Non-uniform B-Spline Subdivision Using Refine and Smooth

  • Thomas J. Cashman
  • Neil A. Dodgson
  • Malcolm A. Sabin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4647)


Subdivision surfaces would be useful in a greater number of applications if an arbitrary-degree, non-uniform scheme existed that was a generalisation of NURBS. As a step towards building such a scheme, we investigate non-uniform analogues of the Lane-Riesenfeld ‘refine and smooth’ subdivision paradigm. We show that the assumptions made in constructing such an analogue are critical, and conclude that Schaefer’s global knot insertion algorithm is the most promising route for further investigation in this area.


Subdivision Scheme Geometric Series Subdivision Surface Smooth Formulation General Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Boehm, W.: Inserting new knots into B-spline curves. Computer-Aided Design 12(4), 199–201 (1980)CrossRefGoogle Scholar
  2. 2.
    Cohen, E., Lyche, T., Riesenfeld, R.: Discrete B-splines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics. Computer Graphics and Image Processing 14(2), 87–111 (1980)CrossRefGoogle Scholar
  3. 3.
    Gasciola, G., Romani, L.: A general matrix representation for non-uniform B-spline subdivision with boundary control. Draft paper (2006)Google Scholar
  4. 4.
    Goldman, R., Schaefer, S.: Global Knot Insertion Algorithms. Presentation at the University of Kaiserslautern (January 17, 2007)Google Scholar
  5. 5.
    Goldman, R., Warren, J.: An extension of Chaiken’s algorithm to B-spline curves with knots in geometric progression. CVGIP: Graphical Models and Image Processing 55(1), 58–62 (1993)CrossRefGoogle Scholar
  6. 6.
    Gregory, J., Qu, R.: Nonuniform corner cutting. Computer Aided Geometric Design 13(8), 763–772 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Lane, J., Riesenfeld, R.: A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 2(1), 35–46 (1980)zbMATHCrossRefGoogle Scholar
  8. 8.
    Lyche, T., Morken, K.: Making the OSLO Algorithm More Efficient. SIAM Journal on Numerical Analysis 23(3), 663–675 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ramshaw, L.: Blossoming: A Connect-the-Dots Approach to Splines. Technical Report 19, Digital Systems Research Center (1987)Google Scholar
  10. 10.
    Sederberg, T., Zheng, J., Sewell, D., Sabin, M.: Non-Uniform Recursive Subdivision Surfaces. In: Proceedings of the 25th annual conference on Computer Graphics and Interactive Techniques, pp. 387–394 (1998)Google Scholar
  11. 11.
    Warren, J.: Binary subdivision schemes for functions over irregular knot sequences. In: Dæhlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces. Vanderbilt U.P, pp. 543–562 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Thomas J. Cashman
    • 1
  • Neil A. Dodgson
    • 1
  • Malcolm A. Sabin
    • 2
  1. 1.Computer Laboratory, University of Cambridge, CB3 0FDEngland
  2. 2.Numerical Geometry Ltd, 26 Abbey Lane, Lode, Cambridge CB5 9EPEngland

Personalised recommendations