Advertisement

Fundamental Splines on Triangulations

  • GÜnther NÜrnberger
  • Frank Zeilfelder
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 145)

Abstract

We analyse properties of the fundamental splines associated with local Lagrange interpolation (cf. Nürnberger and Zeilfelder [14-16]) for C1 splines on triangulations. These splines are zero except at one point of an interpolation set and form a basis of the spline space. It is proved that the support of the fundamental splines is small. Moreover, we show that the interpolation methods can be further simplified for cubic C1 splines on separable triangulations. In this case, the supports of the fundamental spline basis are even smaller. Finally, we describe an algorithm for modifying an arbitrary triangulation such that the resulting triangulation is separable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. de Boor: B-form basics, In: Geometric Modeling, G. Farin, (ed.), 131–148, SIAM, Philadelphia, 1987.Google Scholar
  2. 2.
    C. K. Chui: Multivariate Splines, CBMS 54, SIAM, Philadelphia, 1988.CrossRefGoogle Scholar
  3. 3.
    R. W. Clough, J. L. Tocher:Finite element stiffness matrices for analysis of plates in bending, In: Proc. Conf. on Matrix Methods in Structural Mechanics, Wright Patterson A. F. B., Ohio, 1965. Google Scholar
  4. 4.
    O. Davydov, G. NÜrnberger, F. Zeilfelder:Bivariate spline interpolation with optimal approximation order, Constr. Approx. 17 (2001), 181–208. CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    O. Davydov, F. Zeilfelder: Scattered data fitting by direct extension of local polynomials with bivariate splines, to appear in: Advances in Comp. Math., 2003. Google Scholar
  6. 6.
    G. Farin: Triangular Bernstein-Bézier patches, Comp. Aided Geom. Design 3 (1986), 83–127. CrossRefMathSciNetGoogle Scholar
  7. 7.
    G. Fraeijs de Veubeke: Bending and stretching of plates, In: Proc. Conf. on Matrix Methods in Structural Mechanics, Wright Patterson A. F. B., Ohio, 1965. Google Scholar
  8. 8.
    J. Haber, F. Zeilfelder, O. Davydov, H.-P. Seidel: Smooth approximation and rendering of large scattered data sets, In: Proceedings of IEEE Visualization 2001, T. Ertl, K. Joy, A. Varshney (eds.), 341–347, 571, IEEE, 2001. Google Scholar
  9. 9.
    G. Nürnberger: Approximation by Spline Functions, Springer, Berlin, 1989. CrossRefzbMATHGoogle Scholar
  10. 10.
    G. Nürnberger, L. L. Schumaker, F. Zeilfelder: Local Lagrange interpolation by bivariate C l cubic splines, In: Mathematical Methods in CAGD: Oslo 2000, T. Lyche, L.L. Schumaker, (eds.), 393-404, Vanderbilt University Press, Nashville, 2001. Google Scholar
  11. 11.
    G. Nürnberger, L. L. Schumaker, F. Zeilfelder: Lagrange Interpolation by C l cubic splines on Triangulations of Separable Quadrangulations, In: Approximation Theory X: Splines, Wavelets, and Applications, C. K. Chui, L.L. Schumaker, J. Stöckler, (eds.), 405-424, Vanderbilt University Press, Nashville, 2002. Google Scholar
  12. 12.
    G. Nürnberger, L. L. Schumaker, F. Zeilfelder: Lagrange interpolation by C l cubic splines on triangulated quadrangulations, to appear in: Advances in Comp. Math., 2003. Google Scholar
  13. 13.
    G. Nürnberger, F. Zeilfelder: Developments in bivariate spline interpolation, J. Comput. Appl. Math. 121 (2000), 125–152. CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    G. Nürnberger, F. Zeilfelder: Local Lagrange interpolation by cubic splines on a class of triangulations, In: Proc. Conf. Trends in Approximation Theory, K. Kopotun, T. Lyche, M. Neamtu, (eds.), 341–350, Vanderbilt University Press, Nashville, 2001.Google Scholar
  15. 15.
    G. Nürnberger, F. Zeilfelder: Local Lagrange interpolation on Powell-Sabin triangulations and terrain modelling, In: Recent Progress in Multivariate Approximation, W. Haußmann, K. Jet-ter, M. Reimer (eds.), 227–244, ISNM 137, Birkhäuser, Basel 2001.CrossRefGoogle Scholar
  16. 16.
    G. Nürnberger, F. Zeilfelder: Lagrange interpolation by bivariate C 1 - splines with optimal approximation order, to appear in: Advances in Comp. Math., 2003.Google Scholar
  17. 17.
    G. Sander: Bornes supérieures et inférieures dans l’analyse matricielle des plaques en flexion-torsion, Bull. Soc. Royale Science Liége 33 (1964), 456–494.MathSciNetGoogle Scholar
  18. 18.
    F. Zeilfelder: Scattered data fitting with bivariate splines, In: Principles of Multiresolution in Geometric Modelling, M. Floater, A. Iske, E. Quak (eds.), 243–286, Springer, Berlin, 2002.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2003

Authors and Affiliations

  • GÜnther NÜrnberger
    • 1
  • Frank Zeilfelder
    • 1
  1. 1.Institute for MathematicsUniversity of MannheimMannheimGermany

Personalised recommendations