Exact mirror equation via Berry's caustic touching theorem: plane and spherical mirrors

In this work, we assume that in free space we have an observer, a smooth mirror, and an object placed at arbitrary positions. The aim is to obtain, within the geometrical optics approximation, an exact set of equations that gives the image position of the object registered by the observer. The general results are applied to plane and spherical mirrors, as an application of the caustic touching theorem introduced by Berry; the regions where the observer can receive zero, one, two, three, and one circle of reflected light rays are determined. Furthermore, we show that under the restricted paraxial approximation, that is, when sin⁡ψ≈ψ and cos⁡ψ≈1, the exact set of equations provides the well-known mirror equation.