Abstract
We study the problem of how to obtain an integer realization of a 3d polytope when an integer realization of its dual polytope is given. We focus on grid embeddings with small coordinates and develop novel techniques based on Colin de Verdière matrices and the Maxwell–Cremona lifting method.
As our main result we show that every truncated 3d polytope with n vertices can be realized on a grid of size polynomial in n. Moreover, for a class \(\mathcal{C}\) of simplicial 3d polytopes with bounded vertex degree, at least one vertex of degree 3, and polynomial size grid embedding, the dual polytopes of \(\mathcal{C}\) can be realized on a polynomial size grid as well.
This work was funded by the German Research Foundation (DFG) under grant SCHU 2458/2-1.
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
Andrews, G.E.: A lower bound for the volume of strictly convex bodies with many boundary lattice points. Trans. Amer. Math. Soc. 99, 272–277 (1961)
Bárány, I., Rote, G.: Strictly convex drawings of planar graphs. Documenta Math. 11, 369–391 (2006)
Buchin, K., Schulz, A.: On the number of spanning trees a planar graph can have. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 110–121. Springer, Heidelberg (2010)
Das, G., Goodrich, M.T.: On the complexity of optimization problems for 3-dimensional convex polyhedra and decision trees. Computational Geometry: Theory and Applications 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)
Demaine, E.D., Schulz, A.: Embedding stacked polytopes on a polynomial-size grid. In: Proc. 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1177–1187. ACM Press (2011)
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)
Hopcroft, J.E., Kahn, P.J.: A paradigm for robust geometric algorithms. Algorithmica 7(4), 339–380 (1992)
Lovász, L.: Steinitz representations of polyhedra and the Colin de Verdière number. J. Comb. Theory, Ser. B 82(2), 223–236 (2001)
Maxwell, J.C.: On reciprocal figures and diagrams of forces. Phil. Mag. Ser. 27, 250–261 (1864)
Mor, A.R., Rote, G., Schulz, A.: Small grid embeddings of 3-polytopes. Discrete & Computational Geometry 45(1), 65–87 (2011)
Onn, S., Sturmfels, B.: A quantitative Steinitz’ theorem. Beiträge zur Algebra und Geometrie 35, 125–129 (1994)
Richter-Gebert, J.: Realization Spaces of Polytopes. Lecture Notes in Mathematics, vol. 1643. Springer (1996)
Rote, G., Santos, F., Streinu, I.: Expansive motions and the polytope of pointed pseudo-triangulations. Discrete and Computational Geometry–The Goodman-Pollack Festschrift 25, 699–736 (2003)
Schulz, A.: Drawing 3-polytopes with good vertex resolution. Journal of Graph Algorithms and Applications 15(1), 33–52 (2011)
Steinitz, E.: Polyeder und Raumeinteilungen. In: Encyclopädie der mathematischen Wissenschaften, vol. 3-1-2 (Geometrie), ch. 12, pp. 1–139. B.G. Teubner, Leipzig (1916)
Whitney, H.: Congruent graphs and the connectivity of graphs. Amer. J. Math. 54, 150–168 (1932)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Igamberdiev, A., Schulz, A. (2013). A Duality Transform for Constructing Small Grid Embeddings of 3D Polytopes. In: Wismath, S., Wolff, A. (eds) Graph Drawing. GD 2013. Lecture Notes in Computer Science, vol 8242. Springer, Cham. https://doi.org/10.1007/978-3-319-03841-4_16
Download citation
DOI: https://doi.org/10.1007/978-3-319-03841-4_16
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03840-7
Online ISBN: 978-3-319-03841-4
eBook Packages: Computer ScienceComputer Science (R0)