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A survey of partial differential equations in geometric design

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Abstract

Computer-aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand, since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling, since they offer a number of features from which these areas can benefit. This work summarizes the uses given to PDE surfaces as a surface generation technique together with some other applications to computer graphics.

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References

  1. Bajaj, C., Xu, G.: Anisotropic diffusion on surfaces and functions on surfaces. ACM Trans. Graph. 22, 4–32 (2003)

    Article  Google Scholar 

  2. Barr, A.: Global and local deformations of solid primitives. Comput. Graph. 18, 21–30 (1984)

    Article  Google Scholar 

  3. Bertalmio, M., Cheng, L., Osher, S., Sapiro, S.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759–780 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting, pp. 417–424. SIGGRAPH 2000, New Orleans (2000)

    Google Scholar 

  5. Bloor, M.I.G., Wilson, M.J.: Generating blend surfaces using partial differential equations. Comput. Aided Des. 21(3), 165–171 (1989)

    Article  MATH  Google Scholar 

  6. Bloor, M.I.G., Wilson, M.J.: Representing PDE surfac es in terms of B-splines. Comput. Aided Des. 22(6), 324–331 (1990)

    Article  MATH  Google Scholar 

  7. Bloor, M.I.G., Wilson, M.J.: Using partial differential equations to generate free-form surfaces. Comput. Aided Des. 22(4), 202–212 (1990)

    Article  MATH  Google Scholar 

  8. Bloor, M.I.G., Wilson, M.J.: Functionality in solids obtained from partial differential equations. Computing 8, 21–42 (1993)

    MathSciNet  Google Scholar 

  9. Bloor, M.I.G., Wilson, M.J.: The efficient parametrization of generic aircraft geometry. J. Aircraft 32(6), 1269–1275 (1995)

    Article  Google Scholar 

  10. Bloor, M.I.G., Wilson, M.J.: Spectral approximations to PDE surfaces. Comput. Aided Des. 28(2), 145–152 (1996)

    Article  Google Scholar 

  11. Bloor, M.I.G., Wilson, M.J.: An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch. Comput. Aided Geom. Des. 22(3), 203–219 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Celniker, G., Gossard, D.C.: Deformable curve and surface finite-elements for free-form shape design. Comput. Graph. 25, 257–266 (1991)

    Article  Google Scholar 

  13. Celniker, G., Welch, W.: Linear constraints for deformable non-uniform B-spline surfaces. In: SI3D ’92: Proceedings of the 1992 symposium on Interactive 3D graphics, pp. 165–170. ACM Press, New York, NY (1992)

    Chapter  Google Scholar 

  14. Dekanski, C.W., Bloor, M.I.G., Wilson, M.J.: The representation of marine propeller blades using the PDE method. J. Ship Res. 38(2), 108–116 (1995)

    Google Scholar 

  15. DeRose, T., Kass, M., Truong, T.: Subdivision surfaces in character animation, pp. 85–94. SIGGRAPH 1999, New York, NY (1999)

  16. Du, H., Qin, H.: Direct manipulation and interactive sculpting of PDE surfaces. Comput. Graph. Forum 19(3), C261–C270 (2000a)

  17. Du, H., Qin, H.: Dynamic PDE Surfaces with Flexible and General Geometric Constraints, pp. 213–222. Pacific Graphics 2000, Hong Kong (2000b)

  18. Du, H., Qin, H.: Integrating Physics-Based Modeling with PDE Solids for Geometric Design, pp. 198–207. Pacific Graphics 2001, Tokyo (2001)

  19. Du, H., Qin, H.: Interactive shape design using volumetric implicit PDEs. In: SMI ’03: Proceedings of the Eighth ACM Symposium on Solid Modeling and Applications, pp. 235–246. ACM Press, New York, NY (2003)

    Chapter  Google Scholar 

  20. Du, H., Qin, H.: A shape design system using volumetric implicit PDEs. Comput. Aided Des. 36(11), 1101–1116 (2004)

    Article  Google Scholar 

  21. Du, H., Qin, H.: Dynamic PDE-based surface design using geometric and physical constraints. Graph. Models 67, 43–71 (2005)

    Article  MATH  Google Scholar 

  22. Enright, D., Marschner, S., Fedkiw, R.: Animation and rendering of complex water surfaces. In: SIGGRAPH ’02: Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques, pp. 736–744. ACM Press, New York, NY (2002)

    Chapter  Google Scholar 

  23. Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, a Practical Guide, 5th edn. Morgan Kaufmann, San Diego, CA (2001)

    Google Scholar 

  24. Farlow, S.J.: Partial Differential Equations for Scientists and Enginneers, 1st edn. Dover Publications Inc., Mineaol, NY (1993)

    Google Scholar 

  25. Fedkiw, R.: Simulating Natural Phenomena for Computer Graphics, Geometric Level Set Methods in Imaging, Vision and Graphics, 5th edn. Springer, New York, NY (2003)

    Book  Google Scholar 

  26. Fedkiw, R., Aslam, T., Jensen, H.: Visual simulation of smoke. In: Siggraph 2001 Annual Conference, pp. 23–30. ACM, New York, NY (2001)

    Google Scholar 

  27. Huband, J., Li, W.: Extracting design parameters from airplane wing data by using Bloor–Wilson PDE surface model. Math. Eng. Ind. 8, 239–252 (2001)

    MATH  Google Scholar 

  28. Kubiesa, S., Ugail, H., Wilson, M.J.: Interactive design using higher order PDEs. Visual Comput. 20, 682–693 (2004)

    Article  Google Scholar 

  29. Lin, V.C., Gossard, D.C., Light, R.A.: Variational geometry in computer-aided design. Comput. Graph. 15, 171–177 (1981)

    Article  Google Scholar 

  30. Lowe, T.W., Bloor, M.I.G., Wilson, M.J.: Functionality in blend design. In: Computer Aided and Computational Design ASME, vol. 1, pp. 43–50. Butterworth, New York (1990)

    Google Scholar 

  31. Mémoli, F., Sapiro, G., Osher, S.: Solving variational problems and partial differential equations mapping into general target manifolds. J. Comput. Phys. 195(1), 263–292 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Monterde, J., Ugail, H.: On harmonic and biharmonic Bézier surfaces. Comput. Aided Geom. Des. 21(7), 607–715 (2004)

    Article  MathSciNet  Google Scholar 

  33. Monterde, J., Ugail, H.: A general fourth order PDE method to generate Bézier surfaces from the boundary. Comput. Aided Geom. Des. 23, 208–225 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Nguyen, D., Fedkiw, R., Jensen, H.: Physically based modeling and animation of fire. In: Siggraph 2002 Annual Conference. ACM, New York, NY (2002)

    Google Scholar 

  35. Qin, H., Terzopoulos, D.: D-NURBS: A physics-based framework for geometric design. IEEE Trans. Vis. Comput. Graph. 2(1), 85–96 (1996)

    Article  Google Scholar 

  36. Schneider, R., Kobbelt, L.: Geometric fairing of irregular meshes for free-form surface design. Comput. Aided Geom. Des. 18(4), 359–379 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Terzopoulos, D., Platt, J., Barr, A., Fleischer, K.: Elastically deformable models. Comput. Graph. 21, 205–214 (1987)

    Article  Google Scholar 

  38. Ugail, H.: Parametric design and optimisation of thin-walled structure for food packaging. J. Optimisation Eng. 4(4), 291–307 (2003a)

    Article  MATH  Google Scholar 

  39. Ugail, H.: On the Spine of a PDE Surface, pp. 366–376. Springer, New York, NY (2003b)

  40. Ugail, H.: Spine based shape parametrisation for PDE surfaces. Computing 72, 195–204 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ugail, H., Bloor, M.I.G., Wilson, M.J.: Techniques for interactive design using the PDE method. ACM Trans. Graph. 18(2), 195–212 (1999)

    Article  Google Scholar 

  42. Ugail, H., Wilson, M.J.: Efficient shape parametrisation for automatic design optimisation using a partial differential equation formulation. Comput. Struct. 81(29), 2601–2609 (2003)

    Article  Google Scholar 

  43. Westgaard, G., Nowacki, H.: Construction of fair surfaces over irregular meshes. In: SMA ’01: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 88–98. ACM Press, New York, NY (2001)

    Chapter  Google Scholar 

  44. Witkin, A., Fleischer, K., Barr, A.: Energy constraints on parameterized models. Comput. Graph. 21, 225–232 (1987)

    Article  Google Scholar 

  45. Xu, G., Pan, Q., Bajaj, C.L.: Discrete surface modelling using partial differential equations. Comput. Aided Geom. Des. 23, 125–145 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. You, L., Comninos, P., Zhang, J.J.: PDE blending surfaces with C2 continuity. Comput. Graph. 28(6), 895–906 (2004)

    Article  Google Scholar 

  47. Zhang, J.J., You, L.: Fast surface modelling using a 6th order PDE. Comput. Graph. Forum 23(3), 311–320 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. EIMC Department, School of Informatics, University of Bradford, Richmond Road, Bradford, BD7 1DP, UK

    Gabriela González Castro, Hassan Ugail & Ian Palmer

  2. Media Technology Research Centre, Department of Computer Science, University of Bath, Bath, BA2 7AY, UK

    Philip Willis

Authors
  1. Gabriela González Castro
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  2. Hassan Ugail
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  3. Philip Willis
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  4. Ian Palmer
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Correspondence to Gabriela González Castro.

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González Castro, G., Ugail, H., Willis, P. et al. A survey of partial differential equations in geometric design. Visual Comput 24, 213–225 (2008). https://doi.org/10.1007/s00371-007-0190-z

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