Skip to main content
SpringerLink
Log in

Intersection Algorithms and CAGD

  • Chapter
Geometric Modelling, Numerical Simulation, and Optimization

Abstract

An approach for calculating intersections between parametric surfaces based on long experience in developing intersection algorithms for industrial use, is presented. In addition two novel methods that help improve quality and performance of intersection algorithms are described:

  • An initial assessment of the intersection complexity to identify most transversal intersections, and to identify surface regions with possible complex intersection topology. To find regions where the surfaces possibly intersect, and regions where surface normals possibly are parallel, the computational power of multi-core CPUs and programmable graphics processors (GPUs) is used for subdivision of the surfaces and their normal fields.

  • Approximate implicitization of surface regions to help analyse singular and near singular intersections.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. L. Abrams, W. Cho, C. Y. Hu, T. Maekawa, N. M. Patrikalakis, E. C. Sherbrooke, and Ye X. Methods for rounded interval arithmetic. Computer Aided Design, 30(8):657–666, 1998.

    Article  MATH  Google Scholar 

  2. C. Bajaj. The emergence of algebraic curves and surfaces in geometric design. In Directions in Geometric Computing, pages 1–28. Information Geometers, 1993.

    Google Scholar 

  3. S. Briseid, T. Dokken, T. R. Hagen, and J. O. Nygaard. Spline surface intersections optimized for GPUs. In V. N. Alexandrov, G. D. van Albada, P. M. A. Sloot, and J. Dongarra, editors, Computational Science-ICCS 2006: 6th International Conference, Reading, UK, May 28–31, 2006, Proceedings, Part IV, volume 3994 of Lecture Notes in Computer Science, pages 204–211, Berlin/Heidelberg, 2006. Springer.

    Google Scholar 

  4. E. Cohen, R. F. Riesenfeld, and G. Elber. Geometric modeling with splines. A K Peters Ltd., Natick, MA, 2001. An introduction, With a foreword by Tom Lyche.

    MATH  Google Scholar 

  5. O.J Dahl and D. Belsnes. Algorithms and Data Structures. Studentlitteratur, Lund, Sweden, 1973.

    MATH  Google Scholar 

  6. C. de Boor, K. Höllig, and M. Sabin. High accuracy geometric Hermite interpolation. Comput. Aided Geom. Design, 4(4):269–278, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Dokken. Reference Manual, B-spline Library. SINTEF (Senter for Industriforskning), Oslo, Norway, 1983.

    Google Scholar 

  8. T. Dokken. Finding intersections of b-spline represented geometries using recursive subdivision techniques. Comput. Aided Geom. Design, 2:189–195, 1985.

    Article  MATH  Google Scholar 

  9. T. Dokken. APS-SS: A subrouting package for modelling of sculptured surfaces. SINTEF (Senter for Industriforskning), Oslo, Norway, 1987.

    Google Scholar 

  10. T. Dokken. The SINTEF Spline Library, version 4.0. SINTEF, Oslo, Norway, 1994.

    Google Scholar 

  11. T. Dokken. Aspect of Intersection algorithms and Approximation, Thesis for the doctor philosophias degree. PhD thesis, University of Oslo, 1997.

    Google Scholar 

  12. T. Dokken. Aspects of algorithms for manifold intersection. In Numerical methods and software tools in industrial mathematics, pages 365–380. Birkhauser, Boston, MA, 1997.

    Google Scholar 

  13. T. Dokken. Approximate implicitization. In Mathematical methods for curves and surfaces (Oslo, 2000), Innov. Appl. Math., pages 81–102. Vanderbilt Univ. Press, Nashville, TN, 2001.

    Google Scholar 

  14. T. Dokken, H. K. Kellermann, and C. Tegnander. An approach to weak approximate implicitization. In Mathematical methods for curves and surfaces (Oslo, 2000), Innov. Appl. Math., pages 103–112. Vanderbilt Univ. Press, Nashville, TN, 2001.

    Google Scholar 

  15. 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 (Oslo, 1988), pages 207–214. Academic Press, Boston, MA, 1989.

    Google Scholar 

  16. T. Dokken and J. B. Thomassen. Overview of approximate implicitization. In Topics in algebraic geometry and geometric modeling, volume 334 of Contemp. Math., pages 169–184. Amer. Math. Soc., Providence, RI, 2003.

    Google Scholar 

  17. G. Farin, J. Hoschek, and M.-S. Kim, editors. Handbook of computer aided geometric design. North-Holland, Amsterdam, 2002.

    MATH  Google Scholar 

  18. R. T. Farouki. Pythagorean-hodograph curves. In Handbook of computer aided geometric design, pages 405–427. North-Holland, Amsterdam, 2002.

    Google Scholar 

  19. 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(5):459–478, 2004.

    MATH  MathSciNet  Google Scholar 

  20. R. T. Farouki and V. T. Rajan. On the numerical condition of polynomials in Bernstein form. Comput. Aided Geom. Design, 4(3):191–216, 1987.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Gálvez, J. Puig-Pey, and A. Iglesias. Differential method for parametric surface intersection. In Computational Science and Its Applications — ICCSA 2004: International Conference, Assisi, Italy, May 14–17, 2004, volume 3044 of Lecture Notes in Comput. Sci., pages 651–660. Springer, Berlin, 2004.

    Google Scholar 

  23. K. Hasund, T. Dokken, and S. Ulfsby. System specifications, sculptured surfaces. Technical Report 16, SINTEF (Sentralinstitutt for industriell forskning), Oslo, Norway, 1980.

    Google Scholar 

  24. M. E. Hohmeyer. Robust and Efficient Surface Intersection for Solid Modelling. PhD thesis, Computer Science Division, University of California, 1992.

    Google Scholar 

  25. K. Höllig and J. Koch. Geometric Hermite interpolation. Comput. Aided Geom. Design, 12(6):567–580, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  26. K. Höllig and J. Koch. Geometric Hermite interpolation with maximal order and smoothness. Comput. Aided Geom. Design, 13(8):681–695, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  27. C.-Y. Hu, T. Maekawa, N. M. Patrikalakis, and X. Ye. Robust interval algorithm for surface intersections. Computer-Aided Design, 29(9):617–627, 1997.

    Article  MATH  Google Scholar 

  28. IEEE. Specification for binary floating point arithmetic for microprocessor systems, international standard, 1989.

    Google Scholar 

  29. ISO. Industrial automation systems and integration-product data representation and exchange — part 42: Integrated generic resources: Geometric and topological representation, 1994.

    Google Scholar 

  30. S. Krishnan and D. Manocha. An efficient surface intersection algorithm based on lower-dimensional formulation. ACM Trans. Graph., 16(1):74–106, 1997.

    Article  Google Scholar 

  31. J. M. Lane and R. F. Riesenfeld. Bounds on a polynomial. BIT, 21(1):112–117, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  32. T. Maekawa, W. Cho, and N. M. Patrikalakis. Computation of self-intersections of offsets of bézier surface patches. Journal of Mechanical Design, ASME Transactions, 119:275–283, 1997.

    Article  Google Scholar 

  33. H. Mukundan, K. H. Ko, T. Maekawa, T. Sakkalis, and N. M. Patrikalakis. Tracing surface intersections with a validated ode system solver. In G. Elber, N. Patrikalakis, and P. Brunet, editors, Proceedings of the Ninth EG/ACM Symposium on Solid Modeling and Applications, Genoa, Italy, June 2004, pages 249–254. Eurographics Press, 2004.

    Google Scholar 

  34. N. M. Patrikalakis and T. Maekawa. Intersection problems. In Handbook of computer aided geometric design, pages 623–649. North-Holland, Amsterdam, 2002.

    Google Scholar 

  35. N. M. Patrikalakis and T. Maekawa. Shape interrogation for computer aided design and manufacturing. Springer-Verlag, Berlin, 2002.

    MATH  Google Scholar 

  36. N. M. Patrikalakis, T. Maekawa, K. H. Ko, and H. Mukundan. Surface to surface intersections. Computer-Aided Design & Applications, 1:449–458, 2004.

    Google Scholar 

  37. J. Peters and X. Wu. Sleves for planar spline curves. In Computer Aided Geometric Design, pages 615–635, 2004.

    Google Scholar 

  38. J. Puig-Pey, A. Gálvez, and A. Iglesias. A new differential approach for parametric-implicit surface intersection. In Computational science-ICCS 2003. Part I, volume 2657 of Lecture Notes in Comput. Sci., pages 897–906. Springer, Berlin, 2003.

    Google Scholar 

  39. T. Sederberg, R. Goldman, and H. Du. Implicitizing rational curves by the method of moving algebraic curves. J. Symbolic Comput., 23(2–3):153–175, 1997. Parametric algebraic curves and applications (Albuquerque, NM, 1995).

    Article  MATH  MathSciNet  Google Scholar 

  40. T. W. Sederberg. Implicit and parametric curves and surfaces for computer aided geometric design. PhD thesis, Purdue University, 1983.

    Google Scholar 

  41. T. W. Sederberg. Planar piecewise algebraic curves. Computer Aided Geometric Design, 1(3):241–255, 1984.

    Article  MATH  Google Scholar 

  42. T. W. Sederberg and R. J. Meyers. Loop detection in surface patch intersections. Comput. Aided Geom. Design, 5:161–171, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  43. T. W. Sederberg and Anderson D. C. and Goldman R. N. Implicit representation of parametric curves and surfaces. Computer Vision, Graphics, and Image Processing, 28:72–84, 1984.

    Article  Google Scholar 

  44. T. W. Sederberg, J. Zheng, K. Klimaszewski, and T. Dokken. Approximate implicitization using monoid curves and surfaces. Graphical Models and Image Processing, 61(4):177–198, 1999.

    Article  Google Scholar 

  45. T. W. Sederberg, J. Zheng, and Song X. A conjecture on tangent intersections of surface patches. Comput. Aided Geom. Design, 21(1):1–2, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  46. T. W. Sederberg and A. K. Zundel. Pyramids that bound surface patches. CVGIP: Graphical Model and Image Processing, 58(1):75–81, 1996.

    Article  Google Scholar 

  47. P. Sinha, E. Klassen, and K.K. Wang. Exploiting topological and geometric properties for selective subdivision. In Symposium on Computational Geometry, pages 39–45. ACM Press, 1985.

    Google Scholar 

  48. V. Skytt. Challenges in surface-surface intersections. In Computational methods for algebraic spline surfaces, pages 11–26. Springer, Berlin, 2005.

    Chapter  Google Scholar 

  49. V. Skytt. A recursive approach to surface-surface intersections. In Mathematical methods for curves and surfaces: Troms, Mod. Methods Math., pages 327–338. Nashboro Press, Brentwood, TN, 2005.

    Google Scholar 

  50. A. K. Zundel and T. W. Sederberg. Surface intersection loop destruction. In The mathematics of surfaces, VII (Dundee, 1996), pages 463–478. Info. Geom., Winchester, 1997.

    Google Scholar 

  51. A.K. Zundel. Surface-Surface Intersection: Loop Destruction Using Bézier Clipping and Pyramidal Bounds. Thesis for Doctor of Philosophy. PhD thesis, Brigham Young University, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Applied Mathematics, SINTEF ICT, P.O. Box 124, Blindern, NO-0314, Oslo, Norway

    Tor Dokken & Vibeke Skytt

Authors
  1. Tor Dokken
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vibeke Skytt
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Applied Mathematics, SINTEF ICT, P.O. Box 124, Blindern, NO-0314, Oslo, Norway

    Geir Hasle , Knut-Andreas Lie  & Ewald Quak ,  & 

  2. Centre for Nonlinear Studies (CENS) Institute of Cybernetics, Tallinn University of Technology, Akadeemia tee 21, EE-12618, Tallinn, Estonia

    Ewald Quak

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dokken, T., Skytt, V. (2007). Intersection Algorithms and CAGD. In: Hasle, G., Lie, KA., Quak, E. (eds) Geometric Modelling, Numerical Simulation, and Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68783-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation