iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://doi.org/10.1007/BF02519040
Convergence of subdivision and degree elevation | Advances in Computational Mathematics Skip to main content
Springer Nature Link
Log in

Convergence of subdivision and degree elevation

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

This paper presents a short, simple, and general proof showing that the control polygons generated by subdivision and degree elevation converge to the underlying splines, box-splines, or multivariate Bézier polynomials, respectively. The proof is based only on a Taylor expansion. Then the results are carried over to rational curves and surfaces. Finally, an even shorter but as simple proof is presented for the fact that subdivided Bézier polygons converge to the corresponding curve.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. W. Boehm, Inserting new knots into B-spline curves, Comp. Aided Design 12(1980)199–201.

    Article  Google Scholar 

  2. A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Memoirs Amer. Math. Soc. 93(1991)453.

    MathSciNet  Google Scholar 

  3. E. Cohen, T. Lyche and R.F. Riesenfeld, Discrete B-splines and subdivision techniques in computeraided geometric design and computer graphics, Comp. Graphics Image Proc. 14(1980)87–111.

    Article  Google Scholar 

  4. E. Cohen and L.L. Schumaker, Rates of convergence of control polygons, Comp. Aided Geom. Design 2(1985)229–235.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Dahmen, N. Dyn and D. Levin, On the convergence rates of subdivision algorithms for box spline surfaces, Constr. Approx. 1(1985)305–322.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. Dahmen, Subdivision algorithms converge quadratically, J. Comp. Appl. Math. 16(1986) 145–158.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. De Boor, Cutting corners always works, Comp. Aided Geom. Design 4(1987)125–131.

    Article  MATH  Google Scholar 

  8. G. Farin, Subsplines über Dreiecken, Diss., TU Braunschweig (1979).

  9. G. Farin, Triangular Bernstein-Bézier patches, Comp. Aided Geom. Design. 3(1986)83–127.

    Article  MathSciNet  Google Scholar 

  10. G. Farin,Curves and Surfaces for Computer Aided Geometric Design—A Practical Guide, 2nd ed. (Academic Press, 1990).

  11. J.A. Gregory and R. Qu, Non-uniform corner cutting, Technical Report, Brunel University (1988).

  12. J.M. Lane and R.F. Riesenfeld, A theoretical development for the computer generation of piecewise polynomial surfaces, IEEE Trans. Pattern Anal. Machine Int. PAMI-2(1980)35–45.

    Article  MATH  Google Scholar 

  13. C. Loop, private communication (1988).

  14. C.A. Micchelli and H. Prautzsch, Uniform refinement of curves, Lin. Alg. Appl. 114(1989)841–870.

    Article  MATH  MathSciNet  Google Scholar 

  15. H. Prautzsch and B. Piper, A fast algorithm to raise the degree of spline curves, Comp. Aided Geom. Design 8(1991)253–265.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Prautzsch, The generation of box spline surfaces—A generalization of the subdivision algorithm, Report No. 07/04, TU Braunschweig (1984).

  17. H. Prautzsch, Unterteilungsalgorithmen für multivariate Splines—Ein geometrischer Zugang, Diss., TU Braunschweig (1984).

  18. H. Prautzsch, Linear subdivision, Lin. Alg. Appl. 143(1991)223–230.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Prautzsch, On convex Bézier triangles, Math. Mod. Numer. Anal. 26(1992)23–36.

    MATH  MathSciNet  Google Scholar 

  20. R. Schaback, Comment in Oberwolfach (June, 1992).

  21. L.L. Schumaker,Spline Functions—Basic Theory (Wiley, 1981).

Download references

Author information

Authors and Affiliations

  1. Fakultät für Informatik, Universität Karlsruhe, 76128, Karlsruhe, Germany

    Hartmut Prautzsch & Leif Kobbelt

Authors
  1. Hartmut Prautzsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leif Kobbelt
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prautzsch, H., Kobbelt, L. Convergence of subdivision and degree elevation. Adv Comput Math 2, 143–154 (1994). https://doi.org/10.1007/BF02519040

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02519040

Keywords

Search

Navigation