Closed-form expressions for the approximation of arclength parameterization for Bézier curves

Mohsen Madi

International Journal of Applied Mathematics and Computer Science (2004)

  • Volume: 14, Issue: 1, page 33-41
  • ISSN: 1641-876X

Abstract

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In applications such as CNC machining, highway and railway design, manufacturing industry and animation, there is a need to systematically generate sets of reference points with prescribed arclengths along parametric curves, with sufficient accuracy and real-time performance. Thus, mechanisms to produce a parameter set that yields the coordinates of the reference points along the curve Q(t) = {x(t), y(t)} are sought. Arclength parameterizable expressions usually yield a parameter set that is necessary to generate reference points. However, for typical design curves, such expressions are not often available in closed form. It is thus desirable to find efficient ways to compensate for this lack of arclength parameterization. In this paper, several methods for approximating arclength parameterizations are studied. These methods are examined for both accuracy and real-time processing requirements. The application of generating reference points uniformly spaced along the paths of several curves is chosen for the illustration and comparison between the presented methods.

How to cite

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Madi, Mohsen. "Closed-form expressions for the approximation of arclength parameterization for Bézier curves." International Journal of Applied Mathematics and Computer Science 14.1 (2004): 33-41. <http://eudml.org/doc/207676>.

@article{Madi2004,
abstract = {In applications such as CNC machining, highway and railway design, manufacturing industry and animation, there is a need to systematically generate sets of reference points with prescribed arclengths along parametric curves, with sufficient accuracy and real-time performance. Thus, mechanisms to produce a parameter set that yields the coordinates of the reference points along the curve Q(t) = \{x(t), y(t)\} are sought. Arclength parameterizable expressions usually yield a parameter set that is necessary to generate reference points. However, for typical design curves, such expressions are not often available in closed form. It is thus desirable to find efficient ways to compensate for this lack of arclength parameterization. In this paper, several methods for approximating arclength parameterizations are studied. These methods are examined for both accuracy and real-time processing requirements. The application of generating reference points uniformly spaced along the paths of several curves is chosen for the illustration and comparison between the presented methods.},
author = {Madi, Mohsen},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {arclength parameterization; reference points; Hermite interpolation},
language = {eng},
number = {1},
pages = {33-41},
title = {Closed-form expressions for the approximation of arclength parameterization for Bézier curves},
url = {http://eudml.org/doc/207676},
volume = {14},
year = {2004},
}

TY - JOUR
AU - Madi, Mohsen
TI - Closed-form expressions for the approximation of arclength parameterization for Bézier curves
JO - International Journal of Applied Mathematics and Computer Science
PY - 2004
VL - 14
IS - 1
SP - 33
EP - 41
AB - In applications such as CNC machining, highway and railway design, manufacturing industry and animation, there is a need to systematically generate sets of reference points with prescribed arclengths along parametric curves, with sufficient accuracy and real-time performance. Thus, mechanisms to produce a parameter set that yields the coordinates of the reference points along the curve Q(t) = {x(t), y(t)} are sought. Arclength parameterizable expressions usually yield a parameter set that is necessary to generate reference points. However, for typical design curves, such expressions are not often available in closed form. It is thus desirable to find efficient ways to compensate for this lack of arclength parameterization. In this paper, several methods for approximating arclength parameterizations are studied. These methods are examined for both accuracy and real-time processing requirements. The application of generating reference points uniformly spaced along the paths of several curves is chosen for the illustration and comparison between the presented methods.
LA - eng
KW - arclength parameterization; reference points; Hermite interpolation
UR - http://eudml.org/doc/207676
ER -

References

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  1. Burchard H.G., Ayers J.A., Frey W.H. and Sapidis N.S. (1994): Approximating with aesthetic constraints, In: Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design (N.S.Sapidis, Ed.). - Philadelphia: SIAM, pp.3-28. Zbl0834.41016
  2. Davis P.J. (1963): Interpolation and Approximation. - New York: Blaisdell Publishing Company. Zbl0111.06003
  3. Farin G. (1993): Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. - Boston: Academic Press. Zbl0694.68004
  4. Farouki R.T. (1992): Pythagorean-hodograph curves in practical Use}, In: Geometry Processing for Design and Manufacturing, (R.E. Barnhill, Ed.). - Philadelphia: SIAM, pp.3-33. Zbl0770.41017
  5. Farouki R.T. (1997): Optimal Parameterizations. - Computer Aided Geometric Design, Vol.14, No.2, pp.153-168. Zbl0906.68156
  6. Farouki R.T. and Sakkalis T. (1991): Pythagorean hodographs. - IBM J. Res. Development, Vol.34, No.5, pp.736-752. 
  7. Farouki R.T. and Shah S. (1996): Real-time CNC interpolators for Pythagorean-hodograph curves. - Computer Aided Geometric Design, Vol.13, No.7, pp.583-600. Zbl0875.68875
  8. Foley J.D., Van Dam A., Feiner S.K. and Hughes J.F. (1992): Computer Graphics: Principles and Practice. - Reading: Addison Wesley. Zbl0875.68891
  9. Guggenheimer H.W. (1963): Differential Geometry. - New York: McGraw-Hill, pp.15-17. Zbl0116.13402
  10. Madi M.M. (1996): Arclength Approximation for Reference-Point Generation. - M.Sc. Thesis, Dept. of Computer Science, University of Manitoba. 
  11. Sharpe R.J. and Thorne R.W. (1982): Numerical method for extracting an arclength parameterization from parametric curves. - Computer-Aided Design, Vol.14, No.2, pp. 79-81. 
  12. Su B-Q. and Liu D-Y. (1989): Computational Geometry: Curve and Surface Modeling. - Boston: Academic Press. 
  13. Young E.C. (1993): Vector and Tensor Analysis. - New York: Marcel Dekker. 

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