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首页> 外文期刊>Discrete Applied Mathematics >Diffuse reflection diameter and radius for convex-quadrilateralizable polygons
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Diffuse reflection diameter and radius for convex-quadrilateralizable polygons

机译：可凸四边形多边形的漫反射直径和半径

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摘要

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-104 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-108 under the usual model of diffuse reflection. In other words, there exists a point inside such a polygon from which 3n-108 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-42 and n-44? 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-42 and n-44? respectively, are sometimes attained in the usual model, as well as in the relaxed model of diffuse reflection.
机译：在本文中，我们研究了可凸四边形多边形的漫反射直径和漫反射半径问题。在通常的漫反射模型中，入射在反射表面上某个点的光线从该点沿所有可能的向内方向反射。从多边形边缘反射的光线可能会掠过该反射边缘，但入射光线无法掠过该反射边缘。简单多边形P的漫反射直径是在最坏的情况下从P内任何点光源s照射任何目标点t所需的最小漫反射数量。我们证明该直径的上限为3n- 104是我们通常用于凸四边形多边形的漫反射模型。据我们所知，这是此类多边形的第n个散射反射直径的第一个上限。我们还表明，在通常的漫反射模型下，具有n个顶点的凸-四边形化简单多边形的漫反射半径最大为3n-108。换句话说，在这种多边形内部存在一个点，从该点开始，3n-108通常的漫反射总是足以照亮整个多边形。为了建立通常模型的边界，我们首先证明直径和半径为n-42和n-44？对于相同类别的多边形，分别使用漫反射模型进行松弛;在松弛模型中，允许入射光线掠过反射边缘，然后再在该边缘的任何内部点转向并反射掉该边缘。我们还表明n-42和n-44的最坏情况下的直径和半径下界？有时，在通常的模型中以及在漫反射的松弛模型中都可以分别获得。

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