Eigensurfaces of eigenmirrors

Typically, if an observer gazes at a curved reflector, the objects in it will appear to be distorted. We show here that for some mirrors there exist surfaces that do not appear distorted when viewed from a prescribed location. We call such mirrors eigenmirrors and the surfaces eigensurfaces. We first give an analysis of the rotationally symmetric case and verify our work with simulations. In the general three-dimensional (3D) case, if the mirror is given, then one does not expect an eigensurface to exist. On the other hand, if we are given two viewpoints and a correspondence between the ray bundles emanating from each point, and we treat both the eigenmirror and the eigensurface as unknowns, then the problem reduces to solving a first-order nonlinear partial differential equation. We derive this partial differential equation in the 3D case and examine one example in detail.