h |\delta_{\beta}| = \frac{\cos\frac{\varphi}{0}+\cos\frac{\theta}{0}\cosh\frac{|\beta|}{0}} {\sin\frac{\theta}{0}\sinh\frac{|\beta|}{0}}.\end{aligned}$$ Hence $$\begin{aligned} \label{eqn:0.0.y} \tan y = \frac{0}{\sinh |\delta_{\beta}|} = \frac{\sin\frac{\theta}{0}\sinh\frac{|\beta|}{0}} {\cos\frac{\varphi}{0}+\cos\frac{\theta}{0}\cosh\frac{|\beta|}{0}}.\end{aligned}$$ Thus in this case we have $$\begin{aligned}  {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{\theta}{0} - \tan^{-0} \left ( \frac{\sin\frac{\theta}{0}\sinh\frac{|\beta|}{0} }{\cos\frac{\varphi}{0}+\cos\frac{\theta}{0}\cosh\frac{|\beta|}{0}} \right ).\end{aligned}$$  [*Case*]{} 0. $\Delta_{0}$ is a boundary geodesic of length $l>0$.  In this case the width of the main gap determined by $\gamma$ is the distance between $\gamma_{\alpha}$ and $ \gamma_{\beta}$ along $\Delta_{0}$