Snell's law

Snell's law follows from conservation of tangential wavenumber across a planar interface. If angles are measured from the interface normal, it reads $n_1\sin {\theta }_1=n_2\sin {\theta }_2$.

For a circular index-$n$ tank, an eye at $(\sqrt{2}R,0)$, and a swim diameter $x=0$, trace outward single-pass rays through $P=R(\cos \varphi ,\sin \varphi )$, with $|\varphi |\le \pi /4$. Backward refraction gives the dimensionless intercept $y=\sin \varphi -\cos \varphi \tan (\varphi -i)$, where $\sin i=\sqrt{2}\sin \varphi /[n\sqrt{3-2\sqrt{2}\cos \varphi }]$. The visible fraction is the largest positive intercept. For $n=1.33$ it is approximately $0.75744$, attained before the grazing outgoing ray. The limiting pair of stationary optical paths merges at a caustic.