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 $O(m \log m)$ time a data structure that uses $O(m)$ storage and supports the following query in $O(\log m)$ time: Given a parabola $\p: y = ax^{2} + bx + c$, does $\p$ 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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International Journal of Computational Geometry and Applications, World Scientific Publishing, 2007, 17 (4), pp.349-360. 〈10.1142/S0218195907002379〉
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Otfried Cheong, Hazel Everett, Hyo-Sil Kim, Sylvain Lazard, René Schott. Parabola separation queries and their application to stone throwing. International Journal of Computational Geometry and Applications, World Scientific Publishing, 2007, 17 (4), pp.349-360. 〈10.1142/S0218195907002379〉. 〈inria-00434090〉

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