Abstract
In this paper a bivariate rational biquartic interpolating spline based on function values with two parameters is constructed , and this spline is with biquartic numerator and bilinear denominator. The interpolating function has a simple and explicit mathematical representation, which is convenient both in practical application and in theoretical study. The interpolating surface is C 1 in the interpolating region when one of the parameters satisfies a simple condition. The interpolating surface can be modified by selecting suitable parameters under the condition that the interpolating data are not changed. It is proved that the values of the interpolating function in the interpolating region are bounded no matter what the parameters might be; this is called the bounded property of the interpolation. The approximation expressions of the interpolation are derived:they do not depend on the parameters.
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.
References
Farin, G.: Curves and surfaces for computer aided geometric design: A practical guide. Academic Press, London (1988)
Chui, C.K.: Multivariate spline. SIAM, Philadelphia (1988)
Bézier, P.E.: The mathematical basis of the UNISURF CAD system. Butterworth, London (1986)
Dierck, P., Tytgat, B.: Generating the Bézier points of β-spline curve. Computer Aided Geometric Design 6, 279–291 (1989)
Piegl, L.: On NURBS: A survey. IEEE Computer Graphics and Application 11, 55–71 (1991)
Mielson, N.G.: CAGD’s Top Ten: What to watch. IEEE Computer Graphics and Automation, 35–37 (1993)
Konno, K., Chiyokura, H.: An approach of designing and controlling free-form surfaces by using NURBS boundary Gregory patches. Computer Aided Geometric Design 13, 825–849 (1996)
Wilcox, L.M.: First and second contributions to surface interpolation. Vision Research 39, 2335–2347 (1999)
Lin, R.S.: Real-time surface interpolator for 3-D parametric surface machining on 3-axis machine tools. Machine tools and Manufacture 40, 1513–1526 (2000)
Jiang, D., Liu, H., Wang, W.: Test a modified surface wind interpolation scheme for complex terrain in a stable atmosphere. Atmospheric Environment 35, 4877–4885 (2001)
Comninos, P.: An interpolating piecewise bicubic surface with shape parameters. Computer and Graphics 25, 463–481 (2001)
Müller, R.: Universal parametrization and interpolation on cubic surfaces. Computer Aided Geometric Design 19, 479–502 (2002)
Wang, R.H.: Multivariats Spline Functions and Their Application. Science press/Kluwer Academic, Beijing (2001)
Duan, Q., Djidjeli, K., Price, W.G., et al.: A rational cubic spline based on function values. Computer and Graphics 22, 479–486 (1998)
Duan, Q., Djidjeli, K., Price, W.G., et al.: The approximation properties of some rational cubic splines. International J. of Computer Mathematics 72, 155–166 (1999)
Duan, Q., Wang, L., Twizell, E.H.: A new bivariate rational interpolation based on function values. Information Sciences 166, 181–191 (2004)
Duan, Q., Zhang, Y., Twizell, E.H.: A bivariate rational interpolation and the properties. Applied Mathematics and Computation 179, 190–199 (2006)
Duan, Q., Zhang, H., Zhang, Y., et al.: Bounded property and point control of a bivariate rational interpolating surface. Computers and Mathematics with Applications 52, 975–984 (2006)
Wang, Q., Tan, J.: Rational quartic spline involving shape parameters. Journal of Information and Computational Science 1, 127–130 (2004)
Wang, Q., Tan, J.: Shape preserving piecewise rational biquartic surface. Journal of Information and Computational Science 3, 295–302 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Deng, S., Fang, K., Xie, J., Chen, F. (2008). Rational Biquartic Interpolating Surface Based on Function Values. In: Pan, Z., Zhang, X., El Rhalibi, A., Woo, W., Li, Y. (eds) Technologies for E-Learning and Digital Entertainment. Edutainment 2008. Lecture Notes in Computer Science, vol 5093. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69736-7_82
Download citation
DOI: https://doi.org/10.1007/978-3-540-69736-7_82
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69734-3
Online ISBN: 978-3-540-69736-7
eBook Packages: Computer ScienceComputer Science (R0)