Ellipses in translation surfaces
We study the topology and geometry of the moduli space of immersions of ellipses into a translation surface. The frontier of this space is naturally stratified by the number of ‘cone points’ that an ellipse meets. The stratum consisting of ellipses that meet three cone points is naturally a two dimensional (non-manifold) polygonal cell complex. We show that the topology of this cell-complex together with the eccentricity and direction of each of its vertices determines the translation surface up to homothety. As a corollary we characterize the Veech group of the translation surface in terms of automorphisms of this polygonal cell complex.
Broughton, Sean A. and Judge, Chris, "Ellipses in translation surfaces" (2012). Faculty Publications - Mathematics. 3.