Composite \(\sqrt{2}\) Subdivision Surfaces

  • Guiqing Li
  • Weiyin Ma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4077)


This paper presents a new unified framework for subdivisions based on a \(\sqrt{2}\) splitting operator, the so-called composite \(\sqrt{2}\) subdivision. The composite subdivision scheme generalizes 4-direction box spline surfaces for processing irregular quadrilateral meshes and is realized through various atomic operators. Several well-known subdivisions based on both the \(\sqrt{2}\) splitting operator and 1-4 splitting for quadrilateral meshes are properly included in the newly proposed unified scheme. Typical examples include the midedge and 4-8 subdivisions based on the \(\sqrt{2}\) splitting operator that are now special cases of the unified scheme as the simplest dual and primal subdivisions, respectively. Variants of Catmull-Clark and Doo-Sabin subdivisions based on the 1-4 splitting operator also fall in the proposed unified framework. Furthermore, unified subdivisions as extension of tensor-product B-spline surfaces also become a subset of the proposed unified subdivision scheme. In addition, Kobbelt interpolatory subdivision can also be included into the unified framework using VV-type (vertex to vertex type) averaging operators.


Subdivision Scheme Splitting Operator Quadrilateral Mesh Subdivision Surface Atomic Operator 
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.


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  1. 1.
    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
  2. 2.
    Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Applied and Computational Harmonic Analysis 3, 186–200 (1996)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertram, M.: Biorthogonal Loop-Subdivision Wavelets. Computing 72, 29–39 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Zorin, D., Schröder, P.: A unified framework for primal/dual quadrilateral subdivision scheme. Computer Aided Geometric Design 18(5), 429–454 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Stam, J.: On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. Computer Aided Geometric Design 18(5), 383–396Google Scholar
  6. 6.
    Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10(6), 356–360Google Scholar
  7. 7.
    Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10(6), 350–355 (1978)CrossRefGoogle Scholar
  8. 8.
    Oswald, P., Schröder, P.: Composite Primal/Dual \(\sqrt{3}\) -Subdivision Schemes. Computer Aided Geometric Design 20(2), 135–164 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kobbelt, L.: \(\sqrt{3}\) -Subdivision. In: SIGGRAPH 2000, pp. 103–112 (2000)Google Scholar
  10. 10.
    Oswald, P.: Designing composite triangular subdivision schemes. Computer Aided Geometric Design 22(7), 659–679 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Maillot, J., Stam, J.: A unified subdivision scheme for polygonal modeling. Computer Graphics Forum (Eurographics 2001) 20(3), 471–479 (2001)CrossRefGoogle Scholar
  12. 12.
    Warren, J., Schaefer, S.: A factored approach to subdivision surfaces. Computer Graphics & Applications 24(3), 74–81 (2004)CrossRefGoogle Scholar
  13. 13.
    Stam, J., Loop, C.: Quad/triangle subdivision. Computer Graphics Forum 22(1), 79–85 (2003)CrossRefGoogle Scholar
  14. 14.
    Velho, L.: Using semi-regular 4–8 meshes for subdivision surfaces. Journal of Graphics Tool 5(3), 35–47 (2000)Google Scholar
  15. 15.
    Velho, L.: Stellar subdivision grammars. In: Eurographics Symposium on Geometry Processing, pp. 188–199 (2003)Google Scholar
  16. 16.
    Peters, J., Reif, U.: The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics 16(4), 420–431 (1997)CrossRefGoogle Scholar
  17. 17.
    Habib, A., Warren, J.: Edge and vertex insertion for a class of subdivision surfaces. Computer Aided Geometric Design 16(4), 223–247 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Velho, L., Zorin, D.: 4-8 Subdivision. Computer Aided Geometric Design 18(5), 397–427 (2001)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Li, G., Ma, W., Bao, H.: Subdivision for Quadrilateral meshes. The Visual Computer 20(2-3), 180–198 (2004)CrossRefGoogle Scholar
  20. 20.
    Li, G., Ma, W., Bao, H.: Interpolatory -Subdivision surfaces. In: Proceedings of Geometric Modeling and Processing 2004, pp. 180–189 (2004)Google Scholar
  21. 21.
    Kobbelt, L.: Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum (Proceedings of EUROGRAPHICS 1996) 15(3), 409–410 (1996)CrossRefGoogle Scholar
  22. 22.
    Li, G., Ma, W.: A method for constructing interpolatory subdivision schemes and blending subdivisions (2005) (Submitted for publication)Google Scholar
  23. 23.
    Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Berlin (2002)MATHGoogle Scholar
  24. 24.
    Warren, J., Weimer: Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann Publisher, San Francisco (2002)Google Scholar
  25. 25.
    de Boor, C., Hollig, K., Riemenschneiger, S.: Box Splines. Springer, New York (1993)MATHGoogle Scholar
  26. 26.
    Sovakar, A., Kobbelt, L.: API design for adaptive subdivision schemes. Computer & Graphics 28(1), 67–72 (2004)CrossRefGoogle Scholar
  27. 27.
    Li, G., Ma, W.: Adaptive unified subdivisions with sharp features (preprint, 2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Guiqing Li
    • 1
  • Weiyin Ma
    • 2
  1. 1.School of Computer Science and EngineeringSouth China University of TechnologyGuangzhouChina
  2. 2.Department of Manufacturing Engineering and Engineering ManagementCity University of Hong KongHong Kong (SAR)China

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