Abstract
We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r≥1 and polynomial degree 3r−1. They are defined on a triangulation with Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set of points. We further consider a number of CAGD applications. We show how to define control points and control polynomials (of degree 2r−1), and we provide an efficient and stable computation of the Bernstein–Bézier form of such splines.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
de Boor, C.: B-form basics. In: Farin, G. (ed.) Geometric Modeling: Algorithms and New Trends, pp. 131–148. SIAM, Philadelphia (1987)
de Boor, C.: Multivariate piecewise polynomials. Acta Numer. 2, 65–109 (1993)
Dierckx, P.: On calculating normalized Powell–Sabin B-splines. Comput. Aided Geom. Des. 15, 61–78 (1997)
Farin, G.: Triangular Bernstein–Bézier patches. Comput. Aided Geom. Des. 3, 83–127 (1986)
Lai, M., Schumaker, L.: Spline Functions on Triangulations. Encyclopedia of Mathematics and Its Applications, vol. 110. Cambridge University Press, Cambridge (2007)
Maes, J., Bultheel, A.: Stable multiresolution analysis on triangles for surface compression. Electron. Trans. Numer. Anal. 25, 224–258 (2006)
Manni, C., Sablonnière, P.: Quadratic spline quasi-interpolants on Powell–Sabin partitions. Adv. Comput. Math. 26, 283–304 (2007)
Powell, M., Sabin, M.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3, 316–325 (1977)
Ramshaw, L.: Blossoming: a connect-the-dots approach to splines. Tech. Rep. 19, Digital Systems Research Center (1987)
Sablonnière, P.: Composite finite elements of class C k. J. Comput. Appl. Math. 12&13, 541–550 (1985)
Sablonnière, P.: Error bounds for Hermite interpolation by quadratic splines on an α-triangulation. IMA J. Numer. Anal. 7, 495–508 (1987)
Seidel, H.: An introduction to polar forms. IEEE Comput. Graph. Appl. 13, 38–46 (1993)
Speleers, H.: A normalized basis for quintic Powell–Sabin splines. Comput. Aided Geom. Des. 27, 438–457 (2010)
Speleers, H.: A normalized basis for reduced Clough–Tocher splines. Comput. Aided Geom. Des. 27, 700–712 (2010)
Speleers, H.: On multivariate polynomials in Bernstein–Bézier form and tensor algebra. J. Comput. Appl. Math. 236, 589–599 (2011)
Speleers, H., Dierckx, P., Vandewalle, S.: Numerical solution of partial differential equations with Powell–Sabin splines. J. Comput. Appl. Math. 189, 643–659 (2006)
Speleers, H., Dierckx, P., Vandewalle, S.: Quasi-hierarchical Powell–Sabin B-splines. Comput. Aided Geom. Des. 26, 174–191 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Larry Schumaker.
H. Speleers is a Postdoctoral Fellow of the Research Foundation Flanders (Belgium).
Rights and permissions
About this article
Cite this article
Speleers, H. Construction of Normalized B-Splines for a Family of Smooth Spline Spaces Over Powell–Sabin Triangulations. Constr Approx 37, 41–72 (2013). https://doi.org/10.1007/s00365-011-9151-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00365-011-9151-x
Keywords
- Smooth Powell–Sabin splines
- Normalized B-splines
- Macro-elements
- Control points
- Control polynomials
- Bernstein–Bézier form