Focal Length and the Mirror Formula

The relation between the object distance $u$, the image distance $v$ and the focal length $f$ of a spherical mirror is derived from the geometry of a single ray leaving the tip of an object on the axis, striking the mirror close to the pole, and reflecting through the focus, together with a second ray through the centre of curvature. Using similar triangles formed by this ray, the object, the image and the axis, and applying the New Cartesian sign convention throughout, the algebra reduces to the compact mirror formula $\frac{1}{v}+\frac{1}{u}=\frac{1}{f}.$ This one formula, together with the sign convention of the previous sub-section, correctly predicts the image for every case: a concave mirror forms a real, inverted, diminished image of an object beyond its centre of curvature; a real, inverted, same-size image of an object exactly at the centre of curvature; a real, inverted, magnified image of an object between the centre of curvature and the focus; no image at all (rays emerge parallel) for an object exactly at the focus; and a virtual, erect, magnified image, behind the mirror, for an object between the focus and the pole. A convex mirror, whatever the object distance, always forms a virtual, erect, diminished image behind the mirror, which is exactly why a convex mirror (never a concave one) is used as a vehicle's rear-view or a shop's wide-angle security mirror -- it always shows an upright image and packs a wider field of view into a smaller mirror. The lateral (linear) magnification produced by a spherical mirror, the ratio of image h …