Year: 2025
Type: article
Abstract: In this paper, we address the maximum cover problem of a rotating field of view (FOV) with a convex polygon $$\mathcal {P}$$ P with n vertices. The problem is defined as determining the optimal rotation angle $$\theta $$ θ of the FOV, characterised by a fixed centre and inner angle $$\phi $$ ϕ , such that the intersection between the FOV and $$\mathcal {P}$$ 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 $$\mathcal {P}$$ P forms a convex polygon whose number of vertices varies with the rotation angle $$\theta $$ θ . We analytically derive the area of the intersection when it takes its simplest form, a quadrilateral, as a two-variable function $$A(\theta ,\phi )$$ A ( θ , ϕ ) . The function $$A(\theta ,\phi )$$ A ( θ , ϕ ) , with the angle of rotation $$\theta $$ θ and the fixed inner angle $$\phi $$ ϕ , denoted as $$A_{\phi }(\theta )$$ A ϕ ( θ ) , 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 $$A_{\theta }(\phi )$$ A θ ( ϕ ) (with a restricted inner angle $$\phi $$ ϕ and a fixed direction $$\theta $$ θ ). We show that $$A_{\theta }(\phi )$$ 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. We develop an algorithm that approximates the direction of the field of view, with precision $$\varepsilon >1$$ ε > 1 , and complexity $$\mathcal {O}(n(\log {n}+(\log {\varepsilon })/\phi ))$$ O ( n ( log n + ( log ε ) / ϕ ) ) .
Institution: University of Liverpool
Language: English
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4410805031