 Solution

For $R=1$ and $z_1=0$, the shortest path on a circular cylinder has lengthIts graph is triangular, with endpoint values zero and maximum $\pi$ at ${\varphi }_1=\pi$. In particular, increasing the angle to a full turn brings the endpoint back to the starting point.

For $R=1$ and $z_1=1$, the corresponding length isThis graph has endpoint values one, a maximum $\sqrt{1+{\pi }^2}$ at $\pi$, and symmetry about $\pi$. Each half is smooth, with horizontal slope at the outer endpoint; at $\pi$ the two halves meet at a corner, because the minimizing winding direction changes.