= 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> of <light> $c(y)=c_0/\sqrt{1-ky}$, where $c_0>0$ and $1-ky>0$, the travel-time integrand in <Fermat's principle> is $c_0^{-1}\sqrt{(1-ky)(1+y'^2)}$. For nonzero $k$, the <Beltrami identity> gives
$$
\frac{1-ky}{1+y'^2}=1-ky_0,
$$
where $y_0$ is the turning height. The corresponding stationary rays are the <parabolas>
$$
y=y_0-\frac{k(x-x_c)^2}{4(1-ky_0)}.
$$
For equal-height endpoints $(\pm x_0,0)$ with $x_0>0$, symmetry gives $x_c=0$ and $(kx_0)^2=4ky_0(1-ky_0)$. Real turning heights therefore require $|k|x_0\leq1$, not merely a one-sided condition on signed $k$. When the inequality is strict, the two algebraic values are $y_0=[1\pm\sqrt{1-k^2x_0^2}]/(2k)$; only rays remaining inside the specified optical layer are admissible. Equality merges the two branches. At $k=0$, the medium has constant <speed> of <light> and the ray joining these endpoints is the straight horizontal segment; the formulas containing $1/k$ must be replaced by that limit.
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