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
$\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$.

在本文中，我们提供了旋转视野的最大覆盖分析的理论基础。我们找到了一种替代方法来表达旋转角度和内部角度的交集面积函数，通过使用限制内部角度和固定方向的函数组合。我们展示了两个区域交集的特殊情况的数学解决方案以及原问题的一种解决方法。为了获得最佳解，我们开发了一个算法，以精确度 ε 逼近视野方向，并具有复杂度 O (n (log (n)+(log (ε))/ϕ))。