The Frechet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Frechet distance between them. This variant is called the Translation Invariant Frechet distance, and algorithms to compute it are well studied. The query version, finding an optimal placement in the plane for a query segment where the Frechet distance becomes minimized, is much less well understood. We study Translation Invariant Frechet distance queries in a restricted setting of horizontal query segments. More specifically, we preprocess a trajectory in $mathcal O(n^2 log^2 n) $ time and $mathcal O(n^{3/2})$ space, such that for any subtrajectory and any horizontal query segment we can compute their Translation Invariant Frechet distance in $mathcal O(text{polylog } n)$ time. We hope this will be a step towards answering Translation Invariant Frechet queries between arbitrary trajectories.
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