Towards Subdivision Surfaces C2 Everywhere

  • Malcolm SabinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10521)


The conditions for subdivision surfaces which are piecewise polynomial in the regular region to have continuity higher than C1 were identified by Reif [7]. The conditions are ugly and although schemes have been identified and implemented which satisfy them, those schemes have not proved satisfactory from other points of view. This paper explores what can be created using schemes which are not piecewise polynomial in the regular regions, and the picture looks much rosier. The key ideas are (i) use of quasi-interpolation (ii) local evaluation of coefficients in the irregular context. A new method for determining lower bounds on the Hölder continuity of the limit surface is also proposed.


Subdivision surfaces Continuity Reproduction 



My thanks go to a very diligent referee who bothered to construct a counterexample disproving a conjecture in my first draft of this paper and also pointed out many places where the original text was not clear. I also thank colleagues: Cedric Gerot for helping me understand that counterexample, Leif Kobbelt and Ulrich Reif for prompt replies to my emails requesting clarifications, and to Ioannis Ivrissimtzis for bringing the moving least squares ideas to my attention.


  1. 1.
    McLain, D.H.: Two dimensional interpolation from random data. Comput. J. 19(2), 178–181 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Catmull, E.E., Clark, J.: Recursively generated B-spline surfaces on topological meshes. CAD 10(6), 350–355 (1978)Google Scholar
  3. 3.
    Doo, D.W.H., Sabin, M.A.: Behaviour of recursive division surfaces near extraordinary points. CAD 10(6), 356–360 (1978)Google Scholar
  4. 4.
    Loop, C.T.: Smooth subdivision surfaces based on triangles. M.S. Mathematics thesis, University of Utah (1987)Google Scholar
  5. 5.
    Dyn, N., Levin, D., Gregory, J.: A butterfly subdivision scheme for surface interpolation with tension control. ACM ToG 9(2), 160–169 (1990)CrossRefzbMATHGoogle Scholar
  6. 6.
    Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision. Memoirs of the American Mathematical Society, no. 453. AMS, Providence (1991)Google Scholar
  7. 7.
    Reif, U.: A degree estimate for subdivision surfaces of higher regularity. In: Proceedings of AMS, vol. 124, pp. 2167–2175 (1996)Google Scholar
  8. 8.
    Reif, U.: TURBS - topologically unrestricted rational B-splines. Constr. Approx. 14(1), 57–77 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Prautzsch, H.: Smoothness of subdivision surfaces at extraordinary points. Adv. Comput. Math. 9(3–4), 377–389 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Prautzsch, H., Reif, U.: Degree estimates for \(C^\text{ k }\)-piecewise polynomial subdivision surfaces. Adv. Comput. Math. 10(2), 209–217 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kobbelt, L.: \(\sqrt{3}\)-subdivision. In: Proceedings of SIGGRAPH 2000, pp. 103–112 (2000)Google Scholar
  12. 12.
    Velho, L.: Quasi 4–8 subdivision. CAGD 18(4), 345–357 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Levin, D.: Presentation at Dagstuhl Workshop (2000)Google Scholar
  14. 14.
    Levin, D.: Mesh-independent surface interpolation. In: Brunnett, G., Hamann, B., Müller, H., Linsen, L. (eds.) Geometric Modeling for Scientific Visualization, pp. 37–49. Springer, Heidelberg (2004)Google Scholar
  15. 15.
    Peters, J., Reif, U.: Shape Characterization of subdivision surfaces - basic principles. CAGD 21, 585–599 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Karciauskas, K., Peters, J., Reif, U.: Shape Characterization of subdivision surfaces - case studies. CAGD 21, 601–614 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Levin, A.: The importance of polynomial reproduction in piecewise-uniform subdivision. In: Martin, R., Bez, H., Sabin, M. (eds.) IMA 2005. LNCS, vol. 3604, pp. 272–307. Springer, Heidelberg (2005). doi: 10.1007/11537908_17. ISBN 3-540-28225-4CrossRefGoogle Scholar
  18. 18.
    Levin, A.: Modified subdivision surfaces with continuous curvature. ACM (TOG) 25(3), 1035–1040 (2006)CrossRefGoogle Scholar
  19. 19.
    Zulti, A., Levin, A., Levin, D., Taicher, M.: C2 subdivision over triangulations with one extraordinary point. CAGD 23(2), 157–178 (2006)zbMATHGoogle Scholar
  20. 20.
    Hormann, K., Sabin, M.A.: A family of subdivision schemes with cubic precision. CAGD 25(1), 41–52 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dyn, N., Hormann, K., Sabin, M.A., Shen, Z.: Polynomial reproduction by symmetric subdivision schemes. J. Approx. Theor. 155(1), 28–42 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Boyé, S., Guennebaud, G., Schlick, C.: Least squares subdivision surfaces. Comput. Graph. Forum 29(7), 2021–2028 (2010)CrossRefGoogle Scholar
  23. 23.
    Augsdörfer, U.H., Dodgson, N.A., Sabin, M.A.: Artifact analysis on B-splines, box-splines and other surfaces defined by quadrilateral polyhedra. CAGD 28(3), 177–197 (2010)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Augsdörfer, U.H., Dodgson, N.A., Sabin, M.A.: Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra. CAGD 28(3), 198–211 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Deng, C., Hormann, K.: Pseudo-spline subdivision surfaces. Comput. Graph. Forum 33(5), 227–236 (2014)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Numerical Geometry Ltd.Ely, CambsUK

Personalised recommendations