Separation-Sensitive Collision Detection for Convex Objects

TL;DR

This paper introduces separation-sensitive kinetic data structures for collision detection between moving convex objects, with performance depending on their separation during motion, improving efficiency for various motion types.

Contribution

It presents novel kinetic data structures that adapt to the separation of convex objects, providing bounds on update complexity for different motion trajectories.

Findings

Performance bounds depend on the ratio D/s during motion.

Certificates update in O(log(D/s)) time.

Variants with hysteresis reduce event frequency.

Abstract

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.