= Refraction at a spherical surface
{title2=$n_1/u+n_2/d=(n_2-n_1)/R$}
With incident <light> travelling to the right, use positive object distance $u$ to the left, image distance $d$ to the right, and curvature radius $R$ toward a right-hand centre. For a ray hitting at small height $Y$, its incoming inclination, outgoing downward inclination and inward-normal downward inclination are $\alpha\simeq Y/u$, $\beta\simeq Y/d$ and $\theta\simeq Y/R$. The <paraxial approximation> to <Snell's law> gives $n_1(\alpha+\theta)=n_2(\theta-\beta)$. Cancelling $Y$ proves the title equation. Virtual conjugates and other centre orientations require corresponding signed distances. Applying the relation at both faces of a <thin lens> proves the <lensmaker's equation>.
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