  Consider two entities with constant but not necessarily equal velocities,
moving on two given piece-wise linear trajectories inside a simple polygon $P$.
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: $(1)$ All visible parts of
the other entity's trajectory. $(2)$ All possible constant velocities of the
other entity to become visible.
  Regarding line-segment trajectories, we present $\mathcal{O}(n \log n)$
running time algorithm which obtains all pairs of sub-trajectories on which the
moving entities become visible to one another, where $n$ is the complexity of
$P$. Regarding the general case, we provide an algorithm with $\mathcal{O}(n
\log n + m(\log m + \log n))$ running time, where $m$ indicates the complexity
of both trajectories. 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
output. Interestingly, our results require only $\mathcal{O}(n + m)$ space for
non-constant complexity trajectories.
