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 = ax2 +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(log2 n) time, using O(n log n) space and O(n log2 n) preprocessing time. This matches the best known bound for circular ray shooting in simple polygons.

• Type: Journal
• Year: 2007
• Where: IJCGA
• Authors: Otfried Cheong, Hazel Everett, Hyo-Sil Kim, Sylvain Lazard, René Schott
• Keywords: Data Structure, IJCGA 2007, Non-intersecting Line Segments, Simple Polygons