Optimal bounding cones of vectors in three dimensions

Abstract

The problem of computing the minimum-angle bounding cone of a set of three-dimensional vectors has numerous applications in computer graphics and 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. No optimal-time exact solution to this problem has been yet given. This paper presents a roadmap for a few strategies that provide optimal or near-optimal (time-wise) solutions to this problem, which are also simple to implement. Specifically, 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 cone is sought. Otherwise, if one is willing to accept an, in average, efficient algorithm, we show that the main ingredient of the algorithm of Shirman and Abi-Ezzi [Comput. Graphics Forum 12 (1993) 261–272] can be implemented to run in optimal Θ ( n ) expected time. Furthermore, if the vectors (as points on the sphere of directions) are known to occupy no more than a hemisphere, we show how to simplify this ingredient (by reducing the dimension of the problem) without affecting the asymptotic expected running time. Both versions of this algorithm are based on computing (as an LP-type problem) the minimum spanning circle (respectively, ball) of a two-dimensional (respectively, three-dimensional) set of points.