= Flat-cylinder travel metric for sine-latitude speed
{title2=$z=\operatorname{arsinh}(\cot\theta)$}
Motion on a sphere with speed proportional to $\sin\theta$ has travel-time line element proportional to $\sqrt{d\theta^2/\sin^2\theta+d\phi^2}$. With $z=\operatorname{arsinh}(\cot\theta)$, this becomes the flat cylinder metric $\sqrt{dz^2+d\phi^2}$. Shortest lifted paths are straight lines. Nonvertical lines give $\cot\theta=\sinh(A\phi+B)$; vertical lines are meridians of constant $\phi$. The poles lie at infinite travel-time distance.
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