Skip to main content
SpringerLink
Log in

Shape preserving rational [3/2] Hermite interpolatory subdivision scheme

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

In this paper, a new Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the Rational [3/2] Bernstein Bezier polynomial. We call it the [3/2]-scheme. The limit function of the [3/2]-scheme interpolates both the function values and their derivatives. The proposed scheme has three shape parameters \(w_{0}, w_{1}\) and \(w_{2}\). It is shown that if \(w_{1}=\frac{w_{0}+w_{2}}{2}\), then the [3/2]-scheme reproduces linear polynomial and is \(C^{1}\) provided \(w_{0}\) and \(w_{2}\) lie in a region of convergence. The scheme also satisfies the shape preserving properties, i.e., monotonicity and convexity. We also compare the [3/2]-scheme with other existing schemes like the [2/2]-scheme and the Merrien scheme introduced recently. An error analysis shows that the [3/2]-scheme is better than the [2/2]-scheme and the Merrien scheme. Further, it is observed that in case \(w_{0}=w_{1}=w_{2}\), the [3/2]-scheme reduces to the Merrien scheme.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Albrecht, G., Romani, L.: Convexity preserving interpolatory subdivision with conic precision. Appl. Math. Comput. 219(8), 4049–4066 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Bebarta, S., Jena, M.K.: Shape preserving hermite subdivision scheme constructed from quadratic polynomial. Int. J. Appl. Comput. Math. 7(6), 1–22 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision, vol. 453. American Mathematical Society, Providence (1991)

    MATH  Google Scholar 

  4. Costantini, P., Manni, C.: A geometric approach for Hermite subdivision. Numer. Math. 115(3), 333–369 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charina, M., Conti, C., Mejstrik, T., Merrien, J.L.: Joint spectral radius and ternary Hermite subdivision. Adv. Comput. Math. 47(2), 1–23 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cotronei, M., Moosmüller, C., Sauer, T., Sissouno, N.: Level-dependent interpolatory Hermite subdivision schemes and wavelets. Constr. Approx. 50(2), 341–366 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Conti, C., Cotronei, M., Sauer, T.: Convergence of level-dependent Hermite subdivision schemes. Appl. Numer. Math. 116, 119–128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Conti, C., Hüning, S.: An algebraic approach to polynomial reproduction of Hermite subdivision schemes. J. Comput. Appl. Math. 349, 302–315 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conti, C., Merrien, J.L., Romani, L.: Dual Hermite subdivision schemes of de Rham-type. BIT Numer. Math. 54(4), 955–977 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dubuc, S., Merrien, J.L.: Convergent vector and Hermite subdivision schemes. Constr. Approx. 23(1), 1–22 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubuc, S., Merrien, J.L.: de Rham transform of a Hermite subdivision scheme. Approximation Theory XII, pp. 121–132 (2008)

  12. Dubuc, S., Merrien, J.L.: Hermite subdivision schemes and Taylor polynomials. Constr. Approx. 29(2), 219–245 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dubuc, S.: Scalar and hermite subdivision schemes. Appl. Comput. Harmonic Anal. 21(3), 376–394 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Dyn, N., Levin, D.: Analysis of Hermite-interpolatory subdivision schemes. Spline Functions and Theory of Wavelets. CRM Proc. Lect. Notes 18, 105–113 (1998)

  16. Goodman, T.N., Ong, B.H.: Shape-preserving interpolation by splines using vector subdivision. Adv. Comput. Math. 22(1), 49–77 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hussain, M.Z., Sarfraz, M., Shaikh, T.S.: Shape preserving rational cubic spline for positive and convex data. Egypt. Inform. J. 12(3), 231–236 (2011)

    Article  Google Scholar 

  18. Hüning, S.: Polynomial reproduction of Hermite subdivision schemes of any order. Math. Comput. Simul. 176, 195–205 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jena, M.K.: A Hermite interpolatory subdivision scheme constructed from quadratic rational Bernstein-Bezier spline. Math. Comput. Simul. 187, 433–448 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jena, M.K., Shunmugaraj, P., Das, P.C.: A non-stationary subdivision scheme for generalizing trigonometric spline surfaces to arbitrary meshes. Comput. Aided Geom. Des. 20(2), 61–77 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jena, H., Jena, M.K.: Construction of trigonometric box splines and the associated non-stationary subdivision schemes. Int. J. Appl. Comput. Math. 7(4), 129 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jena, H., Jena, M.K.: A hybrid non-stationary subdivision scheme based on triangulation. Int. J. Appl. Comput. Math. 7(4), 1–32 (2021)

    MathSciNet  MATH  Google Scholar 

  23. Kocić, L.M., Milovanović, G.V.: Shape preserving approximations by polynomials and splines. Comput. Math. Appl. 33(11), 59–97 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kuijt, F., van Damme, R.M.: Shape preserving C2 interpolatory subdivision schemes. Universiteit Twente, Onderafdeling der Toegepaste Wiskunde (1998)

    MATH  Google Scholar 

  25. Kuijt, F., Van Damme, R.: Shape preserving interpolatory subdivision schemes for nonuniform data. J. Approx. Theory 114(1), 1–32 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Merrien, J.L., Sablonnière, P.: Rational splines for Hermite interpolation with shape constraints. Comput. Aided Geom. Des. 30(3), 296–309 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Merrien, J.L., Sauer, T.: Extended Hermite subdivision schemes. J. Comput. Appl. Math. 317, 343–361 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Merrien, J.L., Sauer, T.: Generalized Taylor operators and polynomial chains for Hermite subdivision schemes. Numer. Math. 142(1), 167–203 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  30. Moosmüller, C., Sauer, T.: Factorization of Hermite subdivision operators from polynomial over-reproduction. J. Approx. Theory 271, 105645 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  31. Moosmüller, C., Dyn, N.: Increasing the smoothness of vector and Hermite subdivision schemes. IMA J. Numer. Anal. 39(2), 579–606 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  32. Moosmüller, C., Hüning, S., Conti, C.: Stirling numbers and Gregory coefficients for the factorization of Hermite subdivision operators. IMA J. Numer. Anal. 41(4), 2936–2961 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  33. Mortenson, M.E.: Geometric Modeling, pp. 128–129. McGraw Hill Education (India) Private Limited, New York (2006)

  34. Sabin, M.: Recent progress in subdivision: a survey. Advances in Multiresolution for Geometric Modelling, pp. 203–230 (2005)

  35. Sarfraz, M.: Visualization of positive and convex data by a rational cubic spline interpolation. Inf. Sci. 146(1–4), 239–254 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Schumaker, L.I.: On shape preserving quadratic spline interpolation. SIAM J. Numer. Anal. 20(4), 854–864 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang, H., Kim, J., Yoon, J.: A shape preserving corner cutting algorithm with an enhanced accuracy. Appl. Math. Lett. 137, 108487 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yang, H., Yoon, J.: A shape preserving C2 non-linear, non-uniform, subdivision scheme with fourth-order accuracy. Appl. Comput. Harmonic Anal. 60, 267–292 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, Z., Zheng, H., Zhou, J.: Convergence analysis of Hermite subdivision schemes of any arity. Appl. Numer. Math. 183, 279–300 (2023)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author’s research is supported by Biju Patnaik Research Fellowship of the Department of Science and Technology (DST), Odisha (ST-SCST-MISC-0014-2019).

Author information

Authors and Affiliations

  1. Department of Mathematics, VSS University of Technology, Sidhi Vihar, Burla, Sambalpur, 768018, India

    Shubhashree Bebarta & Mahendra Kumar Jena

Authors
  1. Shubhashree Bebarta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mahendra Kumar Jena
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mahendra Kumar Jena.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bebarta, S., Jena, M.K. Shape preserving rational [3/2] Hermite interpolatory subdivision scheme. Calcolo 60, 8 (2023). https://doi.org/10.1007/s10092-022-00503-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-022-00503-3

Keywords

Mathematics Subject Classification

Search

Navigation