“Free-form shape design using triangulated surfaces” by Welch and Witkin

  • ©William Welch and Andrew P. Witkin




    Free-form shape design using triangulated surfaces



    We present an approach to modeling with truly mutable yet completely controllable free-form surfaces of arbitrary topology. Surfaces may be pinned down at points and along curves, cut up and smoothly welded back together, and faired and reshaped in the large. This style of control is formulated as a constrained shape optimization, with minimization of squared principal curvatures yielding graceful shapes that are free of the parameterization worries accompanying many patch-based approaches. Triangulated point sets are used to approximate these smooth variational surfaces, bridging the gap between patch-based and particle-based representations. Automatic refinement, mesh smoothing, and re-triangulation maintain a good computational mesh as the surface shape evolves, and give sample points and surface features much of the freedom to slide around in the surface that oriented particles enjoy. The resulting surface triangulations are constructed and maintained in real time.


    1. Timothy Barth. Higher order solution of the euler equations on unstructured grids using quadratic reconstruction. In 28th Aerospace Sciences Meeting. AIAA-90-0013, 1990.]]
    2. Marshall Bern and David Eppstein. Mesh generation and opti-mal triangulation. Technical Report CSL-92-1, XEROX Palo Alto Research Center, March 1992.]]
    3. D.L. Bonner, M.J. Jakiela, M. Watanabe, and N. Kishi. Pseu-doedge: nonintersected parametric quilt modeling of multiply connected objects. Computer Aided Design, 25(7):438-452, July 1993.]]
    4. George Celniker and Dave Gossard. Deformable curve and surface finite-elements for free-form shape design. Computer Graphics, 25(4), July 1991. (Proceedings Siggraph ’91).]]
    5. George Celniker and William Welch. Linear constraints for nonuniform B-spline surfaces. In Proceedings, Symposium on Interactive 3D Graphics, 1992.]]
    6. Paul Chew. Guaranteed quality mesh generation for curved surfaces. In Proceedings of the ACM Symposium on Compu-tational Geometry, 1993.]]
    7. R. Courant and D. Hilbert. Methods of Mathematical Physics, volume volume I. Wiley, 1937.]]
    8. M. Eppstein. On the influence of parameterization in paramet-ric interpolation. SIAM J. Numer. Anal., 13:261-268, 1976.]]
    9. Gerald Farin. Curves and Surfaces for Computer Aided Geo-metric Design. Academic Press, 1990.]]
    10. Helaman Ferguson, Alyn Rockwood, and Jordan Cox. Topo-logical design of sculptured surfaces. Computer Graphics, 26(2), July 1992. (Proceedings Siggraph ’92).]]
    11. D. A. Field. Laplacian smoothing and delaunay triangulations. Comm. Appl. Numer. Methods, 4:709-712, 1984.]]
    12. G. Forsythe and W. Wasow. Finite Difference Methods for Partial Differential Equations, chapter 19, pages 179-182. John Wiley and Sons, 1960.]]
    13. Richard Franke and Gregory Nielson. Scattered data inter-polation and applications: a tutorial and survey. In Hans Ha-gen and Dieter Roller, editors, Geometric Modeling. Springer-Verlag, 1991.]]
    14. Gene Golub and Charles Van Loan. Matrix Computations. Johns Hopkins University Press, 1989.]]
    15. Victor Guillemin and Alan Pollack. Differential Topology. Prentice-Hall, 1974.]]
    16. Huges Hoppe, Tony DeRose, TomDuchamp, John McDonald, and Werner Stuetzle. Mesh optimization. In Proceedings of Siggraph 93, 1993.]]
    17. Geertjan Huiskamp. Difference formulas for the surface Laplacian on a triangulated surface. Journal of Computational Physics, 95:477-496, 1991.]]
    18. Michael Kallay. Constrained optimization in surface design. In Modeling in Computer Graphics. Springer Verlag, 1993.]]
    19. Michael Kass, Andrew Witkin, and Dimetri Terzopoulos. Snakes: Active contour models. International Journal Com-puter Vision, 1(4), 1987.]]
    20. N. J. Lott and D. I. Pullin. Method for fairing b-spline surfaces. Computer-Aided Design, 20(10), 1988.]]
    21. Steve Mann, Charles Loop, Michael Lounsbery, D. Meyers, J. Painter, Tony Derose, and K. Sloan. A survey of parametric scattered data fitting using triangular interpolants. In Curve and Surface Modeling. SIAM.]]
    22. Henry Moreton and Carlo S~ equin. Functional minimization for fair surface design. Computer Graphics, 26(2), July 1992. (Proceedings Siggraph ’92).]]
    23. Henry P. Moreton. Minimum Curvature Variation Curves, Net-works, and Surfaces for Fair Free-form Shape Design. PhD thesis, University of California, Berkeley, 1993.]]
    24. Gregory Nielson. A transfinite, visually continuous, triangu-lar interpolant. In Gerald Farin, editor, Geometric Modelling, pages 235-246. SIAM, 1987.]]
    25. Barrett O’Neill. Elementary Differential Geometry. Academic Press, 1966.]]
    26. J~ org Peters. Smooth interpolation of a mesh of curves. Con-structive Approximation, 7:221-246, 1991.]]
    27. John Platt. Constraint Methods for Neural Networks and Com-puter Graphics. PhD thesis, California Institute of Technol-ogy, 1989.]]
    28. Vaughan Pratt. Direct least-squares fitting of algebraic sur-faces. Computer Graphics (SIGGRAPH ’87 Proceedings), 21(4):145-152, July 1987.]]
    29. M. M. Rai and D. A. Anderson. Application of adaptive grids to fluid-flow problems with asymptotic solutions. AIAA J., 20:496-502, 1982.]]
    30. D.G. Schweikert. An interpolation curve using a spline in ten-sion. Journal of Math and Phys., 45:312-317, 1966.]]
    31. Thomas Sederberg and Eugene Greenwood. A physically based approach to 2D shape blending. Computer Graphics, 26(2), July 1992. (Proceedings Siggraph ’92).]]
    32. Michael Spivak. A Comprehensive Introduction to Differential Geometry. Publish or Perish, Inc., 1979.]]
    33. Gilbert Strang. Introduction to Applied Mathematics. Wellesley-Cambridge Press, 1986.]]
    34. Richard Szeliski and David Tonnesen. Surface modeling with oriented particle systems. Computer Graphics, 26(2), July 1992. (Proceedings Siggraph ’92).]]
    35. D. Terzopoulos. Multi-level reconstruction of visual surfaces. MIT Artificial Intelligence Memo Number 671, April 1981.]]
    36. J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin. Numeri-cal Grid Generation: Foundations and Applications. North-Holland, 1985.]]
    37. Joe F. Thompson. A survey of dynamically-adaptive grids in the numerical solution of partial differential equations. Ap-plied Numerical Mathematics, 1:3-27, 1985.]]
    38. Greg Turk. Generating textures on arbitrary surfaces using reaction-diffusion. Computer Graphics (SIGGRAPH ’91 Pro-ceedings), 25(4):289-298, July 1991.]]
    39. Greg Turk. Re-tiling polygonal surfaces. Computer Graphics, 26(2):55-64, July 1992. (Proceedings Siggraph ’92).]]
    40. William Welch. Free-Form shape design using triangulated surfaces. PhD thesis, Carnegie Mellon University, (in prepa-ration) 1994.]]
    41. William Welch and Andrew Witkin. Variational surface mod-eling. Computer Graphics, 26(2), July 1992. (Proceedings Siggraph ’92).]]
    42. Andrew Witkin and Paul Heckbert. Using particles to sample and control implicit surfaces. In these proceedings, July 1994.]]

ACM Digital Library Publication:

Overview Page: