Geodesic
Closed-form expressions for the approximation of arclength parameterization for Bezier curves
International Journal of Applied Mathematics and Computer Science, Tome 14 (2004) no. 1, pp. 33-41.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: arclength parameterization, Hermite interpolation, reference points
Mots-clés : parametryzacja długości łuku, interpolacja Hermite'a, punkt odniesienia 
@article{IJAMCS_2004_14_1_a4,
     author = {Madi, M.},
     title = {Closed-form expressions for the approximation of arclength parameterization for {Bezier} curves},
     journal = {International Journal of Applied Mathematics and Computer Science},
     pages = {33--41},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IJAMCS_2004_14_1_a4/}
}
TY  - JOUR
AU  - Madi, M.
TI  - Closed-form expressions for the approximation of arclength parameterization for Bezier curves
JO  - International Journal of Applied Mathematics and Computer Science
PY  - 2004
SP  - 33
EP  - 41
VL  - 14
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IJAMCS_2004_14_1_a4/
LA  - en
ID  - IJAMCS_2004_14_1_a4
ER  - 
%0 Journal Article
%A Madi, M.
%T Closed-form expressions for the approximation of arclength parameterization for Bezier curves
%J International Journal of Applied Mathematics and Computer Science
%D 2004
%P 33-41
%V 14
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IJAMCS_2004_14_1_a4/
%G en
%F IJAMCS_2004_14_1_a4
Madi, M. Closed-form expressions for the approximation of arclength parameterization for Bezier curves. International Journal of Applied Mathematics and Computer Science, Tome 14 (2004) no. 1, pp. 33-41. http://geodesic.mathdoc.fr/item/IJAMCS_2004_14_1_a4/

[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.

[2] Davis P.J. (1963): Interpolation and Approximation. — New York: Blaisdell Publishing Company.

[3] Farin G. (1993): Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide. — Boston: Academic Press.

[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.

[5] Farouki R.T. (1997): Optimal Parameterizations. — Computer Aided Geometric Design, Vol. 14, No. 2, pp. 153–168.

[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.

[8] Foley J.D., Van Dam A., Feiner S.K. and Hughes J.F. (1992): Computer Graphics: Principles and Practice.—Reading: Addison Wesley.

[9] Guggenheimer H.W. (1963): Differential Geometry. — New York: McGraw-Hill, pp. 15–17.

[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.