Diffuse reflection diameter and radius for convex-quadrilateralizable polygons

Abstract: In this paper we study the diffuse reflection diameter and diffuse reflection radius problems for convex-quadrilateralizable polygons. In the usual model of diffuse reflection, a light ray incident at a point on the reflecting surface is reflected from that point in all possible inward directions. A ray reflected from a polygonal edge may graze that reflecting edge but an incident ray cannot graze the reflecting edge. The diffuse reflection diameter of a simple polygon P is the minimum number of diffuse reflections that may be needed in the worst case to illuminate any target point t from any point light source s inside P. We show that the diameter is upper bounded by ^(3n-10)⁄₄ in our usual model of diffuse reflection for convex-quadrilateralizable polygons. To the best of our knowledge, this is the first upper bound on diffuse reflection diameter within a fraction of n for such a class of polygons. We also show that the diffuse reflection radius of a convex-quadrilateralizable simple polygon with n vertices is at most ^(3n-10)⁄₈ under the usual model of diffuse reflection. In other words, there exists a point inside such a polygon from which ^(3n-10)⁄₈ usual diffuse reflections are always sufficient to illuminate the entire polygon. In order to establish these bounds for the usual model, we first show that the diameter and radius are ^(n-4)⁄₂ and ⌊^(n-4)⁄₄⌋ respectively, for the same class of polygons for a relaxed model of diffuse reflections; in the relaxed model an incident ray is permitted to graze a reflecting edge before turning and reflecting off the same edge at any interior point on that edge. We also show that the worst-case diameter and radius lower bounds of ^(n-4)⁄₂ and ⌊^(n-4)⁄₄⌋ respectively, are sometimes attained in the usual model, as well as in the relaxed model of diffuse reflection.