Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Blossoming begets $ B$-spline bases built better by $ B$-patches

Authors: Wolfgang Dahmen, Charles A. Micchelli and Hans-Peter Seidel
Journal: Math. Comp. 59 (1992), 97-115
MSC: Primary 41A15; Secondary 41A63, 65D07
MathSciNet review: 1134724
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The concept of symmetric recursive algorithm leads to new, s-dimensional spline spaces. We present a general scheme for constructing a collection of multivariate B-splines with $ k - 1$ continuous derivatives whose linear span contains all polynomials of degree at most k. This scheme is different from the one developed earlier by Dahmen and Micchelli and, independently, by Höllig, which was based on combinatorial principles and the geometric interpretation of the B-spline. The new spline space introduced here seems to offer possibilities for economizing the computation for evaluating linear combinations of B-splines.

References [Enhancements On Off] (What's this?)

  • [1] W. Dahmen, Multivariate B-splines--recurrence relations and linear combinations of truncated powers, Multivariate Approximation Theory (W. Schempp and K. Zeller, eds.), Birkhäuser, Basel, 1979, pp. 64-82. MR 560664 (82c:65008)
  • [2] W. Dahmen and C A. Micchelli, Recent progress in multivariate splines, Approximation Theory IV (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), Academic Press, 1983, pp. 27-121. MR 754343 (85h:41013)
  • [3] -, On the linear independence of multivariate B-splines I. Triangulations of simploids, SIAM J. Numer. Anal. 19 (1982), 993-1012. MR 672573 (85c:41016a)
  • [4] K. Höllig, Multivariate splines, SIAM J. Numer. Appl. 19 (1982), 1013-1031. MR 672574 (84i:41013)
  • [5] C. A Micchelli, A constructive approach to Kergin interpolation in $ {{\mathbf{R}}^k}$: Multivariate Bsplines and Lagrange interpolation, Rocky Mountain J. Math. 10 (1980), 485-497. MR 590212 (84i:41002)
  • [6] L. Ramshaw, Béziers and B-splines as multiaffine maps, Theoretical Foundations of Computer Graphics and CAD, Springer, 1988, pp. 757-776. MR 944723 (89k:41014)
  • [7] H.-P. Seidel, A new multiaffine approach to B-splines, Comput. Aided Geom. Design 6 (1989), 23-32. MR 983469 (90b:41023)
  • [8] -, Symmetric recursive algorithms for surfaces: B-patches and the de Boor algorithm for polynomials over triangles, Constr. Approx. 7 (1991), 257-279. MR 1101066 (92c:41010)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 41A15, 41A63, 65D07

Retrieve articles in all journals with MSC: 41A15, 41A63, 65D07

Additional Information

Keywords: Symmetric recursive algorithms, polar forms, multivariate B-splines, approximation, stability
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society