Trajectory Range Visibility

We study the problem of Trajectory Range Visibility, determining the sub-trajectories on which two moving entities become mutually visible. Specifically, we consider two moving entities with not necessarily equal velocities and moving on a given piece-wise linear trajectory inside a simple polygon. Deciding whether the entities can see one another with given constant velocities, and assuming the trajectories only as line segments, was solved by P. Eades et al. in 2020. However, we obtain stronger results and support queries on constant velocities for non-constant complexity trajectories. Namely, given a constant query velocity for a moving entity, we specify all visible parts of the other entity's trajectory and all possible constant velocities of the other entity to become visible. Regarding line-segment trajectories, we obtain $\mathcal{O}(n \log n)$ time to specify all pairs of mutually visible sub-trajectories s.t. $n$ is the number of vertices of the polygon. Moreover, our results for a restricted case on non-constant complexity trajectories yield $\mathcal{O}(n + m(\log m + \log n))$ time, in which $m$ is the overall number of vertices of both trajectories. Regarding the unrestricted case, we provide $\mathcal{O}(n \log n + m(\log m + \log n))$ running time. We offer $\mathcal{O}(\log n)$ query time for line segment trajectories and $\mathcal{O}(\log m + k)$ for the non-constant complexity ones s.t. $k$ is the number of velocity ranges reported in the answer. Interestingly, our results require only $\mathcal{O}(n + m)$ space for non-constant complexity trajectories.