Nothing Special   »   [go: up one dir, main page]

Skip to main content
Log in

Smoothness of Subdivision Surfaces with Boundary

  • Published:
Constructive Approximation Aims and scope

Abstract

Subdivision rules for meshes with boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise \(C^{\ell }\)-continuous boundary limit curves and ensure \(C^{\ell }\)-continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359–397, 2000), we present in this paper general necessary and sufficient conditions for \(C^{\ell }\)-continuity of subdivision schemes for surfaces with boundary, and specialize these to practically applicable sufficient conditions for \(C^1\)-continuity. We use these conditions to show that certain boundary rules for Loop and Catmull–Clark are in fact \(C^1\) continuous.

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

Access this article

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Biermann, H., Levin, A., Zorin, D.: Piecewise smooth subdivision surfaces with normal control. In: SIGGRAPH 2000 Conference Proceedings, Annual Conference Series. ACM SIGGRAPH, Addison Wesley, July 2000

  2. Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10(6), 350–355 (1978)

    Article  Google Scholar 

  3. DeRose, T., Kass, M., Truong, T.: Subdivision surfaces in character animation. Proceedings of SIGGRAPH 98, Orlando, Florida, p. 85–94, July 1998. ISBN 0-89791-999-8

  4. Doo, D.: A subdivision algorithm for smoothing down irregularly shaped polyhedrons. In: Proceedings on Interactive Techniques in Computer Aided Design, p. 157–165. Bologna (1978)

  5. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern geometry–methods and applications. Part I. Springer-, New York, second edition, 1992. The geometry of surfaces, transformation groups, and fields, Translated from the Russian by Robert G. Burns

  6. Grundel, S.: Eigenvalue optimization in \(C^2\) subdivision and boundary subdivision. PhD thesis, New York University, Courant Institute of Mathematical Sciences, New York, NY, 10011 (2011)

  7. Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., Stuetzle, W.: Piecewise smooth surface reconsruction. In: Computer Graphics Proceedings, Annual Conference Series, p. 295–302. ACM Siggraph (1994)

  8. Lee, J.M.: Smooth Manifolds. Springer, Berlin (2003)

    Book  Google Scholar 

  9. Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)

  10. Munkres, J.: Elementary Differential Topology. Princeton University Press, Princeton (1966)

    MATH  Google Scholar 

  11. Nasri, A.H.: Polyhedral subdivision methods for free-form surfaces. ACM Trans. Graph. 6(1), 29–73 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Nasri, A.H.: Boundary-corner control in recursive-subdivision surfaces. Comput. Aided Des. 23(6), 405–410 (1991)

    Article  MATH  Google Scholar 

  13. Nasri, A.H.: Surface interpolation on irregular networks with normal conditions. Comput. Aided Geom. Des. 8, 89–96 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Peters, J., Fan, J.: On the complexity of smooth spline surfaces from quad meshes. Comput. Aided Geom. Des. 27(1), 96–105 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Peters, J., Reif, U.: Analysis of generalized B-spline subdivision algorithms. SIAM J. Numer. Anal. 35(2), 728–748 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Peters, J., Reif, U.: Subdivision surfaces, volume 3 of Geometry and Computing. Springer, Berlin, 2008. With introductory contributions by Nira Dyn and Malcolm Sabin

  17. Reif, U.: A unified approach to subdivision algorithms near extraordinary vertices. Comput. Aided Geom. Des. 12, 153–174 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schweitzer, J.E.: Analysis and application of subdivision surfaces. PhD thesis, University of Washington, Seattle (1996)

  19. Wallner, J., Dyn, N.: Convergence and \(C^1\) analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Des. 22(7), 593–622 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Wallner, J., Nava Yazdani, E., Weinmann, A.: Convergence and smoothness analysis of subdivision rules in Riemannian and symmetric spaces. Adv. Comput. Math. 34(2), 201–218 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Xie, G., Yu, T.P.-Y.: Smoothness analysis of nonlinear subdivision schemes of homogeneous and affine invariant type. Constr. Approx. 22(2), 219–254 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Zorin, D.: Subdivision and multiresolution surface representations. PhD thesis, Caltech, Pasadena (1997)

  23. Zorin, D.: A method for analysis of \(c^1\)-continuity of subdivision surfaces. SIAM J. Numer. Anal. 37(4), 1677–1708 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  24. Zorin, D.: Smoothness of stationary subdivision on irregular meshes. Constr. Approx. 16(3), 359–397 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Sara Grundel.

Additional information

Communicated by Wolfgang Dahmen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biermann, H., Grundel, S. & Zorin, D. Smoothness of Subdivision Surfaces with Boundary. Constr Approx 42, 1–29 (2015). https://doi.org/10.1007/s00365-015-9292-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-015-9292-4

Keywords

Mathematics Subject Classification

Navigation