Refraction at a spherical surface ($n_1/u+n_2/d=(n_2-n_1)/R$)

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.