# 3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects

1 PRISME - Geometry, Algorithms and Robotics
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : We present a new data structure for a set of $n$ convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) {\em vertical ray shooting} --- preprocess a set $\K$ of $n$ non-intersecting convex simply-shaped flat objects in 3-space, whose $xy$-projections are fat, for efficient vertical ray shooting queries, (ii) {\em point enclosure} --- preprocess a set $\Cn$ convex simply-shap- ed fat objects in the plane, so that the $k$ objects containing a query point $p$ can be reported efficiently, (iii) {\em bounded-size range searching} --- preprocess a set $\Cn$ convex fat polygons, so that the $k$ objects intersecting a not-too-large' query polygon can be reported efficiently, and (iv) {\em bounded-size segment shooting} --- preprocess a set $\Cn (iii), so that the first object (if exists) hit by a not-too-long' oriented query segment can be found efficiently. For the first three problems we construct data structures of size$O(\lambda_s(n) \log^3 n)$, where$s$is the maximum number of intersections between the boundaries of the ($xy$-projections) of any pair of objects, and$\lambda_s(n)$is the maximum length of$(n,s)$Davenport-Schinzel sequences. The data structure for the fourth problem is of size$O(\lambda_s(n) \log^2 n)$. The query time in the first problem is$O(\log^4 n)$, the query time in the second and third problems is$O(\log^3 n + k \log^2 n)$, and the query time in the fourth problem is$O(\log^3 n)$. We also present a simple algorithm for computing a depth order for a set$\K$as in (i), that is based on the solution to the vertical ray shooting problem. (A depth order for$\K$, if exists, is a linear order of$\K$, such that, if$K_1,K_2 \in \K$and$K_1$lies vertically above$K_2$, then$K_1$precedes$K_2\$.) The algorithm is able to determine whether such an order exists, unlike the algorithm of Agarwal~et~al. \cite{AgKS} that might output a false order when a depth order does not exist, and it is often more efficient in practical situations than the latter algorithm.
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Type de document :
Rapport
RR-2583, INRIA. 1995
Domaine :
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https://hal.inria.fr/inria-00074101
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Soumis le : mercredi 24 mai 2006 - 14:31:47
Dernière modification le : samedi 27 janvier 2018 - 01:31:32
Document(s) archivé(s) le : jeudi 24 mars 2011 - 14:13:02

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• HAL Id : inria-00074101, version 1

### Citation

Matthew Katz. 3-D Vertical Ray Shooting and 2-D Point Enclosure, Range Searching, and Arc Shooting Amidst Convex Fat Objects. RR-2583, INRIA. 1995. 〈inria-00074101〉

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