Equations, inequations and inequalities characterizing the configurations of two real projective conics

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Abstract

Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well–adapted to the study of the relative position of two conics defined by equations depending on parameters.

Keywords

Arrangements of conics Rigid isotopy Relative position of two conics Classical invariant theory 
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References

  1. 1.
    Basu, S., Pollack, R., Roy, M.F.: Algorithms in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin Heidelberg New York (2003)Google Scholar
  2. 2.
    Berger, M.: Géométrie, 2nd edn. Nathan (1990)Google Scholar
  3. 3.
    Bröcker, T., tom Dieck, T.: Representations of compact Lie groups. GTM, vol. 98, 1st edn. Springer, Berlin Heidelberg New York (1985)Google Scholar
  4. 4.
    Brown, C.: SLFQ (Simplifying Large Formulas with QEPCAD B). http://www.cs.usna. edu/~qepcad/SLFQ/Home.htmlGoogle Scholar
  5. 5.
    Casey, J.: A Treatise of the Analytical Geometry of the Point, Line, Circle and Conic Sections, 2nd edn. Dublin University Press (1893). Digitalized by Gallica, http://gallica.bnf.frGoogle Scholar
  6. 6.
    Cox D., Little J. and O’Shea D. (1997). Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, Berlin Heidelberg New York Google Scholar
  7. 7.
    Degtyarev, A.: Quadratic transformations RP2 → RP2. In: Topology of Real Algebraic Varieties and Related Topics. Amer. Math. Soc. Transl. Ser. 2, vol. 173, pp. 61–71. Amer. Math. Soc., Providence, RI (1996)Google Scholar
  8. 8.
    Devillers, O., Fronville, A., Mourrain, B., Teillaud, M.: Algebraic methods and arithmetic filtering for exact predicates on circle arcs. Comput. Geom. 22(1–3), 119–142 (2002). 16th ACM Symposium on Computational Geometry (Hong Kong, 2000)Google Scholar
  9. 9.
    Etayo F., González-Vega L. and del Rio N. (2006). A new approach for characterizing the relative position of two ellipses depending on one parameter. Comput. Aided Geom. Desi. 23(4): 324–350 CrossRefGoogle Scholar
  10. 10.
    Glenn, O.E.: A Treatise on the Theory of Invariants. Ginn and Company (1915). Available as EText 9933 of project Gutenberg, at http://www.gutenberg.netGoogle Scholar
  11. 11.
    Goresky, M., MacPherson, R.: Stratified Morse Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 14. Springer, Berlin Heidelberg New York (1988)Google Scholar
  12. 12.
    Gudkov, D.A.: Plane real projective plane curves. In: Viro, O.Y. (ed.) Topology and Geometry – Rokhlin Seminar. Lecture Notes in Mathematics vol. 1346, pp. 341–347. Springer, Berlin Heidelberg New York (1988)Google Scholar
  13. 13.
    Gudkov, D.A., Polotovskii, G.M.: Stratification of the space of 4th order curves. the joining of stratums. Uspekhi Mat. Nauk 42(4), 152 (1987) (in Russian)Google Scholar
  14. 14.
    Gudkov, D.A., Polotovskii, G.M.: Stratification of the space of plane algebraic 4th order curves with respect to algebraic–geometric types. I. Deposited in VINITI, No 5600–B87 (1987). 1–55 (in Russian)Google Scholar
  15. 15.
    Gudkov, D.A., Polotovskii, G.M.: Stratification of the space of plane algebraic 4th order curves with respect to algebraic–geometric types. II. Deposited in VINITI, No 6331–B90 (1990). 1–33 (in Russian)Google Scholar
  16. 16.
    Korchagin A.B. and Weinberg D.A. (2002). The isotopy classification of affine quartic curves. Rocky Mountain J. Math. 32(1): 255–347 CrossRefMathSciNetGoogle Scholar
  17. 17.
    Levy H. (1964). Projective and Related Geometries. The Macmillan Co., New York MATHGoogle Scholar
  18. 18.
    Olver P.J. (1999). Classical Invariant Theory. Cambridge University Press, Cambridge MATHGoogle Scholar
  19. 19.
    Uhlig F. (1976). A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil. Linear Algebra Appl. 14(3): 189–209 CrossRefMathSciNetGoogle Scholar
  20. 20.
    Wang, W., Krasauskas, R.: Interference analysis of conics and quadrics. In: Topics in algebraic geometry and geometric modeling. Contemp. Math., vol. 334, pp. 25–36. Amer. Math. Soc., Providence, RI (2003)Google Scholar
  21. 21.
    Wang W., Wang J. and Kim M.S. (2001). An algebraic condition for the separation of two ellipsoids. Comput. Aided Geom. Design 18(6): 531–539 CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Dpto. Matemáticas, estadística y computaciónUniversidad de CantabriaSantanderSpain

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