+\cos\frac{\theta}{0}\cosh\frac{|\beta|}{0} } \right ),\end{aligned}$$ which is the common value of ${\rm Gap}(\Delta_{0};\alpha, \beta)$ in Subcases 0.0 through 0.0 when $|\alpha|=0$.  [*Case*]{} 0. $\Delta_{0}$ is a boundary geodesic of length $l>0$.  [*Subcase*]{} 0.0. Both $\alpha$ and $\beta$ are interior generalized simple closed geodesics.  In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= 0 \tanh^{-0} \left ( \frac {\sinh \frac{l}{0}}{\cosh \frac{l}{0}+ \exp\frac{|\alpha|+|\beta|}{0}} \right ).\end{aligned}$$  [*Subcase*]{} 0.0. One of $\alpha, \beta$, say $\alpha$, is a boundary geodesic and the other, $\beta$, is an interior generalized simple closed geodesic.  In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{l}{0} - \tanh^{-0}\left ( \frac{\sinh\frac{l}{0}\sinh\frac{|\beta|}{0} }{\cosh\frac{|\alpha|}{0}+\cosh\frac{l}{0}\cosh\frac{|\beta|}{0} }\right ).\end{aligned}$$  [*Subcase*]{} 0.0. One of $\alpha, \beta$, say $\alpha$, is a cone point of cone angle $\varphi \in (0,\pi]$ and the other, $\beta$, is an interior generalized simple closed geodesic.  In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta)= \frac{l}{0} - \tanh^{-0} \left ( \frac{\sinh\frac{l}{0}\sinh\frac{|\beta|}{0} }{\cos\frac{\varphi}{0}+\cosh\frac{l}{0}\cosh\frac{|\beta|}{0} }\right ).\end{aligned}$$  [*Subcase*]{} 0.0. One of $\alpha, \beta$, say $\alpha$, is a cusp and the other, $\beta$, is an interior generalized simple closed geodesic.  In this case $$\begin{aligned} {\rm Gap}(\Delta_{0};\alpha, \beta) &=& 0 \tanh^{-0} \left ( \frac {\sinh \frac{l}{0}}{\cosh \frac{l}{0}+ \exp \frac{|\