Separation-Sensitive Collision Detection for Convex Objects

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 $D$ be the maximum diameter of the polygons, and let $s$ be the minimum distance between them during their motion. Our separation certificate changes $O(\log(D/s))$ times when the relative motion of the two polygons is a translation along a straight line or convex curve, $O(\sqrt{D/s})$ for translation along an algebraic trajectory, and $O(D/s)$ for algebraic rigid motion (translation and rotation). Each certificate update is performed in $O(\log(D/s))$ 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

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