  A translation surface is given by polygons in the plane, with sides
identified by translations to create a closed Riemann surface with a flat
structure away from finitely many singular points. Understanding geodesic flow
on a surface involves understanding saddle connections. Saddle connections are
the geodesics starting and ending at these singular points and are associated
to a discrete subset of the plane. To measure the behavior of saddle
connections of length at most $R$, we obtain precise decay rates as $R\to
\infty$ for the difference in angle between two almost horizontal saddle
connections.
