Lateral Magnification in Single Spherical Surface

For an extended object $OO'$ refracted at a single spherical surface, forming an image $II'$, the lateral magnification $m=II'/OO'$ can be found using a ray from $O'$ aimed directly at the centre of curvature $C$ -- since this ray meets the surface exactly along its own normal, it passes through completely undeviated, creating the similar triangles $\triangle COO'\sim\triangle CII'$. Working through this similar-triangle geometry with the sign convention gives $m=\dfrac{R-v}{R-u}$; substituting from the refraction-at-a-spherical-surface equation to eliminate $R$ gives the equivalent, often more directly useful form purely in terms of refractive indices and distances, $\boxed{m=\dfrac{n_1v}{n_2u}}$. As with mirrors and thin lenses, the sign of …