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PARABOLA SEPARATION QUERIES AND THEIR APPLICATION TO STONE THROWING
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Given two sets A and B of m non-crossing line segments in the plane, we show how to compute in O(m log 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 = 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 of complexity n, we can answer such "stone-throwing" queries in O( log 2 n) time, using O(n log n) storage and O(n log 2 n) preprocessing time. This matches the best known bound for circular ray shooting in simple polygons.
World Scientific Pub Co Pte Lt
Title: PARABOLA SEPARATION QUERIES AND THEIR APPLICATION TO STONE THROWING
Description:
Given two sets A and B of m non-crossing line segments in the plane, we show how to compute in O(m log 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 = 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 of complexity n, we can answer such "stone-throwing" queries in O( log 2 n) time, using O(n log n) storage and O(n log 2 n) preprocessing time.
This matches the best known bound for circular ray shooting in simple polygons.
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