“On the Velocity of an Implicit Surface” by Stam and Schmidt

  • ©Jos Stam and Ryan Schmidt




    On the Velocity of an Implicit Surface



    In this article we derive an equation for the velocity of an arbitrary time-evolving implicit surface. Strictly speaking, only the normal component of the velocity is unambiguously defined. This is because an implicit surface does not have a unique parametrization. However, by enforcing a constraint on the evolution of the normal field we obtain a unique tangential component. We apply our formulas to surface tracking and to the problem of computing velocity vectors of a motion blurred blobby surface. Other possible applications are mentioned at the end of the article.


    1. Blinn, J. F. 1982. A Generalization of Algebraic Surface Drawing. ACM Trans. Graph. 1, 3, 235–256.
    2. Bouthors, A. and Nesme, M. 2007. Twinned meshes for dynamic triangulation of implicit surfaces. In Proceedings of the Graphics Interface Conference. 3–9.
    3. Brochu, T. and Bridson, R. 2009. Robust topological operations for dynamic explicit surfaces. SIAM J. Sci. Comput. 31, 4, 2472–2493.
    4. Enright, D., Fedkiw, R., Ferziger, J., and Mitchell, I. 2002. A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116.
    5. Levet, F., Granier, X., and Schlick, C. 2007. Marchingparticles: Fast generation of particles for the sampling of implicit surfaces. Comput. Graph. Geom. 9, 1, 18–49.
    6. Lorensen, W. and Kline, H. 1987. Marching Cubes: A High Resolution 3D Surface Construction Algorithm. ACM Comput. Graph. 21, 4, 163–169.
    7. Meyer, M., Kirby, R. M., and Whitaker, R. 2007. Topology, accuracy, and quality of isosurface meshes using dynamic particles. IEEE Trans. Visualiz. Comput. Graph. 12, 5.
    8. Moore, E. H. 1920. On the reciprocal of the general algebraic matrix. Bull. Amer. Math. Soc. 26, 394–395.
    9. Mullan, M., Whitaker, R., and Hart, J. 2004. Procedural level sets. Presented at the NSF/DARPA CARGO Meeting.
    10. Ohtake, Y., Belyaev, A., and Pasko, A. 2003. Dynamic mesh optimization for polygonized implicit surfaces with sharp features. Vis. Comput. 19, 2-3, 115–126.
    11. Rodrian, H.-C. and Moock, H. 1996. Dynamic triangulation of animated skeleton-based implicit surfaces. In Proceedings of the Implicit Surfaces.
    12. Smets-Solanes, J.-P. 1996. Vector field based texture mapping of animated implicit objects. Comput. Graph. Forum 15, 3, 289–300.
    13. Witkin, A. and Heckbert, P. 1994. Using particles to sample and control implicit surfaces. ACM Comput. Graph. 28, 4, 227–234.
    14. Wyvill, G., McPheeters, C., and Wyvill, B. 1986. Data structure for soft objects. The Vis. Comput. 2, 4, 227–234.

ACM Digital Library Publication: