A Direct Method for Determining the Lower Convex Hull of a Finite Point Set in 3D

Abstract

Determining the convex hull, its lower convex hull, and Voronoi diagram of a point set is a basic operation for many applications of pattern recognition, image processing, and data mining. To date, the lower convex hull of a finite point set is determined from the entire convex hull of the set. There arises a question “How can we determine the lower convex hull of a finite point set without relying on the entire convex hull?” In this paper, we show that the lower convex hull is wrapped by lower facets starting from an extreme edge of the lower convex hull. Then a direct method for determining the lower convex hull of a finite point set in 3D without the entire convex hull is presented. The actual running times on the set of random points (in the uniform distribution) show that our corresponding algorithm runs significantly faster than the incremental convex hull algorithm and some versions of the gift-wrapping algorithm.KeywordsConvex HullExtreme EdgeGift-wrapping AlgorithmLower Convex HullPattern RecognitionVoronoi Diagram