Parabola Separation Queries and their Application to Stone Throwing

Given two sets A and B of m non-intersecting line segments in the plane, we show how to compute in O(m log m) time a data structure that uses O(m) space and allows to answer 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 with complexity n, we can answer such “stone throwing ” queries in O(log 2 n) time, using O(nlog n) space and O(n log 2 n) preprocessing time. This matches the best known bound for circular ray shooting in simple polygons