Skip to main content
SpringerLink
Log in

On Constrained Nonlinear Hermite Subdivision

  • Published:
Constructive Approximation Aims and scope

Abstract

We determine shape-preserving regions and we describe a general setting to generate shape-preserving families for the 2-points Hermite subdivision scheme introduced by Merrien (Numer. Algorithms 2:187–200, [1992]). This general construction includes the shape-preserving families presented in Merrien and Sablonníere (Constr. Approx. 19:279–298, [2003]) and Pelosi and Sablonníere (C 1 GP Hermite Interpolants Generated by a Subdivision Scheme, Prépublication IRMAR 06–23, Rennes, [2006]). New special families are presented as particular examples. Nonstationary and nonuniform versions of such schemes, which produce smoother limits, are discussed.

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

Similar content being viewed by others

References

  1. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision. Mem. Amer. Math. Soc., vol. 93. American Mathematical Society, Providence (1991)

    Google Scholar 

  2. Costantini, P.: Curve and surface construction using variable degree polynomial splines. Comput. Aided Geom. Des. 24, 426–446 (2000)

    Google Scholar 

  3. Costantini, P., Manni, C.: Geometric construction of generalized cubic splines. Rend. Mat. 26, 327–338 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Costantini, P., Lyche, T., Manni, C.: On a class of weak Tchebycheff systems. Numer. Math. 101, 333–354 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Daubechies, I., Guskov, I., Sweldens, W.: Regularity of irregular subdivision. Constr. Approx. 15, 381–426 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Delbourgo, R., Gregory, J.A.: Shape preserving piecewise rational interpolation. SIAM J. Sci. Stat. Comput. 6, 967–976 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numer. 11, 73–144 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dubuc, S.: Scalar and Hermite subdivision schemes. Appl. Comput. Harmon. Anal. 21, 376–394 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dubuc, S., Merrien, J.-L.: A 4-point Hermite subdivision scheme. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces. Innov. Appl. Math., pp. 113–122. Vanderbilt University Press, Nashville (2001)

    Google Scholar 

  10. Goodman, T.N.T.: Shape preserving interpolation by curves. In: Levesley J., Anderson, I.J., Mason, J.C. (eds.) Algorithms for Approximation IV, pp. 24–35. University of Huddersfield Proceedings, Huddersfield (2002)

    Google Scholar 

  11. Goodman, T.N.T., Mazure, M.L.: Blossoming beyond extended Chebyshev spaces. J. Approx. Theory 109, 48–81 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gori, L., Pitolli, F.: A class of totally positive refinable functions. Rend. Mat. 20, 305–322 (2000)

    MATH  MathSciNet  Google Scholar 

  13. Koch, P.E., Lyche, T.: Interpolation with exponential B-splines in tension. In Farin, G. et al. (eds.) Geometric Modelling. Comput. Suppl. vol. 8, pp. 173–190. Springer, Berlin (1993)

    Google Scholar 

  14. Kvasov, B.I.: Algorithms for shape preserving local approximation with automatic selection of tension parameters. Comput. Aided Geom. Des. 17, 17–37 (2000)

    Article  MathSciNet  Google Scholar 

  15. Lamberti, P., Manni, C.: Tensioned quasi-interpolation via geometric continuity. Adv. Comput. Math. 20, 105–127 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Lyche, T., Merrien, J.L.: C 1 Interpolatory subdivision with shape constraints for curves. SIAM J. Numer. Anal. 44, 1095–1121 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Manni, C.: C 1 comonotone Hermite interpolation via parametric cubics. J. Comput. Appl. Math. 69, 143–157 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Manni, C., Pelosi, F.: Quasi-interpolants with tension properties from and in CAGD. Computing 72, 143–160 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Manni, C., Sablonníere, P.: C 1 comonotone Hermite interpolation via parametric surfaces. In: Dahlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in Computer Aided Geometric Design III, pp. 333–342. Vanderbilt University Press, Nashville (1995)

    Google Scholar 

  20. Mazure, M.-L.: Chebyshev–Bernstein bases. Comput. Aided Geom. Des. 16, 649–669 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Merrien, J.-L.: A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2, 187–200 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Merrien, J.-L.: Interpolants d’Hermite C 2 obtenus par subdivision. M2AN Math. Model. Numer. Anal. 33, 55–65 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Merrien, J.-L., Sablonníere, P.: Monotone and convex C 1 Hermite interpolants generated by a subdivision scheme. Constr. Approx. 19, 279–298 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Peña, J.M. (ed.): Shape Preserving Representations in Computer-Aided Geometric Design. Nova Science Publishers, New York (1999)

    MATH  Google Scholar 

  25. Pelosi, F., Sablonníere, P.: C 1 GP Hermite Interpolants Generated by a Subdivision Scheme. J. Comput. Appl. Math. (2007, to appear)

  26. Sablonníere, P.: Bernstein-type bases and corner cutting algorithms for C 1 Merrien’s curves. Adv. Comput. Math. 20, 229–246 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Schweikert, D.G.: An interpolation curve using a spline in tension. J. Math. Phys. 45, 312–317 (1966)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze Matematiche ed Informatiche, Università di Siena, Siena, Italy

    Paolo Costantini

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy

    Carla Manni

Authors
  1. Paolo Costantini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carla Manni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carla Manni.

Additional information

Communicated by Wolfgang Dahmen.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Costantini, P., Manni, C. On Constrained Nonlinear Hermite Subdivision. Constr Approx 28, 291–331 (2008). https://doi.org/10.1007/s00365-007-9001-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-007-9001-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation