Reflections in an Elliptical Region

A ray is reflected multiple times inside an ellipse. The black dots are the two foci of the ellipse. Studying billiards in simple geometric domains is a fascinating area of research. Much nontrivial mathematics results from investigating billiards, despite its simplicity. Closed periodic paths are especially relevant in Gutzwiller's trace formula (and its extensions); they comprise a beautiful piece of physics that links classical and quantum mechanics.