Mariş’s Theorem

During a literature search (to answer my question concerning symmetries of “ground states” in variational problem, I came across a very nice theorem due to Mihai Mariş. The theorem itself is, more than anything else, a statement about the geometry of Euclidean spaces. I will give a rather elementary write-up of (a special case of) the theorem here. (The proof presented here can equally well be applied to get the full strength of the theorem as presented in Maris’s paper; I give just the special case for clarity of the discussion.)

Snell’s law and geometry

Today’s post is somewhat inspired by this question on math.stackexchange. To begin with, recall Snell’s Law from geometric optics. It gives a rule for describing the propagation of light between two media of different indices of refraction: namely that across an interface where on one side the index of refraction is and on the other side , the angles of incidence and refraction and , as measured from the normal to the interface, satisfies the relation

.

Here I pose two problems:

1. Given an observer situated above the interface, and a fish below the interface, find the “apparent position” of the fish according to the observer.
2. Given an observer situated above the interface, and the “apparent position” of a fish below the interface, find the actual position of the fish.

Without loss of generality we can assume that the interface is the -axis, and the observer is situated at the point in the plane.