Deformation of Dynamic Surfaces

  • L. H. You
  • Jian J. Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3992)


Based on dynamic fourth order partial differential equations, we present an iterative finite difference algorithm. With C++ language and OpenGL graphics library, we implement the finite difference algorithm into a user interface and develop shape control parameters, density, damping coefficient, boundary tangents and external forces into user handles for dynamic manipulation of surface deformations. Using the developed user interface, we investigate how these user handles influence the deformation of dynamic surfaces.


Surface deformation dynamic fourth order partial differential equations iterative finite difference algorithm influences of user handles 


  1. 1.
    Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan-Kaufmann, San Francisco (2001)Google Scholar
  2. 2.
    Kang, H., Kak, A.: Deforming virtual objects interactively in accordance with an elastic model. Computer-Aided Design 28(4), 251–262 (1996)CrossRefGoogle Scholar
  3. 3.
    Léon, J.C., Veron, P.: Semiglobal deformation and correction of free-form surfaces using a mechanical alternative. The Visual Computer 13, 109–126 (1997)CrossRefGoogle Scholar
  4. 4.
    Guillet, S., Léon, J.C.: Parametrically deformed free-form surfaces as part of a variational model. Computer-Aided Design 30(8), 621–630 (1998)zbMATHCrossRefGoogle Scholar
  5. 5.
    Terzopoulos, D., Platt, J., Barr, A., Fleischer, K.: Elastically deformable models. Computer Graphics 21(4), 205–214 (1987)CrossRefGoogle Scholar
  6. 6.
    Terzopoulos, D., Fleischer, K.: Deformable models. The Visual Computer 4, 306–331 (1988)CrossRefGoogle Scholar
  7. 7.
    Terzopoulos, D., Fleischer, K.: Modeling inelastic deformation: viscoelasticity, plasticity, fracture. Computer Graphics 22(4), 269–278 (1988)CrossRefGoogle Scholar
  8. 8.
    Metaxas, D., Terzopoulos, D.: Dynamic deformation of solid primitives with constraints. Computer Graphics (SIGGRAPH 1992) 26(2), 309–312 (1992)CrossRefGoogle Scholar
  9. 9.
    Celniker, G., Gossard, D.: Deformable curve and surface finite-elements for free-form shape design. Computer Graphics (SIGGRAPH 1991) 25(4), 257–266 (1991)CrossRefGoogle Scholar
  10. 10.
    Güdükbay, U., Özgüç, B.: Animation of deformable models. Computer-Aided Design 26(12), 868–875 (1994)CrossRefGoogle Scholar
  11. 11.
    Guan, Z.D., Ling, J., Tao, N., Ping, X., Tang, R.X.: Study and application of physics-based deformable curves and surfaces. Computers & Graphics 21(3), 305–313 (1997)CrossRefGoogle Scholar
  12. 12.
    Terzopoulos, D., Qin, H.: Dynamic NURBS with geometric constraints for interactive sculpting. ACM Transactions on Graphics 13(2), 103–136 (1994)zbMATHCrossRefGoogle Scholar
  13. 13.
    Qin, H., Terzopoulos, D.: Dynamic NURBS swung surfaces for physics-based shape design. Computer-Aided Design 27(2), 111–127 (1995)zbMATHCrossRefGoogle Scholar
  14. 14.
    Bloor, M.I.G., Wilson, M.J.: Generating blend surfaces using partial differential equations. Computer-Aided Design 21(3), 165–171 (1989)zbMATHCrossRefGoogle Scholar
  15. 15.
    Bloor, M.I.G., Wilson, M.J.: Complex PDE surface generation for analysis and manufacture. Computing Suppl. 10, 61–77 (1995)Google Scholar
  16. 16.
    Bloor, M.I.G., Wilson, M.J., Mulligan, S.J.: Generating blend surfaces using a perturbation method. Mathematical and Computer Modelling 31(1), 1–13 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Mimis, A.P., Bloor, M.I.G., Wilson, M.J.: Shape parameterization and optimization of a two-stroke engine. Journal of Propulsion and Power 17(3), 492–498 (2001)CrossRefGoogle Scholar
  18. 18.
    Bloor, M.I.G., Wilson, M.J.: Method for efficient shape parametrisation of fluid membranes and vesicles. Physical Review E (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics) 61(4), 4218–4429 (2000)Google Scholar
  19. 19.
    Sevant, N.E., Bloor, M.I.G., Wilson, M.J.: Aerodynamic design of a flying wing using response surface methodology. Journal of Aircraft 37(4), 562–569 (2000)CrossRefGoogle Scholar
  20. 20.
    Ugail, H., Wilson, M.J.: Efficient shape parametrisation for automatic design optimisation using a partial differential equation formulation. Computers and Structures 81, 2601–2609 (2003)CrossRefGoogle Scholar
  21. 21.
    Ugail, H.: Spine based shape parameterisation for PDE surfaces. Computing 72, 195–206 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    You, L.H., Zhang, J.J., Comninos, P.: Blending surface generation using a fast and accurate analytical solution of a fourth order PDE with three shape control parameters. The Visual Computer 20, 199–214 (2004)CrossRefGoogle Scholar
  23. 23.
    You, L.H., Comninos, P., Zhang, J.J.: PDE blending surfaces with C 2 continuity. Computers & Graphics 28, 895–906 (2004)CrossRefGoogle Scholar
  24. 24.
    Du, H., Qin, H.: A shape design system using volumetric implicit PDEs. Computer-Aided Design 36, 1101–1116 (2004)CrossRefzbMATHGoogle Scholar
  25. 25.
    Du, H., Qin, H.: Dynamic PDE-based surface design using geometric and physical constraints. Graphical Models 67, 43–71 (2005)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • L. H. You
    • 1
  • Jian J. Zhang
    • 1
  1. 1.National Centre for Computer AnimationBournemouth UniversityUnited Kingdom

Personalised recommendations