Skip to main content
Log in

Two blossoming proofs of the Lane-Riesenfeld algorithm

  • Published:
Computing Aims and scope Submit manuscript

Abstract

The standard proof of the Lane-Riesenfeld algorithm for inserting knots into uniform B-spline curves is based on the continuous convolution formula for the uniform B-spline basis functions. Here we provide two new, elementary, blossoming proofs of the Lane-Riesenfeld algorithm for uniform B-spline curves of arbitrary degree.

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

Access this article

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.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Barry, P., Goldman, R.: Factored knot insertion. In: Knot Insertion and Deletion Algorithms for B-spline Curves and Surfaces (Goldman, R. and Lyche, T., eds). 1993, pp. 65–88.

  • Barry, P., Goldman, R.: Knot insertion algorithms. In: Knot Insertion and Deletion Algorithms for B-spline Curves and Surfaces (Goldman, R. and Lyche, T., eds). 1993, pp. 89–133.

  • W. Boehm (1980) ArticleTitleInserting new knots into B-spline curves Comput. Aided Geom. Des. 12 199–201

    Google Scholar 

  • de Casteljau, P.: Formes a poles. Hermes 1985.

  • E. Cohen T. Lyche R. Riesenfeld (1980) ArticleTitleDiscrete B-splines and subdivision techniques in computer aided geometric design and computer graphics Comput. Graph. Image Process. 14 87–111 Occurrence Handle10.1016/0146-664X(80)90040-4

    Article  Google Scholar 

  • R. Goldman (1990) ArticleTitleBlossoming and knot insertion algorithms for B-spline curves Comput. Aided Geom. Des. 7 69–81 Occurrence Handle0713.65006 Occurrence Handle10.1016/0167-8396(90)90022-J

    Article  MATH  Google Scholar 

  • R. Goldman (2002) Pyramid Algorithms: A dynamic programming approach to curves and surfaces for geometric modeling Morgan Kaufmann Publishers/Academic Press San Diego

    Google Scholar 

  • R. Goldman J. Warren (1993) ArticleTitleAn extension of Chaikin's Algorithm to B-spline curves with knots in geometric progression CVGIP: Graph. Models Image Process. 55 58–62 Occurrence Handle10.1006/cgip.1993.1004

    Article  Google Scholar 

  • J. Lane R. Riesenfeld (1980) ArticleTitleA theoretical development for the computer generation and display of piecewise polynomial surfaces IEEE Trans. Pattern Anal. Mach. Intell. 2 35–46 Occurrence Handle0436.68063 Occurrence Handle10.1109/TPAMI.1980.4766968

    Article  MATH  Google Scholar 

  • Ramshaw, L.: Blossoming: A connect-the-dots approach to splines. SRC Research Report 19 (1987).

  • Ramshaw, L.: Beziers and B-splines as multiaffine maps. In: Theoretical foundations of computer graphics and CAD (Earnshaw, R. A., ed.) NATO ASI Series F, vol. 40. New York: Springer 1988, pp. 757–776.

  • L. Ramshaw (1989) ArticleTitleBlossoms are polar forms Comput. Aided Geom. Des. 6 323–358 Occurrence Handle0705.65008 Occurrence Handle10.1016/0167-8396(89)90032-0 Occurrence Handle1030618

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Goldman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Vouga, E., Goldman, R. Two blossoming proofs of the Lane-Riesenfeld algorithm. Computing 79, 153–162 (2007). https://doi.org/10.1007/s00607-006-0194-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-006-0194-y

AMS Subject Classifications

Keywords

Navigation