Arrangements of Pseudocircles: On Circularizability

Abstract

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set of 2131 digon-free intersecting arrangements of 6 pseudocircles we identify the three non-circularizable examples. We also show non-circularizability of eight additional arrangements of 6 pseudocircles which have a group of symmetries of size at least 4. Most of our non-circularizability proofs depend on incidence theorems like Miquel’s. In other cases we contradict circularizability by considering a continuous deformation where the circles of an assumed circle representation grow or shrink in a controlled way. The claims that we have all non-circularizable arrangements with the given properties are based on a program that generated all arrangements up to a certain size. Given the complete lists of arrangements, we used heuristics to find circle representations. Examples where the heuristics failed were examined by hand.

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Notes

  1. 1.

    This name refers to the logo of the Krupp AG, a German steel company. Krupp was the largest company in Europe at the beginning of the 20th century.

  2. 2.

    We recommend the Sage Reference Manual on Graph Theory [30] and its collection of excellent examples.

  3. 3.

    For more details, we refer to the Sage Reference Manual on Algebraic Numbers and Number Fields [29].

  4. 4.

    If the three planes \(E_i,E_j,E_k\) intersect in a common line, we still take the expression as a definition for \(I_{ijk}\), i.e., the homogeneous coordinates are all zero.

References

  1. 1.

    Agarwal, P.K., Nevo, E., Pach, J., Pinchasi, R., Sharir, M., Smorodinsky, S.: Lenses in arrangements of pseudo-circles and their applications. J. ACM 51, 139–186 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Albenque, M., Knauer, K.: Convexity in partial cubes: the hull number. Discrete Math. 339, 866–876 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Björner, A., Las Vergnas, M., White, N., Sturmfels, B., Ziegler, G.M.: Oriented Matroids, Encyclopedia of Mathematics and Its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  4. 4.

    Bogomolny, A.: Cut-the-knot: four touching circles. http://www.cut-the-knot.org/Curriculum/Geometry/FourTouchingCircles.shtml#Explanation

  5. 5.

    Bokowski, J., Richter, J.: On the finding of final polynomials. Eur. J. Comb. 11, 21–34 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bokowski, J., Sturmfels, B.: An infinite family of minor-minimal nonrealizable 3-chirotopes. Mathematische Zeitschrift 200, 583–589 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Edelsbrunner, H., Ramos, E.A.: Inclusion-exclusion complexes for pseudodisk collections. Discrete Comput. Geom. 17, 287–306 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Felsner, S., Goodman, J.E.: Pseudoline arrangements. In: Toth, O’Rourke, Goodman, (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, Boca Raton (2018)

    Google Scholar 

  9. 9.

    Felsner, S., Scheucher, M.: Webpage: Homepage of Pseudocircles. http://www3.math.tu-berlin.de/pseudocircles

  10. 10.

    Felsner, S., Scheucher, M.: Arrangements of Pseudocircles: Triangles and Drawings (2017). arXiv:1708.06449

  11. 11.

    Felsner, S., Weil, H.: A theorem on higher Bruhat orders. Discrete Comput. Geom. 23, 121–127 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in \({\mathbb{R}}^d\). Discrete Comput. Geom. 1, 219–227 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Goodman, J.E., Pollack, R.: Allowable sequences and order types in discrete and computational geometry. In: Pach, J. (ed.) New Trends in Discrete and Computational Geometry, pp. 103–134. Springer, New York (1993)

    Google Scholar 

  14. 14.

    Grünbaum, B.: Arrangements and Spreads, CBMS Regional Conference Series in Mathematics, AMS, vol. 10 (1972) (reprinted 1980)

  15. 15.

    Kang, R.J., Müller, T.: Arrangements of pseudocircles and circles. Discrete Comput. Geom. 51, 896–925 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Knuth, D.E.: Axioms and Hulls, LNCS 606. Springer, New York (1992)

    Google Scholar 

  17. 17.

    Krasser, H.: Order types of point sets in the plane, PhD thesis, Graz University of Technology, Austria (2003)

  18. 18.

    Levi, F.: Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 78, 256–267 (1926)

    MATH  Google Scholar 

  19. 19.

    Linhart, J., Ortner, R.: An Arrangement of Pseudocircles Not Realizable with Circles. Beiträge zur Algebra und Geometrie 46, 351–356 (2005)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)

    Google Scholar 

  21. 21.

    Matoušek, J.: Intersection graphs of segments and \(\exists {\mathbb{R}}\) (2014). arXiv:1406.2636

  22. 22.

    Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Topology and Geometry—Rohlin Seminar, LNM 1346, pp. 527–543. Springer (1988)

  23. 23.

    Richter-Gebert, J.: Mnëv’s Universality Theorem Revisited. Séminaire Lotharingien de Combinatoire 34 (1995)

  24. 24.

    Richter-Gebert, J.: Perspectives on Projective Geometry—A Guided Tour through Real and Complex Geometry. Springer, New York (2011)

    Google Scholar 

  25. 25.

    Schaefer, M., Štefankovič, D.: Fixed points, nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60, 172–193 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, sequences A250001 and A288567. http://oeis.org

  27. 27.

    Snoeynik, J., Hershberger, J.: Sweeping arrangements of curves. In: Goodman, J.E., Pollack, R.D., Steiger, W.L. (eds.) Discrete & Computational Geometry, DIMACS DMTCS Series, vol. 6, pp. 309–349. AMS (1991)

  28. 28.

    Stein, W.A., et al.: Sage Mathematics Software (Version 8.0), The Sage Development Team (2017). http://www.sagemath.org

  29. 29.

    Stein, W.A., et al.: Sage Reference Manual: Algebraic Numbers and Number Fields (Release 8.0) (2017). http://doc.sagemath.org/pdf/en/reference/graphs/graphs.pdf

  30. 30.

    Stein, W.A., et al.: Sage Reference Manual: Graph Theory (Release 8.0). (2017). http://doc.sagemath.org/pdf/en/reference/number_fields/number_fields.pdf

  31. 31.

    Suvorov, P.: Isotopic but not rigidly isotopic plane systems of straight lines. In: Topology and Geometry – Rohlin Seminar, LNM 1346, pp. 545–556 Springer (1988)

  32. 32.

    Tutte, W.T.: A census of planar triangulations. Can. J. Math. 14, 21–38 (1962)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Manfred Scheucher.

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Dedicated to the memory of Ricky Pollack.

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Partially supported by the DFG Grants FE 340/11-1 and FE 340/12-1. Manfred Scheucher was partially supported by the ERC Advanced Research Grant No. 267165 (DISCONV). We gratefully acknowledge the computing time granted by TBK Automatisierung und Messtechnik GmbH and by the Institute of Software Technology, Graz University of Technology. We also thank the anonymous reviewers for valuable comments.

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Felsner, S., Scheucher, M. Arrangements of Pseudocircles: On Circularizability. Discrete Comput Geom (2019). https://doi.org/10.1007/s00454-019-00077-y

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