Digitization of straight line segments closeness and convexity

Abstract

In her thesis “Digitisation Functions in Computer Graphics” M. van Lierop presents a framework in which one can mathematically describe digitization of straight line segments, and desirable properties of this digitization, such as translation invariance, closeness, minimality, and convexity (convexity is sometimes called “the subset line property”). In the second chapter of her thesis she proves that a line function that has these four properties does not exist, and the remaining question is whether closeness and convexity are mutually exclusive properties (even if one drops minimality or translation invariance). In this paper we will answer this question affirmatively.