On computing approximate convex hulls

Abstract

The determination of the convex hull of a finite set of points is important in several problems arising in such areas as computer graphics, design automation and operation research. Computing the convex hull may also be needed as a preliminary step in some other geometrical problems. A typical example of this is the determination of the smallest ellipsoid containing all given points it is enough to consider the convex hull [4]. Several algorithms have been presented for computing the convex hull of n planar points in O(n log n) worst case time [3,5,6]. The O(n log n) worst case time bound has also been achieved in three dimensions [5]. On the other hand, it has been showed that !J(n log n) is the lower bound for planar convex hull computation [6,7]. The expected complexity of computing convex hulls has been considered by Bentley and Shamos [2], who showed that the hull of n points in two and three dimensions can be computed in linear time for a large class of probability distributions. Recently, still another approach has been taken for convex hull computation: Bentley, Faust and Preparata [I] presented a linear time algorithm for