Invariant-Based Characterization of the Relative Position of Two Projective Conics

Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 151)


In this paper, we give predicates of bidegree at most (6, 6) in the input for characterizing the relative position of two projective conics. By relative position we mean the morphology of the intersection, the rigid isotopy class and which conic is inside the other when applicable. The predicates are derived by analyzing the algebraic invariant theory of pencils of conics and related constructions.


Projective Conic Pencil Gene Rigid Isotopy Class General Algebraic System Algebraic Invariant Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    D. Avritzer and R. Miranda. Stability of pencils of quadrics in <$>\mathbb{P}^4<$>. The Boletin de la Sociedad Matematica Mexicana, III Ser. 5(2):281–300, 1999.MATHMathSciNetGoogle Scholar
  2. [2]
    S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry, Volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2003.MATHGoogle Scholar
  3. [3]
    E. Briand. Duality for couples of conics. Unpublished, 2005.Google Scholar
  4. [4]
    E. Briand. Equations, inequations and inequalities characterizing the configurations of two real projective conics. Applicable Algebra in Engineering, Communication and Computing, 18(1–2):21–52, 2007.CrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Bromwich. Quadratic Forms and Their Classification by Means of Invariant Factors. Cambridge Tracts in Mathematics and Mathematical Physics, 1906.Google Scholar
  6. [6]
    J. Cremona. Classical invariants and 2-descent on elliptic curves. Journal of Symbolic Computation, 31(1/2):71–87, 2001.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    C. D'Andrea and A. Dickenstein. Explicit formulas for the multivariate resultant. Journal of Pure and Applied Algebra, 164:59–86, 2001.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    O. Devillers, A. Fronville, B. Mourrain, and M. Teillaud. Algebraic methods and arithmetic filtering for exact predicates on circle arcs. Comput. Geom. Theory Appl., 22:119–142, 2002.MATHMathSciNetGoogle Scholar
  9. [9]
    I. Dolgachev. Lectures on Invariant Theory. Cambridge University Press, 2003. London Mathematical Society Lecture Note Series, Volume 296.Google Scholar
  10. [10]
    L. Dupont, D. Lazard, S. Lazard, and S. Petitjean. Near-optimal parameterization of the intersection of quadrics: II. A classification of pencils. Journal of Symbolic Computation, 43(3):192–215, 2008.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    E. Elliott. An Introduction to the Algebra of Quantics. Clarendon Press, Oxford, 1913.Google Scholar
  12. [12]
    F. Etayo, L. González-Vega, and N. del Rio. A new approach to characterizing the relative position of two ellipses depending on one parameter. Computer Aided Geometric Design, 23(4):324–350, 2006.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    I. Gelfand, M. Kapranov, and A. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston, 1994.MATHCrossRefGoogle Scholar
  14. [14]
    O. Glenn. A Treatise on the Theory of Invariants. Ginn and Company, Boston, 1915.Google Scholar
  15. [15]
    J.H. Grace and A. Young. The Algebra of Invariants. Cambridge University Press, 1903.Google Scholar
  16. [16]
    D.A. Gudkov. Plane real projective quartic curves. In Topology and Geometry - Rohlin Seminar, Volume 1346 of Lecture Notes in Math., pages 341–347. Springer-Verlag, 1988.Google Scholar
  17. [17]
    D. Hilbert. Über die Theorie der algebraischen Formen. Math. Ann., 36:473–534, 1890.CrossRefMathSciNetGoogle Scholar
  18. [18]
    D. Hilbert. Über die vollen Invariantensysteme. Math. Ann., 42:313–373, 1893.CrossRefMathSciNetGoogle Scholar
  19. [19]
    H. Kraft and C. Procesi. Classical Invariant Theory, A Primer, 2000. Lecture Notes.Google Scholar
  20. [20]
    T. Lam. The Algebraic Theory of Quadratic Forms. W.A. Benjamin, Reading, MA, 1973.MATHGoogle Scholar
  21. [21]
    H. Levy. Projective and Related Geometries. The Macmillan Co., New York, 1964.MATHGoogle Scholar
  22. [22]
    Y. Liu and F.-L. Chen. Algebraic conditions for classifying the positional relationships between two conics and their applications. J. Comput. Sci. Technol., 19(5):665–673, 2004.CrossRefGoogle Scholar
  23. [23]
    P.J. Olver. Classical Invariant Theory. Cambridge University Press, 1999.Google Scholar
  24. [24]
    D. Pervouchine. Orbits and Invariants of Matrix Pencils. PhD thesis, Moscow State University, 2002.Google Scholar
  25. [25]
    B. Sturmfels. Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation. Springer-Verlag, 1993.MATHGoogle Scholar
  26. [26]
    J. Todd. Projective and Analytical Geometry. Pitman, London, 1947.MATHGoogle Scholar
  27. [27]
    J. A. Todd. Combinant forms associated with a pencil of conics. Proc. Lond. Math. Soc., II Ser. 50:150–168, 1948.MATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    C. Tu, W. Wang, B. Mourrain, and J. Wang. Using signature sequences to classify intersection curves of two quadrics. Computer Aided Geometric Design, 2008, to appear.Google Scholar
  29. [29]
    H.W. Turnbull. The Theory of Determinants, Matrices and Invariants. Blackie (London, Glasgow), 1929.Google Scholar
  30. [30]
    F. Uhlig. A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil. Linear Algebra and Its Applications, 14:189–209, 1976.MATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    W. Wang and R. Krasauskas. Interference analysis of conics and quadrics. In Topics in Algebraic Geometry and Geometric Modeling, Volume 334 of Contemp. Math., pages 25–36. Amer. Math. Soc., 2003.Google Scholar
  32. [32]
    W. Wang, J. Wang, and M.-S. Kim. An algebraic condition for the separation of two ellipsoids. Computer Aided Geometric Design, 18(6):531–539, 2001.MATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    H. Weyl. The Classical Groups, Their Invariants and Representations. Princeton University Press, 1946.Google Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.LORIA-INRIA, Campus scientifiqueVandœuvre-lès-Nancy cedexFrance

Personalised recommendations