Imaging properties of freeform quadric reflectors via the theory of eigenmirrors

A surface <i>S</i> is said to be an eigensurface of a mirror <i>M</i> if the reflection of <i>S</i> in <i>M</i> appears undistorted. In that case, <i>M</i> is the eigenmirror corresponding to the eigensurface <i>S</i>. 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 <i>ψ</i>(<i>x</i>²,<i>y</i>²,<i>z</i>²)=0, where <i>ψ</i> is a quadric polynomial. Lastly, we investigate some more general hypotheses and a connection to dynamical systems.