Skip to main content
SpringerLink
Log in

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]2 and (u,v)∈[0,1]2. The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Arnborg and H. Feng, Algebraic decomposition of regular curves, J. Symbolic Comput. 5 (1988) 131–140.

    Article  MATH  MathSciNet  Google Scholar 

  2. D.S. Arnon, Topologically reliable display of algebraic curves, ACM Computer Graphics 17 (1983) 219–227.

    Article  Google Scholar 

  3. D.S. Arnon, G.E. Collins and S. McCallum, Cylindrical algebraic decomposition I: The basic algorithm, SIAM J. Comput. 13 (1984) 865–877.

    Article  MATH  MathSciNet  Google Scholar 

  4. D.S. Arnon, G.E. Collins and S. McCallum, Cylindrical algebraic decomposition II: An adjacency algorithm for the plane, SIAM J. Comput. 13 (1984) 878–889.

    Article  MATH  MathSciNet  Google Scholar 

  5. D.S. Arnon and S. McCallum, A polynomial-time algorithm for the topological type of a real algebraic curve, J. Symbolic Comput. 5 (1988) 213–236.

    MATH  MathSciNet  Google Scholar 

  6. C. Bajaj, C.M. Hoffmann, R.E. Lynch and J.E.H. Hopcroft, Tracing surface intersections, Comput. Aided Geom. Design 5 (1988) 285–307.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Cellini, P. Gianni and C. Traverso, Algorithms for the shape of semialgebraic sets: A new approach, in: Lecture Notes in Computer Science, Vol. 539 (Springer, New York, 1991) pp. 1–18.

    Google Scholar 

  8. G. Farin, Curves and Surfaces for Computer Aided Geometric Design (Academic Press, 1997).

  9. R.T. Farouki, The characterization of parametric surface sections, Computer Vision, Graphics, Image Processing 33 (1986) 209–236.

    Article  MATH  Google Scholar 

  10. R.T. Farouki, Closing the gap between CAD model and downstream application (Report on the SIAM Workshop on Integration of CAD and CFD, UC Davis, April 12–13, 1999), SIAM News 32 (1999) 1–3.

    Google Scholar 

  11. R.T. Farouki and T.N.T. Goodman, On the optimal stability of the Bernstein basis, Math. Comput. 65 (1996) 1553–1566.

    Article  MATH  MathSciNet  Google Scholar 

  12. R.T. Farouki, C.Y. Han, J. Hass and T.W. Sederberg, Topologically consistent trimmed surface approximations based on triangular patches, Comput. Aided Geom. Design 21 (2004) 459–478.

    MATH  MathSciNet  Google Scholar 

  13. R.T. Farouki and V.T. Rajan, On the numerical condition of polynomials in Bernstein form, Comput. Aided Geom. Design 4 (1987) 191–216.

    Article  MATH  MathSciNet  Google Scholar 

  14. R.T. Farouki and V.T. Rajan, Algorithms for polynomials in Bernstein form, Comput. Aided Geom. Design 5 (1988) 1–26.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Gonzalez-Vega and I. Necula, Efficient topology determination of implicitly defined algebraic plane curves, Comput. Aided Geom. Design 19 (2002) 719–743.

    Article  MATH  MathSciNet  Google Scholar 

  16. T.A. Grandine and F.W. Klein, A new approach to the surface intersection problem, Comput. Aided Geom. Design 14 (1997) 111–134.

    Article  MATH  MathSciNet  Google Scholar 

  17. H. Hong, An efficient method for analyzing the topology of plane real algebraic curves, Math. Comput. Simulation 42 (1996) 571–582.

    Article  MATH  MathSciNet  Google Scholar 

  18. J.M. Lane and R.F. Riesenfeld, Bounds on a polynomial, BIT 21 (1981) 112–117.

    Article  MATH  MathSciNet  Google Scholar 

  19. M.H.A. Newman, Topology of Plane Sets (Cambridge Univ. Press, Cambridge, 1939).

    Google Scholar 

  20. M.-F. Roy and A. Szpirglas, Complexity of the computation of cylindrical decomposition and topology of real algebraic curves using Thom's lemma, in: Lecture Notes in Mathematics, Vol. 1420 (Springer, New York, 1990) pp. 223–236.

    Google Scholar 

  21. T. Sakkalis, The topological configuration of a real algebraic curve, Bull. Austral. Math. Soc. 43 (1991) 37–50.

    Article  MATH  MathSciNet  Google Scholar 

  22. T.W. Sederberg, Implicit and parametric curves and surfaces for computer aided geometric design, Ph.D. thesis, Purdue University (1983).

  23. E.C. Sherbrooke and N.M. Patrikalakis, Computation of the solutions of nonlinear polynomial systems, Comput. Aided Geom. Design 10 (1993) 379–405.

    Article  MATH  MathSciNet  Google Scholar 

  24. X.W. Song, T.W. Sederberg, J. Zheng, R.T. Farouki and J. Hass, Linear perturbation methods for topologically consistent representations of free-form surface intersections, Comput. Aided Geom. Design 21 (2004) 303–319.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Davis, CA, 95616, USA

    Joel Hass

  2. Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA, 95616, USA

    Rida T. Farouki & Chang Yong Han

  3. College of Mechanical and Energy Engineering, Zhejiang University, Hangzhou, China

    Xiaowen Song

  4. Department of Computer Science, Brigham Young University, Provo, UT, 84602, USA

    Thomas W. Sederberg

Authors
  1. Joel Hass
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rida T. Farouki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang Yong Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiaowen Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Thomas W. Sederberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joel Hass.

Additional information

Communicated by T.N.T. Goodman

Mathematics subject classification (2000)

65D17

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hass, J., Farouki, R.T., Han, C.Y. et al. Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme. Adv Comput Math 27, 1–26 (2007). https://doi.org/10.1007/s10444-005-7539-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-005-7539-5

Keywords

Search

Navigation