Reflections in a moving mirror

Euclid's law of reflection, r = i, rests on three assumptions: Fermat's principle of least time, constancy of speed of light, and that the mirror is stationary. For a moving mirror, the corresponding law is known as the relativistic reflection law; however, it was first inferred prior to relativity. While modern derivations rely on Lorentz transformations or Maxwell's equations, the original derivation was an elegant application of geometric optics. In this paper, we develop a new geometric approach which shows that the relativistic reflection from a plane mirror is equivalent to the ordinary reflection from a certain hyperbolic mirror, and that the incident and reflected rays are the focal rays of the hyperbola. We demonstrate that incident concentric spherical waves reflect as non-concentric spherical waves, and thereby derive the Doppler effect formula. We conclude with a class project for students to derive a generalization of Snell's law of refraction for a particular type of a moving interface.