Cylinders Through Five Points: Complex and Real Enumerative Geometry

  • Daniel Lichtblau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4869)


It is known that five points in ℝ3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspond to actual cylinders in ℝ3. We partially classify the case of no real solutions in terms of the geometry of the five given points. We also investigate the special case where the five given points are coplanar, as it differs from the generic case for both complex and real valued solution cardinalities.


Enumerative geometry Gröbner bases nonlinear systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Becker, E., Marinari, M., Mora, T., Traverso, C.: The shape of the Shape Lemma. In: ISSAC 1994. Proceedings of the 1994 International Symposium on Symbolic and Algebraic Computation, pp. 129–133. ACM Press, New York (1994)CrossRefGoogle Scholar
  2. 2.
    Becker, T., Kredel, H., Weispfenning, V.: Gröbner bases: a computational approach to commutative algebra. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  3. 3.
    Borcea, C., Goaoc, X., Lazard, S., Petitjean, S.: Common tangents to spheres in ℝ3. Discrete & Computational Geometry 35(2), 287–300 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bottema, O., Veldkamp, G.: On the lines in space with equal distances to n given points. Geometriae Dedicata 6, 121–129 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bryant, J., Lichtblau, D.: Cylindersthroughfivepoints (2007),
  6. 6.
    Chaperon, T., Goulette, F.: Extracting cylinders in full 3-d data using a random sampling method and the Gaussian image. In: VMV 2001. Proceedings of the 6th International Fall Workshop Vision, Modeling, and Visualization, Stuttgart, Germany, Aka GmbH, pp. 35–42 (November 2001)Google Scholar
  7. 7.
    Chaperon, T., Goulette, F.: A note on the construction of right circular cylinders through five 3d points. Technical report, Centre de Robotique, Ecole des Mines de Paris (2003)Google Scholar
  8. 8.
    Devillers, O., Mourrain, B., Preparata, F., Trebuchet, P.: On circular cylinders by four or five points in space. Discrete and Computational Geometry 28, 83–104 (2003)MathSciNetGoogle Scholar
  9. 9.
    Durand, C.B.: Symbolic and numerical techniques for constraint solving. PhD thesis, Purdue University, Department of Computer Science, Major Professor-Christoph M. Hoffmann (1998)Google Scholar
  10. 10.
    Hoffmann, C.M., Yuan, B.: On spatial constraint solving approaches. In: Richter-Gebert, J., Wang, D. (eds.) ADG 2000. LNCS (LNAI), vol. 2061, pp. 1–15. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Kalkbrenner, M.: Solving systems of algebraic equations using Gröbner bases. In: Davenport, J.H. (ed.) ISSAC 1987 and EUROCAL 1987. LNCS, vol. 378, pp. 282–292. Springer, Heidelberg (1989)Google Scholar
  12. 12.
    Lazard, D., Rouillier, F.: Solving parametric polynomial systems. Journal of Symbolic Computation 42(6), 636–667 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lichtblau, D.: Cylinders through five points: computational algebra and geometry (2006),
  14. 14.
    Lichtblau, D.: Simulating perturbation to enumerate parametrized systems. In: Manuscript (2006)Google Scholar
  15. 15.
    Macdonald, I.G., Pach, J., Theobald, T.: Common tangents to four unit balls in ℝ3. Discrete and Computational Geometry 26(1), 1–17 (2001)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Manubens, M., Montes, A.: Improving the DISPGB algorithm using the discriminant ideal. Journal of Symbolic Computation 41, 1245–1263 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Manubens, M., Montes, A.: Minimal canonical comprehensive Gröbner systems. In: ArXiv Mathematics e-prints (2007)Google Scholar
  18. 18.
    Megyesi, G.: Lines tangent to four unit spheres with coplanar centres. Discrete and Computational Geometry 26(4), 493–497 (2001)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Mishra, B.: Algorithmic algebra. Springer, New York (1993)zbMATHGoogle Scholar
  20. 20.
    Montes, A., Recio, T.: Automatic discovery of geometry theorems using minimal canonical comprehensive groebner systems. In: ArXiv Mathematics e-prints (2007)Google Scholar
  21. 21.
    Rusin, D.: The Mathematical Atlas,
  22. 22.
    Schömer, E., Sellen, J., Teichmann, M., Yap, C.: Smallest enclosing cylinders. Algorithmica 27, 170–186 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sturmfels, B.: Polynomial equations and convex polytopes. American Mathematical Monthly 105(10), 907–922 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Weispfenning, V.: Comprehensive Gröbner bases. Journal of Symbolic Computation 14(1), 1–29 (1992)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Daniel Lichtblau
    • 1
  1. 1.Wolfram Research, Inc., 100 Trade Center Dr., Champaign, IL 61820USA

Personalised recommendations