Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

NURBS approximation of surface/surface intersection curves

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

Abstract

We use a combination of both symbolic and numerical techniques to construct degree boundedC k-continuous, rational B-spline ε-approximations of real algebraic surface-surface intersection curves. The algebraic surfaces could be either in implicit or rational parametric form. At singular points, we use the classical Newton power series factorizations to determine the distinct branches of the space intersection curve. In addition to singular points, we obtain an adaptive selection of regular points about which the curve approximation yields a small number of curve segments yet achievesC k continuity between segments. Details of the implementation of these algorithms and approximation error bounds are also provided.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. C. Bajaj, Geometric modeling with algebraic surfaces,The Mathematics of Surfaces III, ed. D. Handscomb (Oxford University Press, 1990) pp. 3–48.

  2. C. Bajaj, The emergence of algebraic curves and surfaces in geometric design,Directions in Geometric Computing, ed. R. Martin (Information Geometers Press, UK, 1993) pp. 1–29.

    Google Scholar 

  3. C. Bajaj and I. Ihm, Algebraic surface design with Hermite interpolation, ACM Trans. Graphics 11(1992)61–91.

    Article  MATH  Google Scholar 

  4. C. Bajaj and A. Royappa, The GANITH algebraic geometry toolkit, in:Proc. 1 st Int. Symp. on the Design and Implementation of Symbolic Computation Systems, Lecture Notes in Computer Science No. 429 (Springer, 1990) pp. 268–269.

  5. C.L. Bajaj and G. Xu, Piecewise rational approximation of real algebraic curves, CAPO Technical Report 92-19, Computer Science Department, Purdue University (1992).

  6. C.L. Bajaj and G. Xu, Converting a rational curve to a standard rational Bézier representation, to appear inGRAPHICS GEMS IV ed. P. Heckbert (Academic Press, New York, 1994).

    Google Scholar 

  7. C. Bajaj, C. Hoffmann, R. Lynch and J. Hopcroft, Tracing surface intersections, Comp. Aided Geom. Design 5(1988)285–307.

    Article  MATH  MathSciNet  Google Scholar 

  8. K. Cheng, Using plane vector fields to obtain all the intersection curves of two general surfaces,Theory and Practice of Geometric Modeling, ed. W. Strasser and H.-P. Seidel (Springer, Berlin, 1989) pp. 187–204.

    Google Scholar 

  9. C. de Boor,A Practical Guide to Splines (Springer, New York/Heidelberg/Berlin, 1978).

    MATH  Google Scholar 

  10. M. do Carmo,Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ, 1976).

    MATH  Google Scholar 

  11. G. Farin,Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide (Academic Press, Boston, 1988).

    Google Scholar 

  12. T. Garrity and J. Warren, On computing the intersection of a pair of algebraic surfaces, Comp. Aided Geom. Design 6(1989)137–153.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Garrity and J. Warren, Geometric continuity, Comp. Aided Geom. Design 8(1991)51–65.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Geisow, Surface interrogations, Ph.D. Thesis, University of East Anglia (1983).

  15. R. Goldman and C. Micchelli, Algebraic aspects of geometric continuity, in:Mathematical Methods in Computer Aided Geometric Design, ed. T. Lyche and L. Schumaker (Academic Press, Boston, 1989) pp. 313–332.

    Google Scholar 

  16. H. Guggenheimer, Computing frames along a trajectory, Comp. Aided Geom. Design 6(1989) 77–78.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Müllenheim, Convergence of a surface/surface intersection algorithm, Comp. Aided Geom. Design 7(1990)415–423.

    Article  MATH  Google Scholar 

  18. G. Müllenheim, On determining start points for a surface/surface intersection algorithm, Comp. Aided Geom. Design 8(1991)401–408.

    Article  MATH  Google Scholar 

  19. H. Padé, Sur la généralisation des fractions continues algébriques, J. de Math. 4(1894)291–329.

    MATH  Google Scholar 

  20. L. Peigl and W. Tiller, Curves and surface construction using rational B-splines, Comp. Aided Design 19(1987)485–498.

    Article  Google Scholar 

  21. M. Pratt and A. Geisow, Surface/surface intersection problems,The Mathematics of Surfaces, ed. J. Gregory (Oxford University Press, 1986) pp. 117–142.

  22. M.A. Sabin, Contouring—the state of the art, NATO ASI Series,Fundamental Algorithms for Computer Graphics, F17(1985) pp. 411–482.

    Google Scholar 

  23. T. Sederberg, J. Zhao and A. Zundel, Rational approximation of algebraic curves, in:Theory and Practice of Geometric Modeling, ed. W. Strasser and H.-P. Seidel (Springer) pp. 33–54.

  24. T. Sederberg and T. Nishita, Geometric Hermite approximation of surface patch intersection curves, Comp. Aided Geom. Design 8(1991)97–114.

    Article  MATH  MathSciNet  Google Scholar 

  25. H.-P. Seidel, Polar forms for geometrically continuous spline curves of arbitrary degree, ACM Trans. Graphics 12(1993)1–34.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Purdue University, 47907, West Lafayette, IN, USA

    Chandrajit L. Bajaj

  2. Computer Center, Chinese Academy of Sciences, 100080, Beijing, PR China

    Guoliang Xu

Authors
  1. Chandrajit L. Bajaj
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoliang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported in part by NSF Grants CCR 92.22467, DMS 91-01424, AFOSR Grant F49620-10138 and NASA Grant NAG-1-1473.

Supported in part by K.C. Wong Education Foundation, Hong Kong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bajaj, C.L., Xu, G. NURBS approximation of surface/surface intersection curves. Adv Comput Math 2, 1–21 (1994). https://doi.org/10.1007/BF02519033

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02519033

Keywords

Search

Navigation