Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a non-vanishing holomorphic quadratic differential on $X$. Denote by $\mathbb{H}$ the universal cover of $X$ and by $\widetilde q$ the pullback of $q$ to $\mathbb{H}$. The space $\mathbb{H}$ endowed by the flat singular metric $|\widetilde q|$ is a Cat(0) metric space.
For every $x,y,z\in (\mathbb{H},|\widetilde q|)$ denote by $\Delta$ the corresponding geodesic triangle (i.e. the subset of $(\mathbb{H},|\widetilde q|)$ composed by $x,y,z$ and the three geodesics connecting them). It seems to me that it makes sense to talk about the interior of $\Delta$: it is the bounded region of $(\mathbb{H},|\widetilde q|)$ delimited by $\Delta$ (and of course if can by empty). Denote by $\Delta'$ the union of $\Delta$ with its interior.
Is there a nice characterization of such $\Delta'$? In particular, is it true that they are always isometric to polygons with possibly one dimensional components (it seems to me that there can not be zeroes of $\widetilde q$ in the interior of $\Delta$)?