  The Fr\'echet distance is a popular similarity measure between curves. For
some applications, it is desirable to match the curves under translation before
computing the Fr\'echet distance between them. This variant is called the
Translation Invariant Fr\'echet distance, and algorithms to compute it are well
studied. The query version, however, is much less well understood.
  We study Translation Invariant Fr\'echet distance queries in a restricted
setting of horizontal query segments. More specifically, we prepocess a
trajectory in $O(n^2 \log^2 n)$ time and space, such that for any subtrajectory
and any horizontal query segment we can compute their Translation Invariant
Fr\'echet distance in $O(\text{polylog} \, n)$ time. We hope this will be a
step towards answering Translation Invariant Fr\'echet queries between
arbitrary trajectories.
