R. Andrew Hicks, PhD, Drexel University

Abstract: We call a surface that appears undistorted when viewed in a curved mirror an {\em eigensurface}, and the mirror an {\em eigenmirror}. Such pairs are described by a first-order nonlinear partial differential equation of the form $ a_0 + a_{1}u_{x} + a_{2}u_{y} + a_{3}u_{x}u_{y} + a_{4}u_{x}^{2} + a_{5}u_{y}^{2} = 0$, where $a_{i}=a_{i}(x,y,u)$, which we call the {\em anti-eikonal equation}. We give examples of symbolic and numerical solutions, including pairs that are geometrically congruent. Ray tracing simulations are included that visually confirm the unusual properties of these surfaces.