Angle Parametrization of Teichmüller space and hyperelliptic surfaces

Let $S_g$ be a closed orientable surface of genus $g\ge 2$, and let ${\mathcal{T}}_g$ be the Teichmüller space of $S_g$. Let ${\mathcal{H}}_g$ denotes the space of all hyperelliptic surfaces of genus $g$. For $g\ge 3$, we have proved that ${\mathcal{T}}_g$ can be parametrized by $6g-5$ angle parameters. We also prove that for $g\ge 2$, ${\mathcal{H}}_g$ can be parametrized by $4g-2$ angle parameters.