Collision Detection Using Bounding Boxes: Convexity Helps

Abstract

We consider the use of bounding boxes to detect collisions among a set of convex objects in R ^d. 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(α ^{1- 1/d} σ_box ^1/2) if each object has aspect ratio at most α and the set has scale factor θ_box. More significantly, the bounding box performance ratio is ($\Theta \left( {\alpha _{{\text{avg}}}^{\frac{{2(1 - 1/d)}} {{3 - 1/d}}} {\mathbf{ }}\alpha _{box}^{\frac{1} {{3 - 1/d}}} {\mathbf{ }}n^{\frac{{1 - 1/d}} {{3 - 1/d}}} } \right)$ ) if only the average aspect ratio α_avg 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 interpreted as collision; and finally, the question is interesting from a theoretical standpoint.