For all $(t,\theta)\in\mathbb{R}_+\times[0,2\pi]$, define
$$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$
Show that $F$ takes values in $\mathcal{H}$ and that $F:\mathbb{R}_+\times[0,2\pi]\rightarrow\mathcal{H}$ is surjective.