Extrapolating acceleration algorithms for finding B-Spline intersections using recursive subdivision techniques

  • Kaihuai Qin
  • Gang Fan
  • Cai Sun


The new algorithms for finding B-Spline or Bézier curves and surfaces intersections using recursive subdivision techniques are presented, which use extrapolating acceleration technique, and have convergent precision of order 2. Matrix method is used to subdivide the curves or surfaces which makes the subdivision more concise and intuitive. Dividing depths of Bézier curves and surfaces are used to subdivide the curves or surfaces adaptively. Therefore the convergent precision and the computing efficiency of finding the intersections of curves and surfaces have been improved by the methods proposed in the paper.


Extrapolating acceleration intersection B-Spline Bézier curve and surface recursive subdivision 


  1. [1]
    Aziz N M, Bata R. Bézier surface/surface intersection. IEEE CG& A, 1990, 10(1): 50–58.Google Scholar
  2. [2]
    Peng Q S. An algorithm for finding the intersection lines between two B-spline surfaces. Computer-Aided Design, 1984, 16(4): 191–196.CrossRefGoogle Scholar
  3. [3]
    Lasser D. Intersection of parametric surfaces in Bernstein-Bézier representation. Computer-Aided Design, 1986, 18(4): 186–192.CrossRefGoogle Scholar
  4. [4]
    Hanna S L, Abel J F, Greenberg D P. Intersection of parametric surfaces by means of look-up tables. IEEE CG& A, 1983, 3(7): 39–48.Google Scholar
  5. [5]
    Yen J, Spach S, Smith M, Pulleyblank R. Parallel boxing in B-spline intersection. IEEE CG & A, 1991, 11(1): 72–79.Google Scholar
  6. [6]
    Casale M S, Bobrow J E. A set operation algorithm for sculptured solid modeled with trimmed patches. CAGD, 1989, 6: 235–247.MATHGoogle Scholar
  7. [7]
    Houghton E G, Emnett R F. Implementation of a divide-and-conquer method for intersection of parametric surface. CAGD, 1985, 2: 173–183.MATHGoogle Scholar
  8. [8]
    Arner P R. Another look at surface/surface intersection. Ph.D. Thesis, University of Utah, Salt Lake City, Utah, USA.Google Scholar
  9. [9]
    Boehm W. Inserting new knots into B-spline curve and surface. Computer-Aided Design, 1980, 12(4): 199–201.CrossRefGoogle Scholar
  10. [10]
    Boehm W, Prautzsch H. The insertion algorithm. Computer-Aided Design, 1985, 17(2): 58–59.CrossRefGoogle Scholar
  11. [11]
    Boehm W. On the efficiency of knot insertion algorithms. CAGD, 1985, 2: 141–143.MATHGoogle Scholar
  12. [12]
    Cohen E, Lyche T, Riesenfeld R. Discrete B-splines and subdivision techniques in computeraided geometric design and computer graphics. Computer Graphics and Image Processing, 1980, 14: 87–111.CrossRefGoogle Scholar
  13. [13]
    Lyche T, Cohen E, Morken K. Knot line refinement algorithms for tensor product B-spline surfaces. CAGD, 1985, 2: 133–139.MATHMathSciNetGoogle Scholar
  14. [14]
    Boehm W. Generating the Bézier points of B-spline curves and surfaces. Computer-Aided Design, 1981, 13(6): 365–366.CrossRefGoogle Scholar
  15. [15]
    Wang G. The subdivision method for finding the intersection between two Bézier curves or surfaces. Zhejiang University Journal, Special Issue on Computational Geometry, 1984: 108–119.Google Scholar
  16. [16]
    Lane J M, Riesenfeld R F. A theoretical development for the computer generation of piecewise polynomial surfaces. IEEE Trans Pattern Analysis and Machine Intelligence, 1980, 2(1): 35–46.MATHCrossRefGoogle Scholar
  17. [17]
    Farin G. Curve and surface design: from geometry to applications. ACM SIGRAPH'88 Course#24, Altanta, 1988.Google Scholar
  18. [18]
    DeBoor C. A practical guide to splines. Springer-Verlag, New York, 1978.Google Scholar
  19. [19]
    Chang G, Wu J. Mathematical foundation of Bézier technique. Computer-Aided Design, 1981, 13(3): 133–136.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Koparkar P A, Mudur S P. Computational techniques for processing parametric surfaces. Computer Vision, Graphics and Image Processing, 1984, 28: 303–322.CrossRefGoogle Scholar
  21. [21]
    Phillips G M, Taylor P J. Theory and applications of numerical analysis. New York: Academic Press, 1973.MATHGoogle Scholar
  22. [22]
    Deng J. Extrapolation methods and their applications. Shanghai Science & Technology Press, 1984.Google Scholar
  23. [23]
    Dokken T. Finding intersections of B-spline represented geometrics using recursive subdivision techniques. CAGD, 1985, 2(1–3): 189–195.MATHGoogle Scholar
  24. [24]
    Filip D, Magedson R, Markot R. Surface algorithm using bounds on derivatives. CAGD, 1986, 3(4): 295–311.MATHMathSciNetGoogle Scholar
  25. [25]
    Pratt M J, Geisow A D. Surface/surface intersection problems. In: Gregory J A ed. The Mathematics of Surfaces, Oxford: Oxford Univ. Press, 1986: 117–142.Google Scholar
  26. [26]
    Barnhill R Eet al. Surface/surface intersection. CAGD, 1987, 4(1–2): 3–16.MATHMathSciNetGoogle Scholar
  27. [27]
    Katz S, Sederberg T. Genus of the intersection curve of two rational surface patches. CAGD, 1988, 5(3): 253–258.MATHMathSciNetGoogle Scholar
  28. [28]
    Barnhill R. E, Kersey S N. A marching method for parametric surface/surface intersection. Computer Aided Geometric Design, 1990, 7: 257–280.MATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    de Montaudouin Y, Tiller W, Vold H. Applications of power series in computational geometry. Computer-Aided Design, 1986, 18(10): 514–524.CrossRefGoogle Scholar
  30. [30]
    Bajaj C L, Hoffmann C M, Hopcroft J E, Lynch R E. Tracing surface intersections. CAGD, 1988, 5: 285–307.MATHMathSciNetGoogle Scholar
  31. [31]
    Chen J J, Ozsoy T M. Predictor-corrector type of intersection algorithms forC 2 parametric surfaces. Computer-Aided Design, 1988, 20(6): 347–352.MATHCrossRefGoogle Scholar
  32. [32]
    Asteasu C, Orbegozo A. Parametric piecewise surfaces intersection. Computer & Graphics, 1991, 15(1): 9–13.CrossRefGoogle Scholar
  33. [33]
    Phillips M B, Odell G M. An algorithm for locating and displaying the intersection of two arbitrary surfaces. IEEE CG& A, 1984, 4(9): 48–58.Google Scholar
  34. [34]
    Comba P. A procedure for detecting intersections of 3-D objects. JACM, 1968, 15(3): 351–366.CrossRefGoogle Scholar
  35. [35]
    Piegl L. Geometric method of intersecting natural quadrics represented in trimmed surface form. Computer-Aided Design, 1989, 21(4).Google Scholar
  36. [36]
    Boehm W, Farin G, Kahmann J. A survey of curve and surface methods in CAGD. Computer-Aided Geometric Design, 1984, 1(1): 1–60.MATHCrossRefGoogle Scholar

Copyright information

© Science Press, Beijing China and Allerton Press Inc. 1994

Authors and Affiliations

  • Kaihuai Qin
    • 1
  • Gang Fan
    • 1
  • Cai Sun
    • 1
  1. 1.Department of Computer Science and TechnologyTsinghua UniversityBeijing

Personalised recommendations