Skip to main content
SpringerLink
Log in

Deriving the beta-constraints for geometric continuity of parametric curves

  • Published:
Rendiconti del Seminario Matematico e Fisico di Milano Aims and scope Submit manuscript

Abstract

Parametric splines curves are typically constructed so that the firstn parametric derivatives agree where the curve segments abut. This type of continuity condition has become known asC n orn th orderparametric continuity. It has previously been shown that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.

We definen th ordergeometric continuity (Gn), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity.G n continuity provides for the introduction ofn quantities known asshape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.

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. B.A. BarskyThe Beta-spline: A Local Representation Based on shape Parameters and Fundamental Geometric Measures, Ph.D. Thesis, University of Utah, Salt Lake City, Utah (December, 1981)

    Google Scholar 

  2. B.A. BarskyComputer Graphics and Geometric Modeling Using Beta-splines, Springer-Verlag, Heidelberg, 1988

    MATH  Google Scholar 

  3. B.A. Barsky, J.C. BeattyLocal control of Bias and Tension in Beta-splines, ACM Transaction on Graphic, Vol. 2, No. 2, April 1983, pp. 109–134. Also published in SIGGRAPH ’83 Conference proceedings, Vol. 17, No. 3, ACM, Detroit, 25–29 July, 1983, pp. 193–218.

    Article  MATH  Google Scholar 

  4. B.A. Barsky, T.D. DeRoseGeometric continuity of Parametric Curves, Technical Report No. UCB/CSD 84/205, Computer Science Division, Electrical Engineering and Computer Sciences Department, University of California, Berkeley, California (October, 1984)

    Google Scholar 

  5. R.H. Bartles, J.C. BeattyBeta-splines with a Difference, Technical Report No. CS-84-40, Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada (May, 1984).

    Google Scholar 

  6. W. BoehmCurvature Continuous Curves and Surfaces, Computer Aided Geometric Design, Vol. 2, No. 4, December 1985, pp. 313–323.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. BoehmTorsion Continuous Curves, pp. 175–184 in Geometric Modeling: Algorithm and New Trends, edited by Gerald Farin, SIAM, 1987.

  8. C. de BoorOn Calculating with Beta-splines, Journal of Approximation Theory, Vol. 6, No. 1, July 1972, pp. 50–62

    Article  MATH  MathSciNet  Google Scholar 

  9. R.C. BuckAdvanced Calculus, McGraw-Hill Book Company, Inc., New York (1956).

    MATH  Google Scholar 

  10. E.E. Catmull, R.J. RomA Class of Local Interpolating Splines, pp. 317–326 in Computer Aided Geometric Design, ed. Robert E. Barnhill and Richard F. Riesenfeld, Academic Press, New York (1974).

    Google Scholar 

  11. E. Cohen, T. Lyche, R.F. RiesenfeldDiscrete Beta-splines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics, Computer Graphics and Image Processing, Vol. 14, No. 2, October 1980, pp. 87–111. Also Technical Report No. UUCS-79-117, Department of Computer Science, University of Utah, October 1979.

    Article  Google Scholar 

  12. M.G. CoxThe Numerical Evaluation of Beta-splines, Report No. NPL-DNACS-4, Division of Numerical Analysis and Computing, National Physical Laboratory, Teddington, Middlesex, England (August, 1971). Also in J. Inst. Maths. Applics., Vol. 10, 1972, pp. 134–149.

    Google Scholar 

  13. A.D. DeRoseGeometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design, Ph.D. Thesis, University of California, Berkely, Ca. (August, 1985). Available as Tech. Report No. UCB/CSD 86/255, Division of Computer Science, Department of Electrical Engineering and Computer Sciences, University of California, Berkeley.

    Google Scholar 

  14. T.D. DeRose, B.A. BarskyGeometric Continuity and Shape Parameters for Catmull-Rom Splines (Extended Abstract), pp. 57–64 in Proceedings of Graphic Interface ’84, Ottawa (27 May–1 June, 1984).

  15. T.D. DeRose, B.A. BarskyAn Intuitive Approach to Geometric Continuity for Parametric Curves and Surfaces, pp. 343–351 in Proceedings of Graphics Interface ’85, Montreal (27–31 May, 1985). Revised version pubblished in Computer-Generated Images—The state of the Art, edited by Nadia Magnenat-Thalmann and Daniel Thalmann, Springer Verlag, 1985, pp. 159–175. Extended abstract in Proceedings of the International Conference on Computational Geometry and Computer-Aided Design, New Orleans (5–8 June, 1985), pp. 71–75.

  16. T.D. DeRose, B.A. BarskyGeometric Continuity, Shape parameters, and Geometric Constructions for Catmull-Rom Splines, ACM Transactions on Graphics Vol. 7, No. 1, January 1988, pp. 1–41.

    Article  MATH  Google Scholar 

  17. M.P. DoCarmoDifferential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1975).

    Google Scholar 

  18. N. Dyn, C.A. MicchelliPiecewise Polynomial Spaces and Geometry Continuity of Curves, IBM Research Report 11390, September 25, 1985.

  19. G. FarinVisually C 2 Cubic Splines, Computer-Aided Design, Vol. 14, No. 3, May, 1982, pp. 137–139.

    Article  Google Scholar 

  20. G. FarinA construction for Visual C 1 Continuity of Polynomial Surface Patches, Computer Graphics and Image Processing, Vol. 20, 1982, pp. 272–282.

    Article  MATH  Google Scholar 

  21. R. J. FatemanAddendum to the MACSYMA Reference Manual for the VAX, Computer Science Division, University of California, Berkeley (1982).

    Google Scholar 

  22. I.D. Faux, M.J. PrattComputational Geometry for Design and Manufacture, Ellis Horwood Ltd. (1979).

  23. A.H. Fowler, C.W. WilsonCubic Spline, A Curve Fitting Routine, Union Cardibe Corporation Report, Y-1400 (Rev. I.) (1966).

  24. T.N.T. GoodmanProperties of β-Splines, Journal of Approximation Theory, Vol. 44, No. 2, June, 1985, pp. 132–153.

    Article  MATH  MathSciNet  Google Scholar 

  25. W.J. Gordon, R.F. RiesenfeldBeta-spline Curves and Surfaces, pp. 95–126 in Computer Aided Geometric Design, ed. Robert E. Barnhill and Richard Riesenfeld, Academic Press, New York (1974).

    Google Scholar 

  26. G. HerronTechniques for Visual Continuity, pp. 163–174 in Geometric Modeling: Algorithm and New Trends, edited by Gerald Farin, SIAM, 1987.

  27. J. KahmannContinuity of curvature Between Adjacent Bèzier Patches, pp. 65–75 in Surfaces in Computer Aided Geometric Design, ed. Robert E. Barnhill and Wolfgang Boehm, North-Holland Publishing Company (1983).

  28. L.C. KnappA Design Scheme Using Coons Surfaces with Nonuniform Beta-spline Curves, Ph.D. Thesis, Syracuse University, Syracuse, New York (September, 1979).

    Google Scholar 

  29. E. KreyszigDifferential Geometry, University of Toronto Press, Toronto (1959).

    MATH  Google Scholar 

  30. J.R. ManningContinuity Conditions for Spline Curves, The Computer Journal, Vol. 17, No. 2, May, 1974, pp. 181–186.

    Article  MATH  Google Scholar 

  31. G.M. NielsonSome Piecewise Polynomial Alternatives to Splines under Tension, pp. 209–235 in Computer Aided Geometric Design, ed. Robert E. Barnhill and Richard F.Riesenfeld, Academic Press, New York (1974).

    Google Scholar 

  32. B.R. PiperVisually Smooth Interpolation with Triangular Bèzier Patches, pp. 221–233, in Geometric Modeling: Algorithms and New Trends, edited by Gerald Farin, SIAM, 1987.

  33. R.F. RiesenfeldApplications of Beta-spline Approximation to Geometric Problem of Computer-Aided Design, Ph.D Thesis, Syracuse University (May, 1973). Available as Tech. Report No. UTEC-CSc-73-126, Department of Computer Science, University of Utah.

  34. R.F. Riesenfeld, E. Cohen, R.D. Fish, S.W. Thomas, E.S. Cobb, B.A. Barsky, D.L. Schweitzer, J.M. LaneUsing the Oslo Algorithm as a Basis for CAD/CAM Geometric Modelling, pp. 345–356 in Proceedings of the Second Annual NCGA Natinal Conference, National Computer Graphics Association, Inc., Baltimore (14–18 June, 1981).

  35. M.A. SabinParametric Splines in Tension, Technical Report No. VTO7MS/160, British Aircraft Corporation, Weybridge, Surrey, England (july 23, 1970).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, Berkeley, USA

    Brian Barsky

  2. University of Washington, Seattle, USA

    Tony D. DeRose

Authors
  1. Brian Barsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tony D. DeRose
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Conferenza tenuta da B.A. Barsky

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barsky, B., DeRose, T.D. Deriving the beta-constraints for geometric continuity of parametric curves. Seminario Mat. e. Fis. di Milano 63, 49–87 (1993). https://doi.org/10.1007/BF02925094

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02925094

Keywords

Search

Navigation