Optimal bounding cones of vectors in three dimensions

The problem of computing the minimum-angle bounding cone of a set of three-dimensional vectors has numerous applications in computer graphics and in geometric modeling. One such application is bounding the tangents of space curves or the vectors normal to a surface in the computation of the intersection of two surfaces. Surprisingly, no optimal-time exact solution to this problem has been yet given. This paper presents a road-map for a few strategies that provide optimal or near-optimal solutions to this problem, which are also simple to implement. Speci cally, if a worst-case running time is required, we provide an O(n log n)time Voronoi-diagram-based algorithm, where n is the number of vectors whose optimum bounding is sought. Otherwise, if one is willing to accept an in average eÆcient algorithm, we show that the main ingredient of the algorithm of Shirman and Abi-Ezzi [8] can be implemented to run in optimal (n) expected time. Furthermore, if the vectors are known to occupy no more than a hemisphere, we show how to simplify this ingredient (by reducing the dimension of the problem) without a ecting the asymptotic expected running time. Both versions of this algorithm are based on computing (as an LP-type problem) the minimum spanning circle (resp., ball) of a 2D (resp., 3D) set of points.