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)×1 ×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.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions (preliminary version). In: 12th Symposium on Computational Geometry, pp. 319–328 (1996)
Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9(3), 251–280 (1990)
Crapo, H., Whiteley, W.: Plane self stresses and projected polyhedraI: The basic pattern. Structural Topology 20, 55–78 (1993)
Das, G., Goodrich, M.T.: On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees. Comput. Geom. Theory Appl. 8(3), 123–137 (1997)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
Eades, P., Garvan, P.: Drawing stressed planar graphs in three dimensions. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 212–223. Springer, Heidelberg (1996)
Gortler, S.J., Gotsman, C., Thurston, D.: Discrete one-forms on meshes and applications to 3d mesh parameterization. Computer Aided Geometric Design 23(2), 83–112 (2006)
Hopcroft, J.E., Kahn, P.J.: A paradigm for robust geometric algorithms. Algorithmica 7(4), 339–380 (1992)
Lipton, R.J., Rose, D., Tarjan, R.: Generalized nested dissection. SIAM J. Numer. Anal. 16(2), 346–358 (1979)
Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9(3), 615–627 (1980)
Maxwell, J.C.: On reciprocal figures and diagrams of forces. Phil. Mag. Ser. 27, 250–261 (1864)
Ribó Mor, A., Rote, G., Schulz, A.: Small grid embeddings of 3-polytopes (2009) (submitted for publication), http://arxiv.org/abs/0908.0488
Richter-Gebert, J.: Realization Spaces of Polytopes. Lecture Notes in Mathematics, vol. 1643. Springer, Heidelberg (1996)
Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pp. 138–148 (1990)
Schramm, O.: Existence and uniqueness of packings with specified combinatorics. Israel J. Math. 73, 321–341 (1991)
Steinitz, E.: Encyclopädie der mathematischen Wissenschaften. In: Polyeder und Raumteilungen, pp. 1–139 (1922)
Tutte, W.T.: How to draw a graph. Proceedings London Mathematical Society 13(52), 743–768 (1963)
Whiteley, W.: Motion and stresses of projected polyhedra. Structural Topology 7, 13–38 (1982)
Whitney, H.: A set of topological invariants for graphs. Amer. J. Math. 55, 235–321 (1933)
Zhang, F.: Matrix Theory. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schulz, A. (2010). Drawing 3-Polytopes with Good Vertex Resolution. In: Eppstein, D., Gansner, E.R. (eds) Graph Drawing. GD 2009. Lecture Notes in Computer Science, vol 5849. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11805-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-11805-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11804-3
Online ISBN: 978-3-642-11805-0
eBook Packages: Computer ScienceComputer Science (R0)