Abstract
There are several prevailing methods for selecting knots for curve interpolation. A desirable criterion for knot selection is whether the knots can assist an interpolation scheme to achieve the reproduction of polynomial curves of certain degree if the data points to be interpolated are taken from such a curve. For example, if the data points are sampled from an underlying quadratic polynomial curve, one would wish to have the knots selected such that the resulting interpolation curve reproduces the underlying quadratic curve; and in this case the knot selection scheme is said to have quadratic precision. In this paper we propose a local method for determining knots with quadratic precision. This method improves on upon our previous method that entails the solution of a global equation to produce a knot sequence with quadratic precision. We show that this new knot selection scheme results in better interpolation error than other existing methods, including the chord-length method, the centripetal method and Foley’s method, which do not possess quadratic precision.
Supported by the National Key Basic Research 973 Program of China (No.2006CB303102) and the National Nature Science Foundation of China (No.60573181,60933008), Shandong Province National Nature Science Foundation(No.Z2006G05).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The theory of splines and their applications. Academic Press, New York (1967)
de Boor, C.: A practical guide to splines. Springer, New York (1978)
Brodlie, K.W.: A review of methods for curve and function drawing. In: Brodile, K.W. (ed.) Mathematical methods in computer graphics and design, pp. 1–37. Academic Press, London (1980)
Su, B., Liu, D.: Computational Geometry, pp. 47–48. Shanghai Science and Technology Press, Shanghai (1982)
Faux, I.D., Pratt, M.J.: Computational Geometry for Design and Manufacture. Ellis Horwood (1979)
Späth, H.: Spline algorithms for curves and surfaces Utilitas Mathematica. Winnipeg, Canada (1974)
Floater, M.S.: Chordal cubic spline interpolation is fourth order accurate. IMA J. Numer. Anal. 26, 25–33 (2006)
Lee, E.T.Y.: Choosing nodes in parametric curve interpolation. CAD 21(6), 363–370 (1989)
Farin, G.: Curves and surfaces for computer aided geometric design: A practical guide. Academic Press, London (1988)
Zhang, C., Cheng, F., Miura, K.: A method for determing knots in parametric curve interpolation. CAGD 15, 399–416 (1998)
Zhang, C., Cheng, F.: Constructing Parametric Quadratic Curves. Journal of Computational and Applied Mathematics 102, 21–36 (1999)
Xie, H., Qin, H.: A Novel Optimization Approach to The Effective Computation of NURBS Knots. International Journal of Shape Modeling 7(2), 199–227 (2001)
Xie, H., Qin, H.: Automatic Knot Determination of NURBS for Interactive Geometric Design. In: Proceedings of International Conference on Shape Modeling and Applications, SMI 2001, pp. 267–277 (2001)
Marin, S.P.: An approach to data parametrization in parametric cubic spline interpolation problems. J. Approx. Theory 41, 64–86 (1984)
Zhang, C., Han, H., Cheng, F.: Determining Knots by Minimizing Energy. Journal of Computer Science and Technology 216, 261–264 (2006)
Hartley, P.J., Judd, C.J.: Parametrization and shape of B-spline curves for CAD. CAD 12(5), 235–238 (1980)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Caiming, Z., Wenping, W., Jiaye, W., Xuemei, L. (2010). Selecting Knots Locally for Curve Interpolation with Quadratic Precision. In: Mourrain, B., Schaefer, S., Xu, G. (eds) Advances in Geometric Modeling and Processing. GMP 2010. Lecture Notes in Computer Science, vol 6130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13411-1_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-13411-1_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-13410-4
Online ISBN: 978-3-642-13411-1
eBook Packages: Computer ScienceComputer Science (R0)