For all $(t,\theta)\in\mathbb{R}_+\times[0,2\pi]$, define
$$F(t,\theta) = \begin{pmatrix} \frac{1}{\sqrt{3}}\operatorname{sh}(t)\cos(\theta) \\ \frac{1}{\sqrt{3}}\operatorname{sh}(t)\sin(\theta) \\ \operatorname{ch}(t) \end{pmatrix}.$$
Calculate, for all $\theta\in[0,2\pi]$, the hyperbolic length of the path
$$\begin{array}{rcl} \gamma:[0,b] & \rightarrow & \mathcal{H} \\ t & \mapsto & F(t,\theta). \end{array}$$