= Parabolic light rays in a reciprocal-square-root speed profile
{title2=$y=y_0-\frac{k(x-x_c)^2}{4(1-ky_0)}$}
For speed $c(y)=c_0/\sqrt{1-ky}$, with $1-ky>0$, the elapsed-time integrand is $\sqrt{(1-ky)(1+(y')^2)}/c_0$. The <Beltrami identity> gives $(1-ky)/(1+(y')^2)=1-ky_0$, where $y_0$ is the turning height. Integration yields the displayed <parabola>. If the endpoints are $(\pm x_0,0)$, symmetry gives $x_c=0$ and $(kx_0)^2=4ky_0(1-ky_0)$. Hence $kx_0\leq1$, and the two algebraic turning heights are $[1\pm\sqrt{1-k^2x_0^2}]/(2k)$. Only heights lying within the specified optical layer are admissible.
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