Abstract
We study polyhedral splines, in particular simplex splines and box splines with their relationship to cone splines. From basic recurrence relations we derive the piecewise polynomial nature and continuity properties of such splines. These are then used to construct spaces of multivariate spline functions whose evaluation and properties are studied.
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References
Böhm, W. (1983) ‘Subdividing multivariate splines’, Computer-aided Design 15, 345–352.
de Boor, C. (1976) ‘Splines as linear combinations of B-splines’, a survey, in G. G. Lorentz, C. K. Chui and L. L. Schumaker (eds.), Approximation Theory II, Academic Press, New York, pp.1–47.
de Boor, C. and Höllig, K. (1982) ‘Recurrence relations for multivariate B-splines’, Proc. Amer. Math. Soc. 85, 397–400.
de Boor, C. and Höllig, K. (1982) ‘B-splines from parallelepipeds’, J. d’Analyse Math. 42, 99–115.
Chui, C. K. (1987) Lectures on Multivariate Splines, Theory and Applications, Centre for Approximation Theory, College Station.
Cohen, E., Lyche, T. and Riesenfeld, R. (1984) ‘Subdivision algorithms for the generation of box splines surfaces’, Computer Aided Geometric Design 2, 131–148.
Cohen, E., Lyche, T. and Riesenfeld, R. (1987) ‘Cones and recurrence relations for simplex splines’, Constructive Approximation 3, 131–141.
Curry, H. B. and Schoenberg, I. J. (1966) ‘Polya frequency functions IV. The fundamental spline functions and their limits’, J. d’Analyse Math. 17, 71–107.
Dahmen, W. (1980) ‘On multivariate B-splines’, SIAM J. Numer. Anal. 17, 179–190.
Dahmem, W., Dyn, N. and Levin D. (1985) ‘On the convergence rates of subdivision algorithms for box-spline surfaces’, Constructive Approximation 1, 305–322.
Dahmen, W. and Micchelli, C. A. (1982) ‘On the linear independence of multivariate B-splines I. Triangulations of simploids’, SIAM J. Numer. Anal. 19, 992–1012.
Dahmen, W. and Micchelli, C. A. (1983) ‘On the linear independence of multivariate B-splines II. Complete configurations’, Math. Comp. 41, 141–164.
Dahmen, W. and Micchelli, C. A. (1983) ‘Translates of multivariate splines’, Linear Algebra Appl. 52, 217–234.
Dahmen, W. and Micchelli, C. A. (1983) ‘Recent progress in multivariate splines’, in C. K. Chui, L. L. Schumaker and J. D. Ward (eds.), Approximation Theory IV, Academic Press, New York, pp. 27–121.
Dahmen, W. and Micchelli, C. A. (1983) ‘Multivariate splines — A new constructive approach’, in R. Barnhill and W. Boehm (eds.), Surfaces in Computer Aided Design, North Holland, Amsterdam, pp. 191–215.
Dahmen, W. and Micchelli, C. A. (1984) ‘Subdivision algorithm for the generation of box-splines surfaces’, Computer Aided Geometric Design 1, 115–129.
Dahmen, W. and Micchelli, C. A. (1988) ‘Convexity of multivariate Bernstein polynomials and box spline surfaces’, Studio Scientiarum Math. Hungaria 23, 265–287.
Frederickson, P. O. (1971), ‘Generalised triangular splines’, Lakehead University, Report 7–71.
Gmelig Meyling, R. H. J. (1985) ‘Least squares approximation by linear combinations of bivariate B-splines’, Mathematisch Institut, Universiteit van Amsterdam, Report 85–23.
Goodman, T. N. T. (1985) ‘Shape preserving approximation by polyhedral splines’, in W. Schempp and K. Zeller (eds.), Multivariate Approximation Theory III, Birkhanser, Basel, 198–205.
Goodman, T. N. T. and Lee, S. L. (1981) ‘Spline approximation operators of Bernstein — Schoenberg type in one and two variables’, J. Approx. Theory 33, 248–263.
Grandine, T. A. (1985), ‘Computing with multivariate simplex splines’, University of Wisconsin, Madison, Ph. D. thesis.
Ha, K. V. (1988) ‘On multivariate simplex B-splines’, Institute for Informatics, University of Oslo, Ph.D. thesis.
Höllig, K. (1982) ‘Multivariate splines’, SIAM J. Numer. Anal. 19, 1013–1031.
Höllig, K. (1986) ‘Box splines’, in C. K. Chui, L. L. Schumaker and J. D. Ward (eds.), Approximation Theory V, Academic Press, New York, pp. 71–95.
Hollig, K. (1989) ‘Box-splines surfaces’, in T. Lyche and L. L. Schumaker (eds.), Mathematical Methods in Computer Aided Geometric Design, Academic Press, Boston.
Jia, R. Q. (1985) ‘Local linear independence of the translates of a box spline’, Constructive Approximation 1, 175–182.
Marsden, M. J. (1970) ‘An identity for spline functions with applications to variation-diminishing spline approximation’, J. Approx. Theory 3, 7–49.
Micchelli, C. A. (1979) ‘On a numerically efficient method for computing multivariate B-splines’, in W. Schempp, K. Zeller (eds.), Multivariate Approximation Theory, Birkhauser, Basel, pp.211–248.
Prautzsch, H. (1984) ‘Unterteilungsalgorithmen fur multivariate splines’, University Braunschweig, Dissertation.
Traas, C. R. (1989) ‘Bivariate simplicial splines in finite element computations’, to appear.
Zwart, P. B. (1973) ‘Multivariate splines with non-degenerate partitions’, SIAM J. Numer. Anal. 10, 665–673.
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© 1990 Kluwer Academic Publishers
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Goodman, T.N.T. (1990). Polyhedral Splines. In: Dahmen, W., Gasca, M., Micchelli, C.A. (eds) Computation of Curves and Surfaces. NATO ASI Series, vol 307. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2017-0_11
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DOI: https://doi.org/10.1007/978-94-009-2017-0_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7404-9
Online ISBN: 978-94-009-2017-0
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