Abstract
We discuss algorithmic steps when dealing with realizability problems in discrete geometry, especially that of finding realizations for a given oriented matroid. After a brief introduction to known methods, we discuss a dynamic inductive realization method, which has proven successful when other methods did not succeed. A useful theorem in this context in the rank 3 case asserts that a one-element extension of a uniform rank 3 oriented matroid depends essentially just on the mutations involving that element. There are problems in computational synthetic geometry of course, where intuition must help. In this context we mention the application of the software Cinderella to automated deduction in computational synthetic geometry, when studying face lattices of polytopes.
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
A. Altshuler, J. Bokowski, and P. Schuchert. Spatial polyhedra without diagonals. Israel J. Math. 86, 373–396, 1994.
A. Altshuler, J. Bokowski, and P. Schuchert. Sphere systems and neighborly spatial polyhedra with 10 vertices. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 35, 15–28, 1994.
A. Bjôrner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler. Oriented Matroids. Cambridge University Press, Cambridge, 1993.
J. Bokowski. Aspects of computational synthetic geometry II: Combinatorial complexes and their geometric realization — An algorithmic approach. Proceedings of the INRIA Workshop on Computer-Aided Geometric Reasoning (H. Crapo, ed.), Antibes, France, 1987.
J. Bokowski. On the geometric flat embedding of abstract complexes with symmetries. Symmetry of Discrete Mathematical Structures and Their Symmetry Groups: A Collection of Essays (K. H. Hofmann and R. Wille, eds.), Research and Exposition in Mathematics 15, 1–48, Heldermann, Berlin, 1991.
J. Bokowski. Handbook of Convex Geometry, Chapter on Oriented Matroids (P. Gruber and J. M. Wills, eds.). Elsevier, North-Holland, Netherlands, 1992.
J. Bokowski, P. Cara, and S. Mock. On a self dual 3-sphere of Peter McMullen. Periodica Mathematica Hungarica 39, 17–32, 1999.
J. Bokowski and A. Guedes de Oliveira. On the generation of oriented matroids. Discrete Comput. Geom. 24, 197–208, 2000.
J. Bokowski, G. Lafaille, and J. Richter-Gebert. Classification of non-stretchable pseudoline arrangements and related properties. Manuscript, 1991.
J. Bokowski, S. Mock, and I. Streinu. The Folkman-Lawrence topological representation theorem: A direct proof in the rank 3 case. Proceedings of the CIRM Conference “Géométries combinatoires: Matroþdes orientés, matroïdes et applicationsℍ, Luminy, France, 1999.
J. Bokowski and J. Richter. On the finding of final polynomials. Eur. J. Comb. 11, 21–34, 1990.
J. Bokowski and J. Richter-Gebert. Reduction theorems for oriented matroids. Manuscript, 1990.
J. Bokowski and B. Sturmfels.On the coordinatization of oriented matroids. Discrete Comput. Geom. 1, 293–306, 1986.
J. Bokowski and B. Sturmfels. Computational Synthetic Geometry. Lecture Notes in Mathematics 1399, Springer, Berlin, 1989.
B. Grünbaum. Arrangements and Spreads. Regional Conf. Ser. Math. 10, Amer. Math. Soc., Providence, RI, 1972.
D. Ljubic, J.-P. Roudneff, and B. Sturmfels. Arrangements of lines and pseudolines without adjacent triangles. J. Comb. Theory, Ser. A 50, 24–32, 1989.
N. E. Mnёv. The universitality theorems on the classification problem of configuration varieties and convex polytope varieties. Topology and Geometry — Rohlin Seminar (O. Ya. Viro, ed.), Lecture Notes in Mathematics 1346, 527–544, Springer, Berlin, 1988.
K.P. Pock. Entscheidungsmethoden zur Realisierung orientierter Matroide. Diplom thesis, TH Darmstadt, 1991.
J. Richter. Kombinatorische Realisierbarkeitskriterien für orientierte Matroide. Mitteilungen aus dem Math. Sem. Gießen 194, 1–113, 1989. Diplom thesis, TH Darmstadt, 1988.
J. Richter-Gebert. On the Realizability Problem of Combinatorial Geometries — Decision Methods. Ph.D. thesis, Darmstadt, 1992.
J. Richter-Gebert. Realization Spaces of Polytopes. Lecture Notes in Mathematics 1643, Springer, Berlin, 1996.
J. Richter-Gebert and U. Kortenkamp. The Interactive Geometry Software Cinderella. Springer, Berlin, 1999.
J. Scharnbacher. Zur Realisation simplizialer orientierter Matroide. Diplom thesis, TH Darmstadt, 1993.
B. Sturmfels. Aspects of computational synthetic geometry I: Algortihmic coordinatization of matroids. Proceedings of the INRIA Workshop on Computer-Aided Geometric Reasoning (H. Crapo, ed.), Antibes, France, 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bokowski, J. (2001). Effective Methods in Computational Synthetic Geometry. In: Richter-Gebert, J., Wang, D. (eds) Automated Deduction in Geometry. ADG 2000. Lecture Notes in Computer Science(), vol 2061. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45410-1_11
Download citation
DOI: https://doi.org/10.1007/3-540-45410-1_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42598-4
Online ISBN: 978-3-540-45410-6
eBook Packages: Springer Book Archive