Drawing 3-Polytopes with Good Vertex Resolution

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. Submitted: December 2009 Reviewed: September 2010 Revised: October 2010 Accepted: November 2010 Final: November 2010 Published: February 2011 Communicated by: D. Eppstein and E. R. Gansner Article type: Regular paper Supported by the German Research Foundation (DFG) under grant SCHU 2458/1-1. E-mail address: [email protected] (André Schulz) 34 André Schulz Drawing 3-Polytopes with Good Vertex Resolution