= Optical rays in a reciprocal-height refractive index
{title2=$\mu(y)=1/y,\quad (x-c)^2+y^2=R^2$}
For the <refractive index> $\mu=1/y$ on the upper half-plane, <Fermat principle> and the <Beltrami identity> give $1/[y\sqrt{1+(dy/dx)^2}]=1/R$. The nonvertical rays are arcs of <circles> centered on $y=0$; vertical lines are the remaining rays. These are also <geodesics> of the <Poincaré half-plane model>. A ray entering the strip $0<y\leq1$ toward decreasing $y$ at angle $\theta$ to the vertical has limiting horizontal displacement $\tan(\theta/2)$, at most one. The line $y=0$ is approached only after infinite <optical path length> because the <refractive index> diverges there.
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