Subdivision Curves on Surfaces and Applications

  • Dimas Martínez Morera
  • Luiz Velho
  • Paulo Cezar Carvalho
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5197)


Subdivision curves have great importance for many CAD/CAM applications.. In this paper we propose a simple method to define subdivision schemes on triangulations. It works by translating to the triangulation a perturbation of a planar binary subdivision. To reproduce this perturbation in the surface we use both, shortest and straightest geodesics, so we call this strategy intrinsic projection method. It can reproduce any binary subdivision scheme, regardless whether it is linear or not.


Subdivision Curve Discrete Geodesic Manifold Triangulation 


  1. 1.
    Bonneau, G., Hahmann, S.: Smooth polylines on polygon meshes. In: Mathematics and Visualization, pp. 69–84. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary Subdivision. American Mathematical Society, Boston (1991)zbMATHGoogle Scholar
  3. 3.
    de Casteljau, P.F.: Outillage Méthodes Calcul. Internes Dokument P2108, SA André Citroën, Paris (February 1959)Google Scholar
  4. 4.
    Dyn, N., Levin, D.: The subdivision experience. In: Laurent, P.J., Le M’ehaut’e, A., Schumaker, L.L. (eds.) Wavelets, Images and Surface Fitting, pp. 229–244. A K Peters, Ltd., Wellesley (1994)Google Scholar
  5. 5.
    Dyn, N., Levin, D., Gregory, J.A.: A 4-point interpolatory subdivision scheme for curve design. Computer Aided Geometric Design 4, 257–268 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Marinov, M., Kobbelt, L.: A robust two-step procedure for quad-dominant remeshing. Computer Graphics Forum 25(3), 537–546 (2006)CrossRefGoogle Scholar
  7. 7.
    Martínez, D., Carvalho, P.C., Velho, L.: Geodesic Bézier curves: A tool for modeling on triangulations. In: Proceedings of SIBGRAPI 2007. XX Brazilian Symposium on Computer Graphics and Image Processing. XX Brazilian Symposium on Computer Graphics and Image Processing, pp. 71–78. IEEE Computer Society, Los Alamitos (2007)Google Scholar
  8. 8.
    Martínez, D., Velho, L., Carvalho, P.C.: Computing geodesics on triangular meshes. Computer and Graphics 29(5), 667–675 (2005)CrossRefGoogle Scholar
  9. 9.
    Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The discrete geodesic problem. SIAM J. Comput. 16, 647–668 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Park, F.C., Ravani, B.: Bezier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME Journal of Mechanical Design 117, 36–40 (1995)CrossRefGoogle Scholar
  11. 11.
    Polthier, K., Schmies, M.: Straightest geodesics on polyhedral surfaces. In: Hege, H.-C., Polthier, K. (eds.) Visualization and Mathematics, pp. 135–150. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Pottmann, H., Hofer, M.: A variational aproach to spline curves on surfaces. Computer Aided Geometric Design 22(7), 693–709 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rodriguez, R.C., Leite, F.S., Jacubiak, J.: A new geometric algorithm to generate smooth interpolating curves on riemannian manifolds. LMS Journal of Computation and Mathematics 8, 251–266 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rössl, C., Kobbelt, L., Seidel, H.: Extraction of feature lines on triangulated surfaces using morphological operators. In: Proceedings of Smart Graphics, Stanford, USA, pp. 71–75. AAAI Press, Menlo Park (2000)Google Scholar
  15. 15.
    Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S., Hoppe, H.: Fast exact and approximate geodesics on meshes. In: Proceedings of ACM SIGGRAPH 2005, pp. 553–560 (2005)Google Scholar
  16. 16.
    Wallner, J.: Smoothness analysis of subdivision schemes by proximity. Constr. Approx. 24(3), 289–318 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wallner, J., Dyn, N.: Convergence and C 1 analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Design 22(7), 593–622 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wallner, J., Pottmann, H.: Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans. Graphics 25(2), 356–374 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Dimas Martínez Morera
    • 1
  • Luiz Velho
    • 1
  • Paulo Cezar Carvalho
    • 1
  1. 1.Instituto Nacional de Matemática Pura e Aplicada–IMPABrazil

Personalised recommendations