Illumination problems on translation surfaces with planar infinities

TL;DR

This paper investigates the illumination problem on translation surfaces with planar infinities, showing the existence of unilluminated regions and linking circle map properties to dark sectors in mirror configurations.

Contribution

It introduces a novel connection between translation surfaces with infinities and illumination problems, demonstrating unilluminated regions and the role of circle map non-bijectivity.

Findings

Unilluminated regions are isometric to unbounded planar sectors.

Presence of at least two infinities leads to unilluminated regions.

Non-bijective circle maps imply unbounded dark sectors.

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-$\pi$ 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.