Chapter 13 - Hulls and Diagrams

Abstract

This chapter discusses hulls and diagrams in Euclidean space or in a grid, with emphasis on 2D. The convex hull of a set is regarded as a “domain of influence” of the set, while the Voronoi diagram of a set of points defines the “domains of influence” of the points. The definitions of digital convexity and digital Voronoi diagrams, and the algorithms based on adjacency grid models, are reviewed in the chapter. A hull operator is also known as closure operator in algebra. The convex hull of a set of n points is a polygon that can have many vertices. The knowledge of vertices of a simple polygon having integer coordinates results in no asymptotic time benefit for convex hull computation. However, methods of convex hull computation exist that are applicable only to grid polygons. This chapter concludes by discussing the algorithms for computing the rectangle R(M) with the smallest possible area that contains a finite set of points or a simple polygon M in the Euclidean plane.