Direct Computation of a Control Vertex Position on any Subdivision Level

  • Loïc Barthe
  • Leif Kobbelt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2768)


In this paper, we present general closed form equations for directly computing the position of a vertex at different subdivision levels for both triangular and quadrilateral meshes. These results are obtained using simple computations and they lead to very useful applications, especially for adaptive subdivision. We illustrate our method on Loop’s and Catmull-Clark’s subdivision schemes.


Subdivision Scheme Quadrilateral Mesh Central Vertex Subdivision Surface Closed Form Equation 
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.
    Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design 10(6), 350–355 (1978)CrossRefGoogle Scholar
  2. 2.
    Doo, D., Sabin, M.A.: Analysis of the behaviour of recursive subdivision surfaces near extraordinary points. Computer Aided Design 10(6), 356–360 (1978)CrossRefGoogle Scholar
  3. 3.
    Zorin, D., Schröder, P.: Subdivision for modeling and animation. In: SIGGRAPH 2000 course notes (2000)Google Scholar
  4. 4.
    Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah (1987) Google Scholar
  5. 5.
    Vasilescu, M., Terzopoulos, D.: Adaptive meshes and shells: Irregular triangulation, discontinuities and hierarchical subdivision. In: Proceedings of Computer Vision and Pattern Recognition, pp. 829–832 (1992)Google Scholar
  6. 6.
    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh refinement techniques. Wiley-Teubner, Chichester (1996)zbMATHGoogle Scholar
  7. 7.
    Zorin, D., Schröder, P., Sweldens, W.: Interactive multiresolution mesh editing. In: Proceedings of SIGGRAPH 1997, pp. 259–268. ACM, New York (1997)CrossRefGoogle Scholar
  8. 8.
    Xu, Z., Kondo, K.: Local subdivision process with Doo-Sabin subdivision surfaces. In: Proceedings of Shape Modeling International, pp. 7–12 (2002)Google Scholar
  9. 9.
    Lee, A., Moreton, H., Hoppe, H.: Displaced subdivision surfaces. In: Proceedings of SIGGRAPH 2000, pp. 85–94. ACM, New York (2000)CrossRefGoogle Scholar
  10. 10.
    Hoppe, H.: View-dependent refinement of progressive meshes. In: Proceedings of SIGGRAPH 1997, pp. 189–198. ACM, New York (1997)CrossRefGoogle Scholar
  11. 11.
    Kamen, Y., Shirman, L.: Triangle Rendering Using Adaptive Subdivision. IEEE Computer Graphics and Applications 18(2), 356–360 (1998)CrossRefGoogle Scholar
  12. 12.
    Hoppe, H.: Smooth view-dependent level-of-detail control and its application in terrain rendering. IEEE Visualization, 35–42 (1998)Google Scholar
  13. 13.
    Dyn, N., Levin, D., Gregory, J.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Transaction on Graphics 9(2), 160–169 (1990)zbMATHCrossRefGoogle Scholar
  14. 14.
    Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. In: Proceedings of SIGGRAPH 1997, pp. 189–192. ACM, New York (1996)Google Scholar
  15. 15.
    Halstead, M., Kass, M., DeRose, T.: Efficient, fair interpolation using Catmull-Clark surfaces. In: Proceedings of SIGGRAPH 1993, pp. 35–43. ACM, New York (1993)CrossRefGoogle Scholar
  16. 16.
    Hoppe, H., DeRose, T., Duchamp, T., Halstead, M.: Piecewise smooth surfaces reconstruction. In: Proceedings of SIGGRAPH 1994, pp. 295–302. ACM, New York (1994)CrossRefGoogle Scholar
  17. 17.
    Stam, J.: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In: Proceedings of SIGGRAPH 1998, pp. 395–404. ACM, New York (1998)CrossRefGoogle Scholar
  18. 18.
    Kobbelt, L.: \(\sqrt{3}\)-subdivision. In: Proceedings of SIGGRAPH 2000, pp. 103–112. ACM, New York (2000)CrossRefGoogle Scholar
  19. 19.
    Warren, J., Weimer, H.: Subdivision methods for geometric design: a constructive approach, pp. 209–212. Morgan Kaufman, San Fransisco (2002)Google Scholar
  20. 20.
    OpenMesh subdivision tool,

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Loïc Barthe
    • 1
  • Leif Kobbelt
    • 1
  1. 1.Computer Graphics GroupRWTH AachenAachenGermany

Personalised recommendations