Throwing stones inside simple polygons

Abstract

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 gamma : y = ax(2) + bx + c, does gamma 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(n log n) space and O(n log(2) n) preprocessing time. This matches the best known bound for circular ray shooting in simple polygons.