Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines

Authors: Elaine Cohen, Tom Lyche and Richard F. Riesenfeld
Journal: Math. Comp. 82 (2013), 1667-1707
MSC (2010): Primary 41A15, 65D07, 65D17, 65D05
Published electronically: February 14, 2013
MathSciNet review: 3042581
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a simplex spline basis for a space of $ C^1$-quadratics on the well-known Powell-Sabin 12-split triangular region. Among its many important desirable properties, we show that it has an associated recurrence relation for evaluation and differentiation. Also developed are a Marsden-like identity, quasi-interpolants, approximation methods exhibiting unconditional stability, a subdivision scheme, and smoothness conditions across macro-element edges.

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

  • 1. P. Alfeld, and L. Schumaker, The dimension of bivariate spline spaces of smoothness $ r$ for degree $ d\ge 4r+1$, Constr. Approx. 3 (1987), 891-911. MR 889554 (88e:41025)
  • 2. C. de Boor, K. Höllig, and S. Riemenschneider, Box Splines, Springer-Verlag, New York, 1993. MR 1243635 (94k:65004)
  • 3. S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 2nd ed., Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
  • 4. E. Cohen, T. Lyche, and R. F. Riesenfeld, Discrete B-splines and subdivision techniques in computer aided geometric design and computer graphics, Computer Graphics and Image Processing 14 (1980), 87-111.
  • 5. W. Dahmen, On multivariate B-splines, SIAM J. Numer. Anal. 17 (1980), 179-191. MR 567267 (81c:41020)
  • 6. W. Dahmen and C. A. Micchelli, On the linear independence of multivariate B-splines I, Triangulations of simploids, SIAM J. Numer. Anal. 19 (1982), 993-1012. MR 672573 (85c:41016a)
  • 7. W. Dahmen, C. A. Micchelli, and H-P Seidel, Blossoming begets B-splines built better by B-patches, Math. Comp 59 (1992), 97-115. MR 1134724 (93b:41014)
  • 8. P. Dierckx, On calculating normalized Powell-Sabin B-splines, Comput. Aided Geom. Design, 15 (1997), 61-78. MR 1484258 (98j:41004)
  • 9. N. Dyn and T. Lyche, A Hermite subdivision scheme for the evaluation of the Powell-Sabin 12-split element, in Approximation Theory IX, Volume 2, Charles Chui and Larry Schumaker (eds.), Vanderbilt University Press, Nashville, 1998, 33-38. MR 1743030
  • 10. T. A. Grandine, The computational cost of simplex spline functions, SIAM J. Numer. Anal. 24 (1987), 887-890. MR 899710 (88j:41032)
  • 11. T. A. Grandine, The stable evaluation of multivariate simplex splines, Math. Comp. 50 (1988), 197-205. MR 917827 (89a:65018)
  • 12. K. Höllig, Multivariate splines, SIAM J. Numer. Anal. 19 (1982), 1013-1031. MR 672574 (84i:41013)
  • 13. Ming-Jun Lai and Larry L. Schumaker, Spline Functions on Triangulations, Cambridge University Press, Cambridge, 2007. MR 2355272 (2008i:41001)
  • 14. J. M. Lane and R. F. Riesenfeld, A theoretical development for the computer generation and display of piecewise polynomial surfaces, IEEE Trans. Pattern Anal. Mach. Intellig 2 (1980), 35-46.
  • 15. T. Lyche and K. Scherer, On the $ p$-norm condition number of the multivariate triangular Bernstein basis, J. Comput. Appl. Math. 119 (2000), 259-273. MR 1774222 (2001h:41009)
  • 16. J. Maes, E. Vanraes, P. Dierckx, and A. Bultheel, On the stability of normalized Powell-Sabin B-splines, J. Comput. Appl. Math. 170 (2004), 181-196. MR 2075830 (2005f:41029)
  • 17. C.A. Micchelli, On a numerically efficient method for computing multivariate B-splines, in Multivariate Approximation Theory, (W. Schempp and K. Zeller, eds.), Birkhäuser Verlag, Basel, 1979, 211-248. MR 560673 (81g:65017)
  • 18. M. Neamtu, What is the natural generalization of univariate splines to higher dimensions?, in Mathematical Methods in CAGD: Oslo 2000, Tom Lyche and Larry L. Schumaker (eds.), Vanderbilt University Press, Nashville, 2001, 355-392. MR 1858970
  • 19. M. Neamtu, Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay, Trans. Amer. Math. Soc. 359 (2007), 2993-3004. MR 2299443 (2008c:65042)
  • 20. M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximation on triangles, ACM Trans. Math. Software 3 (1977), 316-325. MR 0483304 (58:3319)
  • 21. L. L. Schumaker and T. Sorokina, Smooth macro-elements on Powell-Sabin-12 splits, Math. Comp. 75 (2006), 711-726. MR 2196988 (2006m:65284)
  • 22. H-P Seidel, Polar forms and triangular B-spline surfaces, in Blossoming: The New Polar Form Approach to Spline Curves and Surfaces, Siggraph'91 Course Notes #26, World Scientific Publishing Company, 1992, 235-286. MR 1239195
  • 23. H. Speleers, P. Dierckx, and S. Vandewalle, On the Lp-stability of quasi-hierarchical Powell-Sabin B-splines, In Approximation Theory XII: San Antonio 2007, M. Neamtu and L.L. Schumaker (eds.), Nashboro Press, 2008, 398-413. MR 2544025 (2010i:65025)
  • 24. H. Speleers, P. Dierckx, and S. Vandewalle, On the local approximation power of quasi-hierarchical Powell-Sabin splines, In Mathematical Methods for Curves and Surfaces,
    M. Dæhlen, M. S. Floater, T. Lyche, J-L. Merrien, K. Mørken and L. L. Schumaker (eds.), Lecture Notes in Computer Science LNCS 5862, 2010, 419-433.
  • 25. H. Speleers, A normalized basis for quintic Powell-Sabin splines, Comput. Aided Geom. Design, 27 (2010), 438-457. MR 2657545 (2011j:65034)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 41A15, 65D07, 65D17, 65D05

Retrieve articles in all journals with MSC (2010): 41A15, 65D07, 65D17, 65D05

Additional Information

Elaine Cohen
Affiliation: School of Computing, University of Utah, Salt Lake City, Utah 84112

Tom Lyche
Affiliation: CMA and Institute of Informatics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway

Richard F. Riesenfeld
Affiliation: School of Computing, University of Utah, Salt Lake City, Utah 84112

Received by editor(s): October 25, 2010
Received by editor(s) in revised form: November 3, 2011
Published electronically: February 14, 2013
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society