Box-trees for collision checking in industrial installations

Haverkort, Herman

doi:10.1016/s0925-7721(04)00022-7

Cited by 5 publications

(5 citation statements)

References 6 publications

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“…Namely, a region's bounding box is the smallest axis-parallel box containing it. Since the bounding box is an over-approximation of the space occupied by a region, it is used in such area as computer-aided design [30], in order to rapidly determine regions that may collide. For instance in order to disassemble mechanical parts, one can ask for the parts in assembly order, which can be removed without first removing some other part.…”

Section: R R | 3 ∩ Qmentioning

confidence: 99%

Reasoning about visibility

Villemaire

Hallé

2012

Journal of Applied Logic

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We consider properties of sequences of spatial regions, as seen from a viewpoint. In particular, we concentrate on two types of regions: (1) general domains in which a region is any subset of the space, and (2) axis-parallel domains, where the regions are boxes in an N-dimensional space. We introduce binary relations allowing to express properties of these sequences and present two approaches to process them. First, we show that constraints on these relations can be solved in polynomial time for general domain and that the same problem is NP-complete in the axis-parallel case. Second, we introduce a modal logic on these relations, called Visibility Logic, and show that model-checking on a finite sequence of regions can be done in polynomial time (both in the general and axis-parallel cases). Finally, we present applications to image processing and firewall filtering.

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Section: R R | 3 ∩ Qmentioning

confidence: 99%

Reasoning about visibility

Villemaire

Hallé

2012

Journal of Applied Logic

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“…By construction, every square in Γ intersects ∂ Q . We now use the following fact proved by Haverkort et al [19]:…”

Section: Proofmentioning

confidence: 99%

Star-quadtrees and guard-quadtrees: I/O-efficient indexes for fat triangulations and low-density planar subdivisions

Berg¹,

Haverkort²,

Thite³

et al. 2010

Computational Geometry

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We present a new I/O-efficient index structure for storing planar subdivisions. This socalled guard-quadtree is a compressed linear quadtree, which is provably efficient for map overlay, point location, and range queries in low-density subdivisions. In particular, we can preprocess a low-density subdivision with n edges, in O (sort(n)) I/Os, to build a guardquadtree that allows us to:(i) compute the intersections between the edges of two such preprocessed subdivisions in O (scan(n)) I/Os, where n is the total number of edges in the two subdivisions;(ii) answer a single point location query in O (log B n) I/Os, and k batched point location queries in O (scan(n) + sort(k)) I/Os; and (iii) answer range queries for any constant-complexity query range Q in O ( 1 ε (log B n) + scan(k ε )) I/Os for every ε > 0, where k ε is the number of edges of the subdivision within distance ε · diam(Q ) from Q .For the special case where the subdivision is a fat triangulation, we show how to obtain the same results with an ordinary (uncompressed) linear quadtree, which we call the starquadtree. The star-quadtree is fully dynamic and needs only O (log B n) I/Os per update.Our algorithms and data structures improve on the previous best known bounds for overlaying general subdivisions, both in the number of I/Os and space usage. The constants in the asymptotic bounds are small, which makes our results applicable in practice. Moreover, our algorithms are simpler than previous approaches and almost all of them are cache-oblivious.

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“…Unfortunately, it seems hard to modify these structures to work for rectangles. Finally, Agarwal et al [2002], as well as Haverkort et al [2002], also developed a number of R-trees that have good worst-case query performance under certain conditions on the input.…”

Section: Introductionmentioning

confidence: 99%