The Maximum Cover with Rotating Field of View

Abstract

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 $ϕ$; find the direction (an angle of rotation $θ$) 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 $A_ϕ(θ)$, with the angle of rotation $θ$ and the fixed inner angle $ϕ$, cannot be approximated directly. We found an alternative way to express it by various compositions of a function $A_θ(ϕ)$ (with a restricted inner angle $ϕ$ and a fixed direction $θ$). We show that $A_θ(ϕ)$ 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 $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/ϕ))$.