Imaging properties of freeform quadric reflectors via the theory of eigenmirrors

A surface S is said to be an eigensurface of a mirror M if the reflection of S in M appears undistorted. In that case, M is the eigenmirror corresponding to the eigensurface S. Rotationally symmetric eigenmirror/eigensurface pairs are easy to construct, but the situation for freeform mirrors requires consideration of a partial differential equation, the anti-eikonal equation. Here, we investigate freeform quadric eigenmirrors, classifying them into five families. In each case, the eigensurfaces are biquadratic, i.e., they satisfy equations of the form ψ(x²,y²,z²)=0, where ψ is a quadric polynomial. Lastly, we investigate some more general hypotheses and a connection to dynamical systems.