Skip to main content
SpringerLink
Log in

Cubic spline quasi-interpolants on Powell–Sabin partitions

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

By using the polarization identity, we propose a family of quasi-interpolants based on bivariate \({\fancyscript{C}}^1\) cubic super splines defined on triangulations with a Powell–Sabin refinement. Their spline coefficients only depend on a set of local function values. The quasi-interpolants reproduce cubic polynomials and have an optimal approximation order.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Busch, J.R.: A note on Lagrange interpolation in \(\mathbb{R}^2\). Revista de la Unión Matemática Argentina 36, 33–38 (1990)

    MATH  MathSciNet  Google Scholar 

  2. de Casteljau, P.: Le Lissage. Hermes, Paris (1990)

  3. Chena, S.-K., Liu, H.-W.: A bivariate \({\fancyscript{C}}^1\) cubic super spline space on Powell–Sabin triangulation. Comput. Math. Appl. 56, 1395–1401 (2008)

    Google Scholar 

  4. Chung, K.C., Yao, T.H.: On a lattices admitting unique Lagrange interpolation. SIAM J. Numer. Math. Anal. 14, 735–743 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dierckx, P.: On calculating normalized Powell–Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Foucher, F., Sablonnière, P.: Approximating partial derivatives of first and second order by quadratic spline quasi-interpolants on uniform meshes. Math. Comput. Simul. 77, 202–208 (2008)

    Article  MATH  Google Scholar 

  7. Goldman, R.N.: Blossoming and knot insertion algorithms for B-spline curves. Comput. Aided Geom. Des. 7, 69–78 (1990)

    Article  MATH  Google Scholar 

  8. Ibáñez, M.J., Lamnii, A., Mraoui, H., Sbibih, D.: Construction of spherical spline quasi-interpolants based on blossoming. J. Comput. Appl. Math. 234, 131–145 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  10. Lamnii, M., Mraoui, H., Tijini, A.: Construction of quintic Powell–Sabin spline quasi-interpolants based on blossoming. J. Comput. Appl. Math. 234, 190–209 (2013)

    Article  MathSciNet  Google Scholar 

  11. Lamnii, M., Mraoui, H., Zidna, A.: Local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation. Appl. Math. Comput. 219, 10456–10467 (2013)

    Article  MathSciNet  Google Scholar 

  12. Lamnii, M., Mraoui, H., Tijini, A., Zidna, A.: A normalized basis for \({\fancyscript{C}}^1\) cubic super spline space on Powell–Sabin triangulation. Math. Comput. Simul. 99, 108–124 (2014)

    Google Scholar 

  13. Lamnii, M., Mazroui, A., Tijini, A.: Raising the approximation order of multivariate quasi-interpolants. BIT Numer. Math. (2014). doi:10.1007/s10543-014-0470-8

    MATH  MathSciNet  Google Scholar 

  14. Liu, H.-W., Chen, S.-K., Chen, Y.P.: Bivariate \({\fancyscript{C}}^1\) cubic spline space over Powell–Sabin type-1 refinement. J. Inf. Comput. Sci. 4, 151–160 (2007)

    Google Scholar 

  15. Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Des. 6, 323–358 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sbibih, D., Serghini, A., Tijini, A.: Polar forms and quadratic spline quasi-interpolants on Powell Sabin partitions. Appl. Numer. Math. 59, 938–958 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sbibih, D., Serghini, A., Tijini, A.: Normalized trivariate B-splines on Worsey-Piper split and quasi-interpolants. BIT Numer. Math. 52, 221–249 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Seidel, H.-P.: Symmetric recursive algorithms for surfaces: B-patches and the de Boor algorithm for polynomials over triangles. Constr. Approx. 7, 257–279 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  20. Speleers, H.: A normalized basis for reduced Clough–Tocher splines. Comput. Aided Geom. Des. 27(9), 700–712 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Speleers, H.: A normalized basis for quintic Powell–Sabin splines. Comput. Aided Geom. Des. 27(6), 438–457 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Speleers, H.: Multivariate normalized Powell–Sabin B-splines and quasi-interpolants. Comput. Aided Geom. Des. 30, 2–19 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  23. Speleers, H.: Construction of normalized B-splines for a family of smooth spline spaces over Powell–Sabin triangulations. Constr. Approx. 37(1), 41–72 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  24. Speleers, H.: A family of smooth quasi-interpolants defined over Powell–Sabin triangulations. Technical Report TW617, KU Leuven (2012)

Download references

Acknowledgments

The authors thank the referees for the useful corrections and comments which improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, University Hassan First, Settat, Morocco

    A. Lamnii

  2. Faculty of Science, University Mohammed First, Oujda, Morocco

    M. Lamnii & H. Mraoui

Authors
  1. A. Lamnii
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Lamnii
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Mraoui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Mraoui.

Additional information

Communicated by Tom Lyche.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lamnii, A., Lamnii, M. & Mraoui, H. Cubic spline quasi-interpolants on Powell–Sabin partitions. Bit Numer Math 54, 1099–1118 (2014). https://doi.org/10.1007/s10543-014-0489-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-014-0489-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation