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(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