Fermat's Principle, Hypothetical Velocity, Lorentz Transformations and a Moving Mirror

In (1) we showed that Fermat’s minimum time approach is equivalent to a hypothetical velocity approach for two rays. This holds for Snell’s law, a fixed mirror and a mirror moving at constant velocity v. The hypothetical velocity H forms a hypotenuse and the two rays make angles AA and BB with a normal (y axis). Hsin(AA) = v(relative A) and Hsin(BB) = v(relative B) where v relative is the relative velocity in the direction of the ray. For A it is 1-v cos(AA) and for B 1+v cos(BB) for c=1. In (2) the author notes that Fermat’s principle yields the same result Einstein obtained in a 1905 paper using Lorentz transformations. In this note, we wish to show how the hypothetical velocity approach (and hence Fermat’s principle) is related to Lorentz transformations for a moving mirror