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.

翻译：我们研究轨迹范围的可见度问题, 确定两个移动实体相互可见的子轨迹。 具体地说, 我们考虑两个移动实体, 其速度不一定等于速度, 并在简单的多边形内移动给定的片断线性轨迹。 决定这些实体能否用给定的恒定速度看到对方, 并假设轨迹仅作为线段, P. Eades 等人在2020年解决了。 然而, 我们获得了更强有力的结果, 并且支持关于非稳定复杂度实体的恒定速度查询。 也就是说, 鉴于一个移动实体的恒定查询速度, 我们指定了其他实体的轨迹的所有可见部分, 以及其它实体的所有恒定速度。 关于线性轨迹, 我们得到的 $ mathal {( n\ log n. n) 时间来指定所有可相互可见的子轨迹的子轨迹( t. at. 美元是多元值的轨迹值 。