Trajectory Range Visibility
Seyed Mohammad Hussein Kazemi, Arash Vaezi, Mohammad Ali Abam,, Mohammad Ghodsi

TL;DR
This paper develops algorithms to determine when and how two entities moving along trajectories within a polygon can see each other, supporting complex trajectories and reporting all possible velocities and visible segments.
Contribution
It extends previous work by supporting non-constant complexity trajectories and providing efficient algorithms with detailed visibility and velocity range reports.
Findings
Algorithm for line segment trajectories runs in O(n log n) time.
Algorithm for complex trajectories runs in O(n log n + m(log m + log n)) time.
Supports efficient querying with O(log n) and O(log m + k) times.
Abstract
Consider two entities with constant but not necessarily equal velocities, moving on two given piece-wise linear trajectories inside a simple polygon . The Trajectory Range Visibility problem deals with determining the sub-trajectories on which two entities become visible to each other. A more straightforward decision version of this problem is called Trajectory Visibility, where the trajectories are line segments. The decision version specifies whether the entities can see one another. This version was studied by P. Eades et al. in 2020, where they supposed given constant velocities for the entities. However, the approach presented in this paper supports non-constant complexity trajectories. Furthermore, we report every pair of constant velocities with which the entities can see each other. In particular, for every constant velocity of a moving entity, we specify: All visible…
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Taxonomy
TopicsData Management and Algorithms · Advanced Combinatorial Mathematics · Computational Geometry and Mesh Generation
