Past exam of the mathematical methods II course of the University of Oxford / 2015 resit / 10 / d

A ray leaving focus $A$ and reaching a boundary point $P$ has incoming unit direction ${\mathbf{u}}_A$. By the law of reflection, its normal component reverses and its tangential component stays fixed. The result just derived shows that the reflected direction is $-{\mathbf{u}}_B$, pointing exactly from $P$ to focus $B$. Thus all such in-plane rays from one focus converge to the other after one reflection, in the ideal geometrical optics model.

Moreover, every reflected path has the same total length $AP+PB=s$. In a uniform medium their propagation times are equal as well, so different reflection points do not introduce a geometrical path-length spread. Finite wavelength, surface errors and incomplete reflecting coverage limit a real focus; the ideal argument concerns rays in the plane of the ellipse. It does not assert that an arbitrary three-dimensional surface with only one elliptical section focuses every out-of-plane ray.