  We show that the limiting gap distribution of slopes of saddle connections on
a Veech translation surface is piecewise real-analytic with finitely many
points of non-analyticity. We do so by taking an explicit parameterization of a
Poincar\'{e} section to the horocycle flow on
$\text{SL}(2,\mathbb{R})/\text{SL}(X,\omega)$ and establishing a key finiteness
result for the first return map under this flow.
