Abstract
A well-known method of estimating the length of a parametric curve in \({\mathbb {R}}^d\) is to sample some points from it and compute the length of the polygon passing through them. In this paper we show that for uniform sampling of regular smooth curves, Richardson extrapolation can be applied repeatedly giving a sequence of derivative-free length estimates of arbitrarily high orders of accuracy. A similar result is derived for the approximation of the area of parametric surfaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, D.N.: A concise introduction to numerical analysis. In: Lecture Notes. IMA, Minnesota (2001)
Casciola, G., Morigi, S.: Reparametrization of nurbs curves. Int. J. Shape Model. 2, 103–116 (1996)
Constantini, P., Farouki, R.T., Manni, C., Sest ini, A.: Computation of optimal composite re-parametrizations. Comput. Aided Geom. Des. 18, 875–897 (2001)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic, New York (1975)
Farouki, R.T.: Optimal parametrizations. Comput. Aided Geom. Des. 8, 153–168 (1997)
Farouki, R.T.: Pythagorean-hodograph curves. In: Handbook of Computer Aided Geometric Design, pp. 405–427. North Holland, Amsterdam (2002)
Farouki, R.T., Sakkalis, T.: Real rational curves are not ’unit speed.’ Comput. Aided Geom. Des. 8, 151–157 (1991)
Floater, M.S.: Arc length estimation and the convergence of polynomial curve interpolation. BIT 45, 679–694 (2005)
Floater, M.S.: Chordal cubic spline interpolation is fourth order accurate. IMA J. Numer. Anal. 26, 25–33 (2006)
Floater, M.S., Rasmussen, A.F.: Point-based methods for estimating the length of a parametric curve. J. Comput. Appl. Math. 196, 512–522 (2006)
Gravesen, J.: Adaptive subdivision and the length and energy of Bézier curves. Comput. Geom. 8, 13–31 (1997)
Guenter, B., Parent, R.: Computing the arc length of parametric curves. IEEE Comput. Graph. Appl. 5, 72–78 (1990)
Kearney, J., Wang, H., Atkinson, K.: Arc-length parameterized spline curves for real-time simulation. In: Curve and Surface Design, Saint-Malo, pp. 387–396 (2002)
Lyness, J.N.: Quadrature over curved surfaces by extrapolation. Math. Comput. 63, 727–740 (1994)
Rasmussen, A.F., Floater, M.S.: A point-based method for estimating surface area. In: Proceedings of the SIAM conference on Geometric Design and Computing, Phoenix (2005)
Sharpe, R.J., Thorne, R.W.: Numerical method for extracting an arc length parameterization from parametric curves. Comput. Aided Des. 14(2), 79–81 (1982)
Vincent, S., Forsey, D.: Fast and accurate parametric curve length computation. J. Graphics Tools 6(4), 29–40 (2002)
Walter, M., Fournier, A.: Approximate arc length parameterization. In: Proceedings of the 9th Brazilian Symposium on Computer Graphics and Image Processing, pp. 143–150 (1996)
Wang, F.-C., Yang, D.C.H.: Nearly arc-length parameterized quintic-spline interpolation for precision machining. Comput. Aided Des. 25(5), 281–288 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Floater, M.S., Rasmussen, A.F. & Reif, U. Extrapolation methods for approximating arc length and surface area. Numer Algor 44, 235–248 (2007). https://doi.org/10.1007/s11075-007-9095-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-007-9095-1