Abstract

Flat surfaces with cone points are central objects in Teichmueller theory. Such surfaces have canonical polygonal cell decompostions called Delaunay partitions. For the generic flat surface, this decomposition is a triangulation. The natural action of SL(2, R) on the flat surfaces induces diagonal flipping of the triangulations. We will show how diagonal flipping can be understood in terms of immersed ellipses that meet 5 or more cone points.