Angle Parametrization of Teichm\"uller space and hyperelliptic surfaces

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