Abstract
Oriented matroids are combinatorial structures that encode the combinatorics of point configurations. The set of all triangulations of a point configuration depends only on its oriented matroid. We survey the most important ingredients necessary to exploit oriented matroids as a data structure for computing all triangulations of a point configuration, and report on experience with an implementation of these concepts in the software package TOPCOM. Next, we briefly overview the construction and an application of the secondary polytope of a point configuration, and calculate some examples illustrating how our tools were integrated into the polymake framework.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Oswin Aichholzer, The path of a triangulation, Proceedings of the 15th Annual ACM Symposium on Computational Geometry (Miami Beach, Florida, USA ), 1999, pp. 14–23.
Louis J. Billera and Bernd Sturmfels, Fiber polytopes, Annals of Mathematics 135 (1992), 527–549.
Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, Encyclopedia of Mathematics, vol. 46, Cambridge University Press, Cambridge, 1993.
Jürgen Bokowski and A. Guedes de Oliveira, On the generation of oriented matroids,Discrete & Computational Geometry 24 (2000), 197–208, Branko Grünbaum Birthday Issue.
Komei Fukuda, cdd—an implementation of the double description method with LP solver, http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html.
Komei Fukuda and Lukas Finschi, Generation of oriented matroids —a graph theoretical approach, Discrete & Computational Geometry 27 (2002), 117 136.
Ewgenij Gawrilow and Michael Joswig, Polymake—a versatile tool for the algorithmic treatment of polytopes and polyhedra,2001.
Izrail M. Gelfand, Mikhail M. Kapranov, and Andrei V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser, Boston, 1994.
Branko Grünbaum, Convex Polytopes, Interscience, London, 1967.
E. Lawler, J. Lenstra, and A. H. G. Rinnooy Kan, Generating all maximal independent sets: NP-hardness and polynomial-time algorithms, SIAM Journal on Computing 9 (1980), 558–565.
Carl Lee, Triangulations of polytopes, CRC Handbook of Discrete and Computational Geometry (Jacob E. Goodman and Joseph O’Rourke, eds.), CRC Press LLC, Boca Raton, 1997, pp. 271–190.
Jesfis A. de Loera, Triangulations of polytopes and computational algebra, Ph.D. thesis, Cornell University, 1995.
Jesfis A. de Loera, Serkan Hogten, Francisco Santos, and Bernd Sturmfels, The polytope of all triangulations of a point configuration, Documenta Mathematika 1 (1996), 103–119.
Tomonari Masada, Hiroshi Imai, and Keiko Imai, Enumeration of regular triangulations, Proceedings of the 12th Annual Symposium on Computational Geometry (Philadelphia, PA, USA), ACM Press, 1996, pp. 224–233.
Julian Pfeifle, Secondary polytope of c4(8) on the carathéodory curve, Electronic Geometry Models (2000), http://www.eg-models.de/2000.09.032.
Julian Pfeifle, Secondary polytope of the 3-cube, Electronic Geometry Models (2000), http://www.eg-models.de/2000.09.031.
Konrad Polthier et al., Javaview, http://www.javaview.de, 2001. 18, Jörg Rambau, Projections of polytopes and polyhedral subdivisions, Berichte aus der Mathematik, Shaker, Aachen, 1996, Dissertation, TU Berlin.
Jörg Rambau, Triangulations of cyclic polytopes and higher Bruhat orders, Mathematika 44 (1997), 162–194.
Jörg Rambau, TOPCOM—Triangulations of Point Configurations and Oriented Matroids,Software under the Gnu Public Licence, http://www.zib.de/rambau/TOPCOM.html, 1999.
Jörg Rambau, Point configuration and a triangulation without flips as constructed by Santos,Electronic Geometry Models (2000), http://www.eg-models.de/2000.08.005.
Jörg Rambau, Topcom: Triangulations of point configurations and oriented matroids,ZIB-Report 02–17, 2002, Proceedings of the International Congress of Mathematical Software, to appear.
Ulrich Reitebuch, Rhombicosidodecahedron,Electronic Geometry Models (2000), http://www.eg-models.de/2000.09.013.
Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner Deforinations of Hypergeometric Differential Equations, Algorithms and Computation in Mathematics, vol. 6, Springer, 2000.
Francisco Santos, A point configuration whose space of triangulations is disconnected, Journal of the American Mathematical Society 13 (2000), 611–637.
Francisco Santos, Non-connected tonic Hilbert schemes,P reprint CO/0204044 vl, arXiv:math, April 2002.
Francisco Santos, Triangulations of oriented matroids, Memoirs of the American Mathematical Society 156 (2002).
Bernd Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, vol. 8, AMS, 1996.
Fumihiko Takeuchi, Combinatorics of triangulations, Ph.D. thesis, Graduate School of Information Science, University of Tokyo, 2001.
Fumihiko Takeuchi and Hiroshi Imai, Enumerating triangulations for products of two simplices and for arbitrary configurations of points, Computing and Combinatorics, 1997, pp. 470–481.
Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, 1995, Updates, Corrections, and more available at http://www.math.tu-berlin.de/~ziegler.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pfeifle, J., Rambau, J. (2003). Computing Triangulations Using Oriented Matroids. In: Joswig, M., Takayama, N. (eds) Algebra, Geometry and Software Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-05148-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-05148-1_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05539-3
Online ISBN: 978-3-662-05148-1
eBook Packages: Springer Book Archive