Self-Intersection Problems and Approximate Implicitization

  • Jan B. Thomassen
Conference paper


We discuss how approximate implicit representations of parametric curves and surfaces may be used in algorithms for finding self-intersections. We first recall how approximate implicitization can be formulated as a linear algebra problem, which may be solved by an SVD. We then sketch a self-intersection algorithm, and discuss two important problems we are faced with in implementing this algorithm: What algebraic degree to choose for the approximate implicit representation, and — for surfaces — how to find self-intersection curves, as opposed to just points.


Parameter Plane Implicit Representation NURBS Curve NURBS Surface Algebraic Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, and J.E.H. Hopcroft, Tracing surface intersections, Computer Aided Geometric Design 5 (1988) 285–307.MathSciNetCrossRefGoogle Scholar
  2. 2.
    R.E. Barnhill, and S.N. Kersey, A marching method for parametric surface/surface intersection, Computer Aided Geometric Design 7 (1990) 257–280.MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms, Second Edition, Springer-Verlag, New York, 1997.Google Scholar
  4. 4.
    T. Dokken, V. Skytt, and A.-M. Ytrehus, Recursive Subdivision and Iteration in Intersections and Related Problems, in Mathematical Methods in Computer Aided Geometric Design, T. Lyche and L. Schumaker (eds.), Academic Press, 1989, 207–214.Google Scholar
  5. 5.
    T. Dokken, Aspects of Intersection Algorithms and Applications, Ph.D. thesis, University of Oslo, July 1997.Google Scholar
  6. 6.
    T. Dokken, and J.B. Thomassen, Overview of Approximate Implicitization, in Topics in Algebraic Geometry and Geometric Modeling, AMS Cont. Math. 334 (2003), 169–184.MathSciNetGoogle Scholar
  7. 7.
    G. Farin, Curves and surfaces for CAGD: A practical guide, Fourth Edition, Academic Press, 1997.Google Scholar
  8. 8.
    R.N. Goldman, T.W. Sederberg, and D.C. Anderson, Vector elimination: A technique for the implicitization, inversion, and intersection of planar rational polynomial curves, Computer Aided Geometric Design 1 (1984) 327–356.CrossRefGoogle Scholar
  9. 9.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, Second Edition, Cambridge University Press, 1992.Google Scholar
  10. 10.
    T.W. Sederberg, D.C. Anderson, and R.N. Goldman, Implicit Representation of Parametric Curves and Surfaces, Comp. Vision, Graphics, and Image Processing 28, 72–84 (1984).CrossRefGoogle Scholar
  11. 11.
    T.W. Sederberg, J. Zheng, K. Klimaszewski, and T. Dokken, Approximate Implicitization Using Monoid Curves and Surfaces, Graph. Models and Image Proc. 61, 177–198 (1999).CrossRefGoogle Scholar
  12. 12.
    V. Skytt, Challenges in surface-surface intersections, these proceedings.Google Scholar
  13. 13.
    E. Wurm, and B. Jüttler, Approximate implicitization via curve fitting, in L. Kobbelt, P. Schröder, H. Hoppe (eds.), Symposium on Geometry Processing, Eurographics, ACM Siggraph, New York 2003, 240–247.Google Scholar
  14. 14.
    E. Wurm, and J.B. Thomassen, Deliverable 3.2.1 — Benchmarking of the different methods for approximate implicitization, Internal report in the GAIA II project, October 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Jan B. Thomassen
    • 1
    • 2
  1. 1.Centre of Mathematics for ApplicationsUniversity of OsloNorway
  2. 2.SINTEF Applied MathematicsNorway

Personalised recommendations