Abstract
We consider the choice of a functional to measure the distance between two parametric curves. We identify properties of such a distance functional that are important for geometric design. Several popular definitions of distance are examined, and new functionals are presented which satisfy the desired properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Bogacki, S.E. Weinstein and Y. Xu, Degree reduction of Bézier curves by uniform approximation with endpoint interpolation, Computer Aided Design 27 (1995) 651–661.
P. Bogacki and S.E. Weinstein, Geometric degree reduction, in: 4th SIAM Conference on Geometric Design, Nashville, TN (1995).
W.L.F. Degen, Best approximations of parametric curves by splines, in: Mathematical Methods in Computer Aided Geometric Design, Vol. II, eds. T. Lyche and L.L. Schumaker (Academic Press, New York, 1992) pp. 171–184.
M. Eck, Least squares degree reduction of Bézier curves, Computer Aided Design 27 (1995) 845–851.
E.F. Eisele, Chebyshev approximation of plane curves by splines, Journal of Approximation Theory 76 (1994) 133–148.
J.D. Emery, The definition and computation of a metric on plane curves, Computer Aided Design 18 (1986) 25–28.
G.M. Ewing, Calculus of Variations with Applications (Dover, New York, 1985).
G. Farrin, Curves and Surfaces for Computer Aided Geometric Design, A Practical Guide (Academic Press, New York, 3rd ed., 1994).
F.N. Fritsch and G.M. Nielson, On the problem of determining the distance between parametric curves, in: Curve and Surface Design, ed. H. Hagen (SIAM, Philadelphia, PA, 1992).
F. Hausdorff, Set Theory (Chelsea, New York, 1957).
J. Hoschek, Approximate conversion of spline curves, Computer Aided Geometric Design 4 (1987) 59–66.
J. Hoschek, F.J. Schneider and P. Wassum, Optimal approximate conversion of spline surfaces (1989).
J. Hoschek and F.J. Schneider, Approximate spline conversion for integral and rational Bézier and B-spline surfaces, in: Geometry Processing for Design and Manufacturing, ed. R.E. Barnhill (SIAM, Philadelphia, PA, 1992).
T. Lyche and K. Mørken, A metric for parametric approximation, in: Curves and Surfaces in Geometric Design, eds. P.J. Laurent, A. Le Méhauté and L.L. Schumaker (A.K. Peters, Wellesley, MA, 1994) pp. 311–318.
D.S. Meek and D.J. Walton, Alignment of planar curves, Image and Vision Computing 12 (1994) 305–311.
F. Morgan, Geometric Measure Theory (Academic Press, San Diego, CA, 1988).
S.E. Weinstein and Y. Xu, Degree reduction of Bézier curves by approximation and interpolation, in: Approximation Theory, ed. G.A. Anastassiou (New York, 1992) pp. 503–512.
H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, 1957).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bogacki, P., Weinstein, S. & Xu, Y. Distances between oriented curves in geometric modeling. Advances in Computational Mathematics 7, 593–621 (1997). https://doi.org/10.1023/A:1018923609019
Issue Date:
DOI: https://doi.org/10.1023/A:1018923609019