Abstract
We analyse properties of the fundamental splines associated with local Lagrange interpolation (cf. Nürnberger and Zeilfelder [14-16]) for C1 splines on triangulations. These splines are zero except at one point of an interpolation set and form a basis of the spline space. It is proved that the support of the fundamental splines is small. Moreover, we show that the interpolation methods can be further simplified for cubic C1 splines on separable triangulations. In this case, the supports of the fundamental spline basis are even smaller. Finally, we describe an algorithm for modifying an arbitrary triangulation such that the resulting triangulation is separable.
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
C. de Boor: B-form basics, In: Geometric Modeling, G. Farin, (ed.), 131–148, SIAM, Philadelphia, 1987.
C. K. Chui: Multivariate Splines, CBMS 54, SIAM, Philadelphia, 1988.
R. W. Clough, J. L. Tocher:Finite element stiffness matrices for analysis of plates in bending, In: Proc. Conf. on Matrix Methods in Structural Mechanics, Wright Patterson A. F. B., Ohio, 1965.
O. Davydov, G. NÜrnberger, F. Zeilfelder:Bivariate spline interpolation with optimal approximation order, Constr. Approx. 17 (2001), 181–208.
O. Davydov, F. Zeilfelder: Scattered data fitting by direct extension of local polynomials with bivariate splines, to appear in: Advances in Comp. Math., 2003.
G. Farin: Triangular Bernstein-Bézier patches, Comp. Aided Geom. Design 3 (1986), 83–127.
G. Fraeijs de Veubeke: Bending and stretching of plates, In: Proc. Conf. on Matrix Methods in Structural Mechanics, Wright Patterson A. F. B., Ohio, 1965.
J. Haber, F. Zeilfelder, O. Davydov, H.-P. Seidel: Smooth approximation and rendering of large scattered data sets, In: Proceedings of IEEE Visualization 2001, T. Ertl, K. Joy, A. Varshney (eds.), 341–347, 571, IEEE, 2001.
G. Nürnberger: Approximation by Spline Functions, Springer, Berlin, 1989.
G. Nürnberger, L. L. Schumaker, F. Zeilfelder: Local Lagrange interpolation by bivariate C l cubic splines, In: Mathematical Methods in CAGD: Oslo 2000, T. Lyche, L.L. Schumaker, (eds.), 393-404, Vanderbilt University Press, Nashville, 2001.
G. Nürnberger, L. L. Schumaker, F. Zeilfelder: Lagrange Interpolation by C l cubic splines on Triangulations of Separable Quadrangulations, In: Approximation Theory X: Splines, Wavelets, and Applications, C. K. Chui, L.L. Schumaker, J. Stöckler, (eds.), 405-424, Vanderbilt University Press, Nashville, 2002.
G. Nürnberger, L. L. Schumaker, F. Zeilfelder: Lagrange interpolation by C l cubic splines on triangulated quadrangulations, to appear in: Advances in Comp. Math., 2003.
G. Nürnberger, F. Zeilfelder: Developments in bivariate spline interpolation, J. Comput. Appl. Math. 121 (2000), 125–152.
G. Nürnberger, F. Zeilfelder: Local Lagrange interpolation by cubic splines on a class of triangulations, In: Proc. Conf. Trends in Approximation Theory, K. Kopotun, T. Lyche, M. Neamtu, (eds.), 341–350, Vanderbilt University Press, Nashville, 2001.
G. Nürnberger, F. Zeilfelder: Local Lagrange interpolation on Powell-Sabin triangulations and terrain modelling, In: Recent Progress in Multivariate Approximation, W. Haußmann, K. Jet-ter, M. Reimer (eds.), 227–244, ISNM 137, Birkhäuser, Basel 2001.
G. Nürnberger, F. Zeilfelder: Lagrange interpolation by bivariate C 1 - splines with optimal approximation order, to appear in: Advances in Comp. Math., 2003.
G. Sander: Bornes supérieures et inférieures dans l’analyse matricielle des plaques en flexion-torsion, Bull. Soc. Royale Science Liége 33 (1964), 456–494.
F. Zeilfelder: Scattered data fitting with bivariate splines, In: Principles of Multiresolution in Geometric Modelling, M. Floater, A. Iske, E. Quak (eds.), 243–286, Springer, Berlin, 2002.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Basel AG
About this paper
Cite this paper
NÜrnberger, G., Zeilfelder, F. (2003). Fundamental Splines on Triangulations. In: Haussmann, W., Jetter, K., Reimer, M., Stöckler, J. (eds) Modern Developments in Multivariate Approximation. International Series of Numerical Mathematics, vol 145. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8067-1_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8067-1_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9427-2
Online ISBN: 978-3-0348-8067-1
eBook Packages: Springer Book Archive