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Discrete & Computational Geometry

, Volume 60, Issue 2, pp 381–405 | Cite as

On the Reconstruction Problem for Pascal Lines

  • Abdelmalek Abdesselam
  • Jaydeep Chipalkatti
Article
  • 48 Downloads

Abstract

Given a sextuple of distinct points ABCDEF on a conic, arranged into an array \(\big [ \begin{array}{ccc} A &{} B &{} C\\ F &{} E &{} D \end{array} \big ],\) Pascal’s theorem says that the points \(AE \cap BF, BD \cap CE, AD \cap CF\) are collinear. The line containing them is called the Pascal of the array, and one gets altogether 60 such lines by permuting the points. In this paper we prove that the initial sextuple can be explicitly reconstructed from four specifically chosen Pascals. The reconstruction formulae are encoded by some transvectant identities which are proved using the graphical calculus for binary forms.

Keywords

Pascal lines Transvectants Invariant theory of binary forms 

Mathematics Subject Classification

14N05 22E70 51N35 

References

  1. 1.
    Abdesselam, A.: On the volume conjecture for classical spin networks. J. Knot Theory Ramif. 21(3), 1250022 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abdesselam, A., Chipalkatti, J.: Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture. Adv. Math. 208(2), 491–520 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baker, H.F.: Principles of Geometry, vol. II. Cambridge University Press, Cambridge (1930)Google Scholar
  4. 4.
    Blinn, J.F.: Lines in space, part 8: line(s) through four lines. IEEE Comput. Graph. Appl. 24(5), 100–106 (2004)CrossRefGoogle Scholar
  5. 5.
    Castravet, A.-M., Tevelev, J.: Hypertrees, projections, and moduli of stable rational curves. J. Reine Angew. Math. 675, 121–180 (2013)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chipalkatti, J.: On the coincidences of Pascal lines. Forum Geom. 16, 1–21 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Clebsch, A.: Theorie der Binären Algebraischen Formen. B.G. Teubner, Leipzig (1872)zbMATHGoogle Scholar
  8. 8.
    Conway, J., Ryba, A.: The Pascal mysticum demystified. Math. Intell. 34(3), 4–8 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cvitanović, P.: Group Theory. Birdtracks, Lie’s, and Exceptional Groups. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  10. 10.
    Dolotin, V., Morozov, A.: Introduction to Non-linear Algebra. World Scientific, Hackensack (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Grace, J.H., Young, A.: The Algebra of Invariants. Chelsea, New York (1962)zbMATHGoogle Scholar
  12. 12.
    Harris, J.: Galois groups of enumerative problems. Duke Math. J. 46(4), 685–724 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Howard, B., Millson, J., Snowden, A., Vakil, R.: The equations for the moduli space of \(n\) points on the line. Duke Math. J. 146(2), 175–226 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Joubert, P.: Sur l’équation du sixième degré. C. R. Acad. Sci. Paris 64, 1025–1029 (1867)Google Scholar
  15. 15.
    Kadison, L., Kromann, M.T.: Projective Geometry and Modern Algebra. Birkhäuser, Boston (1996)zbMATHGoogle Scholar
  16. 16.
    Kraft, H.: A result of Hermite and equations of degree 5 and 6. J. Algebra 297(1), 234–253 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kung, J.P.S., Rota, G.-C.: The invariant theory of binary forms. Bull. Am. Math. Soc. (N.S.) 10(1), 27–85 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lafforgue, V.: Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale (2012). arXiv:1209.5352
  19. 19.
    Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts, vol. 44. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
  20. 20.
    Pedoe, D.: How many Pascal lines has a sixpoint? Math. Gaz. 25(264), 110–111 (1941)CrossRefGoogle Scholar
  21. 21.
    Pedoe, D.: Geometry: A Comprehensive Course. Dover Books on Advanced Mathematics, 2nd edn. Dover, New York (1988)zbMATHGoogle Scholar
  22. 22.
    Richter-Gebert, J., Lebmeir, P.: Diagrams, tensors and geometric reasoning. Discrete Comput. Geom. 42(2), 305–334 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Salmon, G.: A Treatise on Conic Sections, 6th edn. Chelsea, New York (2005)zbMATHGoogle Scholar
  24. 24.
    Schreck, P., Mathis, P., Marinković, V., Janičić, P.: Wernick’s list: a final update. Forum Geom. 16, 69–80 (2016)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Seidenberg, A.: Lectures in Projective Geometry. Van Nostrand, New York (1962)zbMATHGoogle Scholar
  26. 26.
    Wernick, W.: Triangle constructions with three located points. Math. Mag. 55(4), 227–230 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of MathematicsMachray Hall, University of ManitobaWinnipegCanada

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