Imagine a polygon-shaped platform $P$ and only one static spotlight outside $P$; which direction should the spotlight face to light most of $P$? This problem occurs in maximising the visibility, as well as in limiting the uncertainty in localisation problems. More formally, we define the following maximum cover problem: "Given a convex polygon $P$ and a Field Of View (FOV) with a given centre and inner angle $\phi$; find the direction (an angle of rotation $\theta$) of the FOV such that the intersection between the FOV and $P$ has the maximum area". In this paper, we provide the theoretical foundation for the analysis of the maximum cover with a rotating field of view. The main challenge is that the function of the area $A_{\phi}(\theta)$, with the angle of rotation $\theta$ and the fixed inner angle $\phi$, cannot be approximated directly. We found an alternative way to express it by various compositions of a function $A_{\theta}(\phi)$ (with a restricted inner angle $\phi$ and a fixed direction $\theta$). We show that $A_{\theta}(\phi)$ has an analytical solution in the special case of a two-sector intersection and later provides a constrictive solution for the original problem. Since the optimal solution is a real number, we develop an algorithm that approximates the direction of the field of view, with precision $\varepsilon$, and complexity $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$.

翻译：想象一个多边形平台 $P$ 和 $P$ 外仅有一个静态聚光灯；聚光灯应朝向哪个方向才能最大程度照亮 $P$？该问题在最大化可见性以及限制定位问题中的不确定性方面具有应用。更形式化地，我们定义以下最大覆盖问题："给定凸多边形 $P$ 和一个具有给定中心点与内角 $\phi$ 的视场 (FOV)；寻找视场的方向（旋转角 $\theta$），使得视场与 $P$ 的交集面积最大化"。本文为旋转视场下的最大覆盖分析提供理论基础。主要挑战在于，面积函数 $A_{\phi}(\theta)$（以旋转角 $\theta$ 和固定内角 $\phi$ 为变量）无法直接近似。我们找到一种替代方法，通过函数 $A_{\theta}(\phi)$（具有受限内角 $\phi$ 和固定方向 $\theta$）的多种组合来表达该函数。我们证明在两扇区交集的特殊情况下 $A_{\theta}(\phi)$ 存在解析解，并进一步为原问题提供构造性解法。由于最优解为实数，我们开发了一种算法来近似视场方向，精度为 $\varepsilon$，复杂度为 $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$。