Coverage restricted to an angle

Given a set $S$ of $n$ antennas and point $q$ on the plane, $q$ is $a$ ‐covered by $S$ if $\exists s_i,s_j\in S$ such that the angle $\measuredangle (s_i,q,s_j)\ge a$ . It is shown how to minimise the transmission range of $S$ to $a$ ‐cover a point in $\mathcal{O}\left(n\right)$ time and space, and how to construct the $\frac{p}{2}$ ‐covered region in $\mathcal{O}(nlogn)$ time and $\mathcal{O}\left(n\right)$ space. Finally, this paper introduces the Coverage Voronoi diagram and an algorithm to construct it in $\mathcal{O}(n^{4}logn)$ time.