Collision Detection Using Bounding Boxes: Convexity Helps

We consider the use of bounding boxes to detect collisions among a set of convex objects in Rd. We derive tight bounds on the ratio between the number of box intersections and the number of object intersections. Confirming intuition, we show that the performance of bounding boxes improves significantly when the underlying objects are all convex. In particular, the ratio is θ(α1-1/d σbox1/2) if each object has aspect ratio at most α and the set has scale factor σbox. More significantly, the bounding box performance ratio is θ(αavg2(1-1/d/3-1/d σbox1/3-1/d n1-1/d/3-1/d) if only the average aspect ratio αavg of the n objects is known. These bounds are the best possible as we show matching lower bound constructions. The case of convex objects is interesting for several reasons: first, in many applications, the objects are either naturally convex or are approximated by their convex hulls for convenience; second, in some applications, the penetration of convex hulls is interpretedas collision; and finally, the question is interesting from a theoretical standpoint.