  We characterize subgroups of the mapping class group that stabilize a
Teichmueller disk in terms of ellipses and strips that are immersed in the
associated translation surface. In particular, we show that the space of
immersed ellipses/strips that meet at least three cone points is naturally a
(non-manifold) 2-dimensional cell complex. The topology of this complex and the
geometry of its 0-cells determine the translation surface and its affine
diffeomorphism group (up to the kernel of the differential).
