Illumination problems on translation surfaces with planar infinities
arXiv:1105.2972
Abstract
In the current article we discuss an illumination problem proposed by Urrutia and Zaks. The focus is on configurations of finitely many two-sided mirrors in the plane together with a source of light placed at an arbitrary point. In this setting, we study the regions unilluminated by the source. In the case of rational- angles between the mirrors, a planar configuration gives rise to a surface with a translation structure and a number of planar infinities. We show that on a surface of this type with at least two infinities, one can find plenty of unilluminated regions isometric to unbounded planar sectors. In addition, we establish that the non-bijectivity of a certain circle map implies the existence of unbounded dark sectors for rational planar mirror configurations illuminated by a light-source.
8 pages divided into two columns each, 7 figures