Abstract
In applications such as Computer Aided Geometric Design it is useful to have operators which approximate a given bivariate function f by a spline-function S which in some sense preserves the shape of f, i. e. the shape of the surface represented by f. It seems reasonable to require that if f is monotone in a given direction or is convex, then so is S. We would also like in general that S is no more ‘bumpy’ than is f.
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, K. Höllig, B-splines from parallelipipeds, J. d’Analyse Math.. 42 (1982/3), 99–115.
W. Boehm, Triangular spline algorithms, to appear in Computer Aided Geometric Design.
G. Chang, P.J. Davis, The convexity of Bernstein polynomials over triangles, J. Approx. Theory 40 (1984), 11–28
G. Chang, J. Hoscheck, these proceedings
E. Cohen, T. Lyche, R. Riesenfeld, Discrete box splines and refinement algorithms, Computer Aided Geometric Design 1 (1984), 131–148
W. Dahmen, N. Dyn, D. Levin, On the convergence rates of subdivision algorithms for box spline surfaces, to appear in Constructive App roximation
W. Dahmen, C.A. Micchelli, Recent progress in multivariate splines, in: Approximation Theory IV, ed. C.K. Chui, L.L. Schumaker, J. Ward, Academic Press, New York (1983), 27–121
W. Dahmen, C.A. Micchelli, Subdivision algorithms for the generation of box spline surfaces, to appear in Computer Aided Geometric Design
P.O. Frederickson, Generalised triangular splines, Mathematics Report 7–71, Lakehead University (1971)
T.N.T. Goodman, Variation diminishing properties of Bernstein polynomials on triangles, to appear in J. Approximation Theory
H. Prautzsch, Unterteilungsalgorithmen fur multivariate Splines - ein geometrischer Zugang, Ph.D. Thesis, Technische Universitat Braunschweig, 1984
M.A. Sabin, The use of piecewise forms for the numerical representation of shape, Ph.D. Dissertation, Hungar. Acad. of Science, Budapest (1977)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Goodman, T.N.T. (1985). Shape Preserving Approximation by Polyhedral Splines. In: Schempp, W., Zeller, K. (eds) Multivariate Approximation Theory III. International Series of Numerical Mathematics, vol 75. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9321-3_19
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9321-3_19
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9995-6
Online ISBN: 978-3-0348-9321-3
eBook Packages: Springer Book Archive