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Subdivision Surfaces and Applications

  • Chiara Eva Catalano
  • Ioannis Ivrissimtzis
  • Ahmad Nasri
Part of the Mathematics and Visualization book series (MATHVISUAL)

After a short introduction on the fundamentals of subdivision surfaces, the more advanced material of this chapter focuses on two main aspects. First, shape interrogation issues are discussed; in particular, artifacts, typical of subdivision surfaces, are analysed. The second aspect is related to how structuring the geometric information: a multi-resolution approach is a natural choice for this geometric representation, and it can be seen as a possible way to structure geometry. Moreover, a first semantic structure can be given by a set of meaningful geometric constraints that the shape has to preserve, often due to the specific application context. How subdivision surfaces can cope with constraint-based modelling is treated in the chapter with a special attention to applications.

Keywords

Subdivision Scheme Subdivision Surface Control Polygon Valence Vertex Subdivision Rule 
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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References

  1. 1.
    Abbas A and A. Nasri. Interpolating meshes of curves by catmull-clark subdivision surfaces with a shape parameter. In The Ninth International Conference on Computer Aided Design and Computer Graphics (CAD-CG’05), CAD-CG, pages 107-112, Hong Kong, 2005. IEEE Press.CrossRefGoogle Scholar
  2. 2.
    N. Alkalai and N. Dyn. Optimizing 3D triangulations: Improving the initial triangulation for the butterfly subdivision scheme. In Neil Dodgson, Michael Floater, and Malcom Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 231-244. Springer, 2004.Google Scholar
  3. 3.
    A. A. Ball and D. J. T. Storry. Conditions for tangent plane continuity over recursively generated B-spline surfaces. ACM Transactions on Graphics, 7(2):83-102, 1988.zbMATHCrossRefGoogle Scholar
  4. 4.
    L. Barthe and L. Kobbelt. Subdivision scheme tuning around extraordinary vertices. Computer Aided Geometric Design, 21(6):561-583, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    H. Biermann, I. Boier-Martin, F. Bernardini, and D. Zorin. Cut-and-paste editing of multiresolution surfaces. In Proc. of SIGGRAPH, pages 312-321, 2002.Google Scholar
  6. 6.
    H. Biermann, I. Boier-Martin, D. Zorin, and F. Bernardini. Sharp features on multiresolution subdivision surfaces. In Conference Proceedings of Pacific Graphics, 2001.Google Scholar
  7. 7.
    H. Biermann, A. Levin, and D. Zorin. Piecewise smooth subdivision surfaces with normal control. In SIGGRAPH 00 Conference Proceedings, pages 113-120, 2000.Google Scholar
  8. 8.
    I. Boier-Martin and F. Bernardini. Subdivision-base representations for surface styling and design. In DIMACS Workshop on Computer Aided Design and Manufacturing, October 2003. DIMACS Center, Rutgers University, Piscataway, New Jersey.Google Scholar
  9. 9.
    G.P. Bonneau, G. Elber, S. Hahmann, and B. Sauvage. Multiresolution analysis. In L. De Floriani and M. Spagnuolo, editors, Shape Analysis and Structuring. Springer, 2007.Google Scholar
  10. 10.
    C.E. Catalano. Feature Based Methods for Free Form Surface Manipulation in Aesthetic Engineering. PhD thesis, Genova, 2004.Google Scholar
  11. 11.
    C.E. Catalano. Introducing design intent in discrete surface modelling. International Journal of Computer Applications in Technology (IJCAT), 23(2/3/4):108-119, 2005. Special Issue on Models and methods for representing and processing shape semantics.CrossRefGoogle Scholar
  12. 12.
    E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design, 10:350-355, 1978.CrossRefGoogle Scholar
  13. 13.
    F. Cirak, M. Ortiz, and P. Schr öder. Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Numerical Methods in Engineering, 47(12):2039-72, April 2000.zbMATHCrossRefGoogle Scholar
  14. 14.
    I. Daubechies, I. Guskov, and W. Sweldens. Regularity of irregular subdivision. Constructive Approximation, 15(3):381-426, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    T. DeRose, M. Kass, and T. Truong. Subdivision surfaces in character animation. In SIGGRAPH 98 Conference Proceedings, pages 85-94, 1998.Google Scholar
  16. 16.
    D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356-360, 1978.CrossRefGoogle Scholar
  17. 17.
    N. Dyn. Subdivision schemes in Computer-Aided Geometric Design. In W. Light, editor, Advances in numerical analysis, volume 2, chapter 2, pages 36-104. Clarendon Press, 1992.Google Scholar
  18. 18.
    N. Dyn. Analysis of convergence and smoothness by the formalism of Laurent polynomials. In A. Iske, E. Quak, and M. S. Floater, editors, Tutorials on Multiresolution in Geometric Modelling, chapter 3, pages 51-68. Springer, 2002.Google Scholar
  19. 19.
    N. Dyn and D. Levin. Subdivision schemes in geometric modelling. Acta Numerica, 11:73-144, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    N. Dyn, D. Levin, and J. A. Gregory. A 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design, 4:257-268, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    N. Dyn, D. Levin, and J. A. Gregory. A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics, 9(2):160-169, 1990.zbMATHCrossRefGoogle Scholar
  22. 22.
    G. Farin, J. Hoschek, and M. Kim, editors. Handbook of Computer Aided Geometric Design. Elsevier, 2002.Google Scholar
  23. 23.
    S. Green and G. Turkiyyah. Second order accurate constraints for subdivision finite elements. Numerical Methods in Engineering, 60(13), 2004.Google Scholar
  24. 24.
    H. Hoppe, T. DeRose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer, and W. Stuetzle. Piecewise smooth surface reconstruction. Computer Graphics, 28:295-302, 1994.Google Scholar
  25. 25.
    I. Ivrissimtzis, N. Dodgson, and M. Sabin. 5-subdivision. In N. Dodgson, M. Floater, and M. Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 285-300. Springer, 2004.Google Scholar
  26. 26.
    I. Ivrissimtzis, M. Sabin, and N. Dodgson. On the support of recursive subdivision. ACM Transactions on Graphics, 23(4):1043-1060, 2004.CrossRefGoogle Scholar
  27. 27.
    I. Ivrissimtzis and H.-P. Seidel. Evolutions of polygons in the study of subdivision surfaces. Computing, 72(1-2):93-104, 2004.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    T. Ju, J. Warren, G. Eichele, C. Thaller, W. Chiu, and J. Carson. A geometric database for gene expression data. In Eurographics Symposium on Geometric Processing. L. Kobbelt, P. Shroeder, H. Hoppe (Editors))p, 2003.Google Scholar
  29. 29.
    A. Khodakovsky and P. Schr öder. Fine level feature editing for subdivision surfaces. In Proc. ACM Solid Modeling, pages 203-211, 1999.Google Scholar
  30. 30.
    L. Kobbelt. Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer G√aphics Forum, 15(3):409-420, 1996.CrossRefGoogle Scholar
  31. 31.
    L. Kobbelt. 3 subdivision. In SIGGRAPH 00, Conference Proceedings, pages 103-112, 2000.Google Scholar
  32. 32.
    L. Kobbelt, M. Botsch, K. Kaehler, C. R össl, R. Schneider, and J. Vorsatz. Geometric modeling based on polygonal meshes. Tutorial T4, Eurographics 2000, 2000.Google Scholar
  33. 33.
    L. Kobbelt and P. Schr öder. A multiresolution framework for variational subdivision. ACM Trans. on Graph., 17(4):209-237, 1998.CrossRefGoogle Scholar
  34. 34.
    J. Kuragano, H. Suzuki, and F. Kimura. Generation of NC tool path for subdivision surfaces. In Proceedings of CAD/Graphics√ 2001, Kunming China, pages 676-682, 2001.Google Scholar
  35. 35.
    U. Labsik and G. Greiner. Interpolatory 3-subdivision. Computer Graphics Forum, 19(3):131-138, 2000.CrossRefGoogle Scholar
  36. 36.
    A. Lee, H. Moreton, and H. Hoppe. Displaced subdivision surfaces. In SIGGRAPH ’00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques, pages 85-94, New York, NY, USA, 2000. ACM Press/Addison-Wesley Publishing Co.CrossRefGoogle Scholar
  37. 37.
    A. Levin. Interpolating nets of curves by smooth subdivision surfaces. In Computer Graphics Proceedings, ACM SIGGRAPH 1999, pages 57-64, 1999.Google Scholar
  38. 38.
    A. Levin. Combined subdivision schemes. PhD thesis, School of Mathematical Science, Tel Aviv University, 2000.Google Scholar
  39. 39.
    N. Litke, A. Levin, and P. Schr öder. Fitting subdivision surfaces. IEEE Visualization, pages 319-324, October 1998.Google Scholar
  40. 40.
    N. Litke, A. Levin, and P. Schr öder. Trimming for subdivision surfaces. Computer Aided Geometric Design, 18(5):463-481, June 1998.Google Scholar
  41. 41.
    C. Loop. Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics, 1987.Google Scholar
  42. 42.
    C. Loop. Bounded curvature triangle mesh subdivision with the convex hull property. The Visual Computer Journal, 18(5-6):316-325, 2002.CrossRefGoogle Scholar
  43. 43.
    C. Loop. Smooth ternary subdivision of triangle meshes. In Proceedings of the 5th Conference on Curves and Surfaces, pages 295-302. Nashboro Press, 2003.Google Scholar
  44. 44.
    W. Ma, X. Ma, S. Tso, and Z. Pan. Subdivision surface fitting from a dense triangular mesh. In Proc. of Geometric Modeling and Processing, pages 94-103, July 1999.Google Scholar
  45. 45.
    M. Marinov, N. Dyn, and D. Levin. Geometrically controlled 4-point interpolatory schemes. In N. Dodgson, M. Floater, and M. Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 301-315. Springer, 2004.Google Scholar
  46. 46.
    A. Nasri. Polyhedral subdivision methods for free-form surfaces. ACM Transactions on Graphics, 6(1):29-73, 1987.zbMATHCrossRefGoogle Scholar
  47. 47.
    A. Nasri. Recursive subdivision of polygonal complexes and its applications in CAGD. Computer Aided Geometric Design, 17:595-619, 2000. Presented also at The 5th Siam Conferene On Geometric Design, Nashville, 1997.Google Scholar
  48. 48.
    A. Nasri. Constructing polygonal complexes with shape handles for curve interpolation by subdivision surfaces. Computer Aided Design, 33:753-765, 2001.CrossRefGoogle Scholar
  49. 49.
    A. Nasri. Interpolating an unlimited number of curves meeting at extraordinary points on subdivision surfaces. Computer Graphics Forum, 22(1):87-97, 2003.CrossRefGoogle Scholar
  50. 50.
    A. Nasri, Abbas A, and I. Hasbini. Skinning Catmull-Clark subdivision surfaces with incompatible cross-sectional curves. In Pacific Graphics 2003, pages 102-111, Canmore, Canada, 2003. IEEE Press. ISBN 0-7695-2028-6.Google Scholar
  51. 51.
    A. Nasri and M. Sabin. Taxonomy of interpolation conditions in recursive subdivision curves. The Visual Computer, 18(4):259-272, 2002.CrossRefGoogle Scholar
  52. 52.
    A. Nasri and M. Sabin. Taxonomy of interpolation conditions in recursive subdivision surfaces. Journal Visual Computer, 18(6):382-403, 2002.CrossRefGoogle Scholar
  53. 53.
    A. Nasri, M. Sabin, R. Abu Zaki, N. Nassiri, and R. Santina. Feature curves with cross curvature control on Catmull-Clark subdivision surfaces. volume 4035 of Lecture Notes in Computer Science, pages 761-768. Springer, 2006.Google Scholar
  54. 54.
    Y. Ohtake. Mesh Viewer. http://www.mpi-sb.mpg.de/∼ohtake/software. Last access October 2006.
  55. 55.
    P. Oswald and P. Schr öder. Composite primal/dual sqrt(3)-subdivision schemes. CAGD, 20(3):135-164, 2003.zbMATHGoogle Scholar
  56. 56.
    H.R. Pakdel and F.F. Samavati. Incremental Catmull-Clark subdivision. In 5th International Conference on 3-D Digital Imaging and Modeling, pages 95-102, Canada, June 2005. IEEE Computer Society Press.CrossRefGoogle Scholar
  57. 57.
    J. Peters and U. Reif. The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics, 16(4):420-431, 1997.CrossRefGoogle Scholar
  58. 58.
    J. Peters and U. Reif. Analysis of algorithms generalizing B-spline subdivision. SIAM Journal on Numerical Analysis, 35(2):728-748, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    H. Prautzsch. Smoothness of subdivision surfaces at extraordinary points. Advances in Computational Mathematics, 9(3-4):377-389, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  60. 60.
    H. Qin, C. Mandal, and B. C. Vemuri. Dynamic Catmull-Clark subdivision surfaces. IEEE Transactions on Visualization and Computer Graphics, 4(3):216-229, 1998.Google Scholar
  61. 61.
    U. Reif. A unified approach to subdivision algorithms near extraordinary vertices. Computer Aided Geometric Design, 12(2):153-174, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    U. Reif and P. Schr öder. Curvature integrability of subdivision surfaces. Advances in Computational Mathematics, 14(2):157-174, 2000.Google Scholar
  63. 63.
    M. Sabin and L. Barthe. Artifacts in recursive subdivision surfaces. In Curve and Surface Fitting, pages 353-362. Nashboro Press, 2003.Google Scholar
  64. 64.
    M. Sabin, N. Dodgson, M. Hassan, and I. Ivrissimtzis. Curvature behaviours at extraordinary points of subdivision surfaces. Computer Aided Design, 35(11):1047-1051, 2003.CrossRefGoogle Scholar
  65. 65.
    S. Schaefer, D. Zorin, and J. Warren. Lofting curve networks with subdivision surfaces. In Proceedings of Eurographics Symposium on Graphics Processing, pages 105-116, 2004.Google Scholar
  66. 66.
    J. Schweitzer. Analysis And Applications of Subdivision Surfaces. PhD thesis, The University of Washington, 1991.Google Scholar
  67. 67.
    T. Sederberg, J. Zheng, A. Bakenov, and A. Nasri. T-Splines and T-NURCCs. ACM Transaction on Graphics, 22(3):477-484, 2003. ACM SIGGRAPH 2003, ACM Press.CrossRefGoogle Scholar
  68. 68.
    T. W. Sederberg, J. Zheng, D. Sewell, and M. Sabin. Non-uniform subdivision surfaces. In ACM Siggraph 1998, volume 17, pages 387-394, 1998.CrossRefGoogle Scholar
  69. 69.
    J. Stam. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In SIGGRAPH 98 Conference Proceedings, pages 395-404, 1998.Google Scholar
  70. 70.
    J. Stam and C. Loop. Quad/triangle subdivision. Computer Graphics Forum, 22(1):79-85,2003.CrossRefGoogle Scholar
  71. 71.
    L. Velho, K. Perlin, L. Ying, and H. Biermann. Procedural shape synthesis on subdivision surfaces. In SIBGRAPI 2001, 2001.Google Scholar
  72. 72.
    J. Warren and H. Weimer. Subdivision Methods for Geometric Design. Morgan Kaufmann, 2001.Google Scholar
  73. 73.
    D. Zorin and D. Kristjansson. Evaluation of piecewise smooth subdivision surfaces. The Visual Computer Journal, 18(5-6):299-315, 2002.CrossRefGoogle Scholar
  74. 74.
    D. Zorin, P. Schr öder, A. DeRose, L. Kobbelt, A. Levin, and W. Sweldens. SIGGRAPH 00 Course Notes, Subdivision for modeling and animation, 2000.Google Scholar
  75. 75.
    D. Zorin, P. Schr öder, and W. Sweldens. Interpolating subdivision for meshes with arbitrary topology. In SIGGRAPH 96 Conference Proceedings, pages 189-192, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Chiara Eva Catalano
    • 1
  • Ioannis Ivrissimtzis
    • 2
  • Ahmad Nasri
    • 3
  1. 1.Istituto di Matematica Applicata e Tecnologie InformaticheItalian National Research CouncilGenovaItaly
  2. 2.Department of Computer ScienceDurham UniversityUK
  3. 3.Department of Computer ScienceAmerican University of BeirutLebanon

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