationSurface
• R \ {0} for HalfDilationSurface
• {k-th root of unity} for k-differentials (or RationalConeSurface in sage-flatsurf)
• {complex number of modulus one} for ConeSurface
• anything for SimilaritySurface

For each half edge, one has to store an element in this group (which is a complex number in general). For translation surfaces we just not store anything since this is always the identity. However, for any other kind of surfaces it is required to have this extra bit of information.