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 in a 2(n - 2) x 1 x 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. © 2010 Springer-Verlag.