Ray in a linear-speed medium ($(x-h{)}^2+y^2=R^2$)

For light speed $c(y)=y>0$, the travel-time integrand is $\sqrt{1+y^{\prime 2}}/y$. The Beltrami identity gives $y\sqrt{1+y^{\prime 2}}=R$, 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, $ds/y\ge |dy|/y$ and the integral diverges. These semicircles are therefore ideal-endpoint hyperbolic geodesics, not finite-time optical minimizers joining zero-speed endpoints.