A UNIFIED APPROACH TO GEOMETRIC MODELING OF CURVES AND SURFACES

L. H. You, Jian J. Zhang

Abstract

A unified approach to geometric modeling of curves and surfaces is given. Both a vector-valued fourth and sixth order partial differential equations (PDEs) of motion are proposed. The fourth order PDE covers all existing PDEs used for surface modeling, and the sixth order PDE considers the curvature effect on curves and surfaces. In order to apply these PDEs to create curves and surfaces in real time, we have presented a composite power series method which guarantees the exact satisfaction of boundary conditions, and represents curves and surfaces with analytical mathematical formulae. We have examined the accuracy and efficiency of the proposed method, and employed it to a number of applications of static and dynamic modeling of curves and surfaces, including free-form surface generation and surface blending. It is found that this method has similar computational accuracy and efficiency to the corresponding closed form solution method, and creates curves and surfaces far more efficiently and accurately than numerical methods. In addition, it can deal with complicated shape modeling problems..

References

  1. Athanasopoulos, M., Ugail, H., Castro, G. G., 2009, Parametric design of aircraft geometry using partial differential equations. Advances in Engineering Software 40, 479-486.
  2. Bloor, M. I. G., Wilson, M. J., 1989a. Generating n-sided patches with partial differential equations. New Advances in Computer Graphics, Proceedings of CG International'89, Springer-Verlag, Tokyo, Japan, 129- 145.
  3. Bloor, M. I. G., Wilson, M. J., 1989b. Generating blend surfaces using partial differential equations. Computer-Aided Design 21(3),165-171.
  4. Bloor, M. I. G., Wilson, M. J., 1990a. Using partial differential equations to generate free-form surfaces. Computer-Aided Design 22(4), 202-212.
  5. Bloor, M. I. G., Wilson, M. J., 1990b. Representing PDE surfaces in terms of B-splines. Computer-Aided Design 22(6), 324-331.
  6. Bloor, M. I. G., Wilson, M. J., 1996. Spectral approximations to PDE surfaces, Computer-Aided design 28(2),145-152.
  7. Celniker, G., Gossard, D., 1991. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics 25(4), 257-266.
  8. Du, H., Qin, H., 2005. Dynamic PDE-based surface design using geometric and physical constraints. Graphical Models 67, 43-71.
  9. Farin, G., 1997. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide. 4th Edition, Academic Press.
  10. Guillet, S., Léon, J. C., 1998. Parametrically deformed free-form surfaces as part of a variational model. Computer-Aided Design 30(8), 621-630.
  11. Hyodo, Y., 1990. The generation of free form surface defined with contour and sectional curves. Journal of the Japan Society of Precision Engineering 56(10), 1912-1916.
  12. Li, Z. C., 1998. Boundary penalty finite element methods for blending surfaces, I. Basic theory. Journal of Computational Mathematics 16, 457-480.
  13. Li, Z. C., 1999. Boundary penalty finite element methods for blending surfaces, II. Biharmonic equations. Journal of Computational and Applied Mathematics 110, 155-176.
  14. Li, Z. C., Chang, C.-S., 1999. Boundary penalty finite element methods for blending surfaces, III. Superconvergence and stability and examples. Journal of Computational and Applied Mathematics 110, 241- 270.
  15. Miura, K. T., 2000. Unit quaternion integral curve: a new type of fair free-form curves. Computer Aided Geometric Design 17(1), 39-58.
  16. Nealen, A., Müller, M., Keiser, R., Boxerman, E., Carlson, M., 2006. Physically based deformable models in computer graphics. Computer Graphics Forum 25(4), 809-836.
  17. Ochiai, Y., Yasutomi, Z., 2000. Improved method generating a free-form surface using integral equations. Computer Aided Geometric Design 17(3), 233-245.
  18. Qin, H., Terzopoulos, D., 1995. Dynamic NURBS swung surfaces for physics-based shape design. Computeraided Design 27(2), 111-127.
  19. Terzopoulos, D., Platt, J., Barr, A., Fleischer, K., 1987. Elastically deformable models. Computer Graphics 21(4), 205-214.
  20. Terzopoulos, D., Fleischer, K., 1988. Modeling inelastic deformation: viscoelasticity, plasticity, fracture. Computer Graphics 22(4), 269-278.
  21. Terzopoulos, D., Qin, H., 1994. Dynamic NURBS with geometric constraints for interactive sculpting. ACM Transactions on Graphics 13(2), 103-136.
  22. Ugail, H., Bloor, M. I. G., Wilson, M. J., 1999a. Techniques for interactive design using the PDE method. ACM Transactions on Graphics 18(2), 195- 212.
  23. Ugail, H., Bloor, M. I. G., Wilson, M. J., 1999b. Manipulation of PDE surfaces using an interactively defined parameterisation. Computers & Graphics 23(4), 525-534.
  24. Vassilev, T. I., 1997. Interactive sculpting with deformable nonuniform B-splines. Computer Graphics Forum 16, 191-199.
  25. Vida, J., Martin, R. R., Varady, T., 1994. A survey of blending methods that use parametric surfaces. Computer-Aided Design 26(5), 341-365.
  26. You, L. H., Zhang, J. J., Comninos, P., 1999. Cloth deformation modelling using a plate bending model. In Proceedings of the 7th International Conference in Central Europo on Computer Graphics, Visualization and Interactive Digital Media'99, 485-491.
  27. You, L. H., Zhang, J. J., Comninos, P., 2000. A volumetric deformable muscle model for computer animation using weighted residual method. Computer Methods in Applied Mechanics and Engineering 190(8-10), 853- 863.
  28. You, L. H., Zhang, J. J., 2003. Fast generation of 3-D deformable moving surfaces. IEEE Transactions on Systems, Man and Cybernetics-Part B: Cybernetics 33(4), 616-625.
  29. You, L. H., Zhang, J. J., 2004. PDE blending surfaces with C 2 continuity. Computers & Graphics 28, 895-906.
  30. Zhang, J. J., You, L. H., Comninos, P., 1999. Computer simulation of flexible fabrics. In Proceedings of the 17th Annual Conference, UK Chapter, EUROGRAPHICS, 27-35.
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Paper Citation


in Harvard Style

You L. and Zhang J. (2011). A UNIFIED APPROACH TO GEOMETRIC MODELING OF CURVES AND SURFACES . In Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2011) ISBN 978-989-8425-45-4, pages 23-30. DOI: 10.5220/0003316300230030


in Bibtex Style

@conference{grapp11,
author={L. H. You and Jian J. Zhang},
title={A UNIFIED APPROACH TO GEOMETRIC MODELING OF CURVES AND SURFACES},
booktitle={Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2011)},
year={2011},
pages={23-30},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0003316300230030},
isbn={978-989-8425-45-4},
}


in EndNote Style

TY - CONF
JO - Proceedings of the International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2011)
TI - A UNIFIED APPROACH TO GEOMETRIC MODELING OF CURVES AND SURFACES
SN - 978-989-8425-45-4
AU - You L.
AU - Zhang J.
PY - 2011
SP - 23
EP - 30
DO - 10.5220/0003316300230030