Flat-cylinder travel metric for sine-latitude speed ($z=\mathrm{arsinh}(\cot \theta )$)

Motion on a sphere with speed proportional to $\sin \theta$ has travel-time line element proportional to $\sqrt{d{\theta }^2/{\sin }^2\theta +d{\varphi }^2}$. With $z=\mathrm{arsinh}(\cot \theta )$, this becomes the flat cylinder metric $\sqrt{d{z}^2+d{\varphi }^2}$. Shortest lifted paths are straight lines. Nonvertical lines give $\cot \theta =\sinh (A\varphi +B)$; vertical lines are meridians of constant $\varphi$. The poles lie at infinite travel-time distance.