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.
@article{bwmeta1.element.bwnjournal-article-amcv14i1p33bwm,
author = {Madi, Mohsen},
title = {Closed-form expressions for the approximation of arclength parameterization for B\'ezier curves},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {14},
year = {2004},
pages = {33-41},
zbl = {1171.65344},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv14i1p33bwm}
}
Madi, Mohsen. Closed-form expressions for the approximation of arclength parameterization for Bézier curves. International Journal of Applied Mathematics and Computer Science, Tome 14 (2004) pp. 33-41. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv14i1p33bwm/
[000] 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. | Zbl 0834.41016
[001] Davis P.J. (1963): Interpolation and Approximation. - New York: Blaisdell Publishing Company. | Zbl 0111.06003
[002] Farin G. (1993): Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. - Boston: Academic Press. | Zbl 0694.68004
[003] 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. | Zbl 0770.41017
[004] Farouki R.T. (1997): Optimal Parameterizations. - Computer Aided Geometric Design, Vol.14, No.2, pp.153-168. | Zbl 0906.68156
[005] Farouki R.T. and Sakkalis T. (1991): Pythagorean hodographs. - IBM J. Res. Development, Vol.34, No.5, pp.736-752.
[006] 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. | Zbl 0875.68875
[007] Foley J.D., Van Dam A., Feiner S.K. and Hughes J.F. (1992): Computer Graphics: Principles and Practice. - Reading: Addison Wesley. | Zbl 0875.68891
[008] Guggenheimer H.W. (1963): Differential Geometry. - New York: McGraw-Hill, pp.15-17. | Zbl 0116.13402
[009] Madi M.M. (1996): Arclength Approximation for Reference-Point Generation. - M.Sc. Thesis, Dept. of Computer Science, University of Manitoba.
[010] 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.
[011] Su B-Q. and Liu D-Y. (1989): Computational Geometry: Curve and Surface Modeling. - Boston: Academic Press.
[012] Young E.C. (1993): Vector and Tensor Analysis. - New York: Marcel Dekker.