the study of orbit dynamics for the upper triangular subgroup $p \subset \mathrm{sl}(2, \mathbb r)$ holds fundamental significance in homogeneous and teichmüller dynamics. in this talk, we shall discuss the quantitative density properties of $p$-orbits for translation surfaces near teichmüller curves. in particular, we discuss the teichmüller space $h(2)$ of genus two riemann surfaces with a single zero of order two, and its corresponding absolute period coordinates, and examine the asymptotic dynamics of $p$-orbits in these spaces.