In this paper, we address the maximum cover problem of a rotating field of view (FOV) with a convex polygon \(\mathcal {P}\) with n vertices. The problem is defined as determining the optimal rotation angle \(\theta \) of the FOV, characterised by a fixed centre and inner angle \(\phi \) , such that the intersection between the FOV and \(\mathcal {P}\) has the maximum possible area. This problem is relevant for applications in visibility optimisation and uncertainty reduction in localisation tasks. We present a theoretical framework and a corresponding algorithm to approximate the value of the maximum, ensuring that the solution is close to the optimal within the specified precision. The intersection between the rotating FOV and \(\mathcal {P}\) forms a convex polygon whose number of vertices varies with the rotation angle \(\theta \) . We analytically derive the area of the intersection when it takes its simplest form, a quadrilateral, as a two-variable function \(A(\theta ,\phi )\) . The function \(A(\theta ,\phi )\) , with the angle of rotation \(\theta \) and the fixed inner angle \(\phi \) , denoted as \(A_{\phi }(\theta )\) , is non-monotonic and has multiple local extreme points inside of a given domain which poses a challenge in identifying them and approximating the global maximum. 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. We develop an algorithm that approximates the direction of the field of view, with precision \(\varepsilon >1\) , and complexity \(\mathcal {O}(n(\log {n}+(\log {\varepsilon })/\phi ))\) .
