Here is a variation on the classical polygon illumination problem. For $c \geq 0$ we say that a mirror has reflection index $c$ if whenever a ray hits the mirror with angle of incidence $\alpha$ then the angle of reflection is $$\alpha'=\begin{cases}c\alpha+(1-c)\frac{\pi}{2} & c\le 1 \\\ c^{-1}\alpha & c\geq 1 \end{cases}$$.

The classical problem is : Let $P$ be a polygon and consider the sides as mirrors. Can we place a source of light in the interior of $P$ which illuminates all of $P$? The answer is not known even if we relax "all of $P$" to "$P$ minus a finite set of points".

If we fix $P$ with $n$ sides, the possible reflection indices we can assign to the sides are parametrized by $\mathbb R_{+}^n$. Can we always find an $x\in \mathbb R_{+}^n$ for which the corresponding "perturbed" polygon can be illuminated by some interior light source? Can we choose $x$ close to $(1,1,\dots,1)$?

I'd be interested to know if something like this has been studied before. The reflection rules don't necessarily need to be as described above. Any kind of modified reflection is of interest, even something like $\alpha'=\tan^{-1}(c+\tan \alpha)$. In this last case we can consider affine transformations of $P$ where the light continues in a straight line, like it is usually done for polygons with angles $q\pi , q\in \mathbb Q$, but this doesn't seem very useful here since for most perturbations the angles are going to be irrational multpiles of $\pi$.

  • 1
    $\begingroup$ Perhaps easier, but still interesting, is a kind of misère version: Is there a polygon $P$ and an assignment of reflection indices so that the unilluminated points (if there are any) form a set of more than measure zero? This is unsolved for classical reflection, as detailed in the MO question "Polygons and Mirrors" mathoverflow.net/questions/41995 $\endgroup$ – Joseph O'Rourke Sep 8 '11 at 18:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.