1,493 research outputs found

    Separation-Sensitive Collision Detection for Convex Objects

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    We develop a class of new kinetic data structures for collision detection between moving convex polytopes; the performance of these structures is sensitive to the separation of the polytopes during their motion. For two convex polygons in the plane, let DD be the maximum diameter of the polygons, and let ss be the minimum distance between them during their motion. Our separation certificate changes O(log(D/s))O(\log(D/s)) times when the relative motion of the two polygons is a translation along a straight line or convex curve, O(D/s)O(\sqrt{D/s}) for translation along an algebraic trajectory, and O(D/s)O(D/s) for algebraic rigid motion (translation and rotation). Each certificate update is performed in O(log(D/s))O(\log(D/s)) time. Variants of these data structures are also shown that exhibit \emph{hysteresis}---after a separation certificate fails, the new certificate cannot fail again until the objects have moved by some constant fraction of their current separation. We can then bound the number of events by the combinatorial size of a certain cover of the motion path by balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999; see also http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission with camera-ready versio

    Continuous Yao Graphs

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    In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points SR2S\subset \mathbb{R}^2 and an angle 0<θ2π0 < \theta \leq 2\pi, we define the continuous Yao graph cY(θ)cY(\theta) with vertex set SS and angle θ\theta as follows. For each p,qSp,q\in S, we add an edge from pp to qq in cY(θ)cY(\theta) if there exists a cone with apex pp and aperture θ\theta such that qq is the closest point to pp inside this cone. We study the spanning ratio of cY(θ)cY(\theta) for different values of θ\theta. Using a new algebraic technique, we show that cY(θ)cY(\theta) is a spanner when θ2π/3\theta \leq 2\pi /3. We believe that this technique may be of independent interest. We also show that cY(π)cY(\pi) is not a spanner, and that cY(θ)cY(\theta) may be disconnected for θ>π\theta > \pi.Comment: 7 pages, 7 figures. Presented at CCCG 201
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