Drawing 3-Polytopes with Good Vertex Resolution

Abstract

We study the problem how to obtain a small drawing of a 3-polytope with Euclidean distance between any two points at least 1. The problem can be reduced to a one-dimensional problem, since it is sufficient to guarantee distinct integer x-coordinates. We develop an algorithm that yields an embedding with the desired property such that the polytope is contained inside a 2(n−2)×2 ×1 box. The constructed embedding can be scaled to a grid embedding whose x-coordinates are contained in [0,2(n−2)]. Furthermore, the point set of the embedding has a small spread, which differs from the best possible spread only by a multiplicative constant.