Algebraic Curves of Low Convolution Degree

  • Jan Vršek
  • Miroslav Lávička
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6920)

Abstract

Studying convolutions of hypersurfaces (especially of curves and surfaces) has become an active research area in recent years. The main characterization from the point of view of convolutions is their convolution degree, which is an affine invariant associated to a hypersurface describing the complexity of the shape with respect to the operation of convolution. Extending the results from [1], we will focus on the two simplest classes of planar algebraic curves with respect to the operation of convolution, namely on the curves with the convolution degree one (so called LN curves) and two. We will present an algebraic analysis of these curves, provide their decomposition, and study their properties.

Keywords

Curve Versus Algebraic Curf Geometric Design Rational Curf Hermite Interpolation 
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.

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References

  1. 1.
    Vršek, J., Lávička, M.: On convolution of algebraic curves. Journal of Symbolic Computation 45(6), 657–676 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Farouki, R.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Heidelberg (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Farouki, R., Sakkalis, T.: Pythagorean hodographs. IBM Journal of Research and Development 34(5), 736–752 (1990)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Farouki, R., Sakkalis, T.: Pythagorean-hodograph space curves. Adv. Comput. Math. 2, 41–66 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kosinka, J., Jüttler, B.: G 1 Hermite interpolation by Minkowski Pythagorean hodograph cubics. Computer Aided Geometric Design 23, 401–418 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kosinka, J., Jüttler, B.: MOS surfaces: Medial surface transforms with rational domain boundaries. In: Martin, R., Sabin, M.A., Winkler, J.R. (eds.) Mathematics of Surfaces 2007. LNCS, vol. 4647, pp. 245–262. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Kosinka, J., Lávička, M.: On rational Minkowski Pythagorean hodograph curves. Computer Aided Geometric Design 27(7), 514–524 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Computer Aided Geometric Design 15, 223–249 (1998)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pottmann, H., Peternell, M.: Applications of Laguerre geometry in CAGD. Computer Aided Geometric Design 15, 165–186 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jüttler, B.: Triangular Bézier surface patches with linear normal vector field. In: Cripps, R. (ed.) The Mathematics of Surfaces VIII. Information Geometers, pp. 431–446 (1998)Google Scholar
  11. 11.
    Peternell, M., Manhart, F.: The convolution of a paraboloid and a parametrized surface. Journal for Geometry and Graphics 7(2), 157–171 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Sampoli, M.L., Peternell, M., Jüttler, B.: Rational surfaces with linear normals and their convolutions with rational surfaces. Computer Aided Geometric Design 23(2), 179–192 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lávička, M., Bastl, B.: Rational hypersurfaces with rational convolutions. Computer Aided Geometric Design 24(7), 410–426 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lee, I.K., Kim, M.S., Elber, G.: Polynomial/rational approximation of Minkowski sum boundary curves. Graphical Models and Image Processing 60(2), 136–165 (1998)CrossRefGoogle Scholar
  15. 15.
    Gravesen, J., Jüttler, B., Šír, Z.: On rationally supported surfaces. Computer Aided Geometric Design 25, 320–331 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Šír, Z., Gravesen, J., Jüttler, B.: Curves and surfaces represented by polynomial support functions. Theoretical Computer Science 392(1-3), 141–157 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Arrondo, E., Sendra, J., Sendra, J.R.: Parametric generalized offsets to hypersurfaces. Journal of Symbolic Computation 23, 267–285 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Arrondo, E., Sendra, J., Sendra, J.R.: Genus formula for generalized offset curves. Journal of Pure and Applied Algebra 136, 199–209 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sendra, J.R., Sendra, J.: Algebraic analysis of offsets to hypersurfaces. Mathematische Zeitschrift 237, 697–719 (2000)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Šír, Z., Bastl, B., Lávička, M.: Hermite interpolation by hypocycloids and epicycloids with rational offsets. Computer Aided Geometric Design 27, 405–417 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jüttler, B.: Hermite interpolation by Pythagorean hodograph curves of degree seven. Math. Comp. 70, 1089–1111 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jüttler, B., Sampoli, M.: Hermite interpolation by piecewise polynomial surfaces with rational offsets. Computer Aided Geometric Design 17, 361–385 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Meek, D.S., Walton, D.J.: Geometric Hermite interpolation with Tschirnhausen cubics. J. Comput. Appl. Math. 81(2), 299–309 (1997)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Brieskorn, E., Knörer, H.: Plane algebraic curves. Birkhaüser, Basel (1986)CrossRefGoogle Scholar
  25. 25.
    Cox, D.A., Little, J., O’Shea, D.: Using algebraic geometry, 2nd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  26. 26.
    Fulton, W.: Algebraic Curves. Benjamin, New York (1969)MATHGoogle Scholar
  27. 27.
    Kim, M.S., Elber, G.: Problem reduction to parameter space. In: Proceedings of the 9th IMA Conference on the Mathematics of Surfaces, pp. 82–98. Springer, Heidelberg (2000)Google Scholar
  28. 28.
    Walker, R.: Algebraic Curves. Princeton University Press, Princeton (1950)MATHGoogle Scholar
  29. 29.
    Lávička, M., Bastl, B., Šír, Z.: Reparameterization of curves and surfaces with respect to their convolution. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, J.-L., Mørken, K., Schumaker, L.L. (eds.) MMCS 2008. LNCS, vol. 5862, pp. 285–298. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  30. 30.
    Cohen, H., Frey, G. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC (2005)Google Scholar
  31. 31.
    Hartshorne, R.: Algebraic Geometry. Springer, Heidelberg (1977)CrossRefMATHGoogle Scholar
  32. 32.
    Moon, H.: Minkowski Pythagorean hodographs. Computer Aided Geometric Design 16, 739–753 (1999)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jan Vršek
    • 1
  • Miroslav Lávička
    • 1
  1. 1.Faculty of Applied Sciences, Department of MathematicsUniversity of West BohemiaPlzeňCzech Republic

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