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DOI : 10.1142/S0218195907002379

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International Journal of Computational Geometry & Applications 17, 4 (2007) 349-360

Parabola separation queries and their application to stone throwing

Otfried Cheong 1, Hazel Everett 2, Hyo-Sil Kim 1, Sylvain Lazard 2, René Schott 3

(2007)

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.

1 :	Department of Electrical Engineering [Korea Advanced Institute of Science and Technology] (KAIST)
	Korea Advanced Institute of Science and Technology
2 :	VEGAS (INRIA Lorraine - LORIA)
	INRIA – CNRS : UMR7503 – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine
3 :	Institut Elie Cartan Nancy (IECN)
	CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine

Équipe de recherche (sauf Inria) : Probabilités et statistique

Domaine

:

Informatique/Géométrie algorithmique

Mots-clés : Ray shooting – simple polygon – parabola separation queries – parabolic arc

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Contributeur : Sylvain Lazard <>

Soumis le : Vendredi 20 Novembre 2009, 16:57:00

Dernière modification le : Jeudi 8 Avril 2010, 16:18:54