= Ray in a linear-speed medium
{title2=$(x-h)^2+y^2=R^2$}
For light speed $c(y)=y>0$, the travel-time integrand is $\sqrt{1+y'^2}/y$. The <Beltrami identity> gives $y\sqrt{1+y'^2}=R$, so nonvertical rays are semicircles centred on $y=0$. Boundary endpoints at $y=0$ have infinite travel time: along any rectifiable path entering positive height, $ds/y\geq |dy|/y$ and the integral diverges. These semicircles are therefore ideal-endpoint <hyperbolic geodesics>, not finite-time optical minimizers joining zero-speed endpoints.
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