Parabola separation queries and their application to stone throwing

Abstract

Given two sets A and B of m non-crossing line segments in the plane, we show how to compute in O(mlog m) time a data structure that uses O(m) storage and supports the following query in O(log m) time: Given a parabola $γ : y = ax^{2} + bx + c$, does γ separate A and B? This structure can be used to build a data structure that stores a simple polygon and allows ray-shooting queries along parabolic trajectories with vertical main axis. For a polygon of complexity~n, we can answer such "stone-throwing" queries in $O(log^{2} n)$ time, using O(nlog n) storage and $O(nlog^{2} n)$ preprocessing time. This matches the best known bound for circular ray shooting in simple polygons.