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

Cohen, Elaine; Lyche, Tom; Riesenfeld, Richard

doi:10.1090/S0025-5718-2013-02664-6

A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines
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by Elaine Cohen, Tom Lyche and Richard F. Riesenfeld

Math. Comp. 82 (2013), 1667-1707

DOI: https://doi.org/10.1090/S0025-5718-2013-02664-6

Published electronically: February 14, 2013

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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

Peter Alfeld and L. L. Schumaker, The dimension of bivariate spline spaces of smoothness $r$ for degree $d\geq 4r+1$, Constr. Approx. 3 (1987), no. 2, 189–197. MR 889554, DOI 10.1007/BF01890563
C. de Boor, K. Höllig, and S. Riemenschneider, Box splines, Applied Mathematical Sciences, vol. 98, Springer-Verlag, New York, 1993. MR 1243635, DOI 10.1007/978-1-4757-2244-4
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
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.
Wolfgang Dahmen, On multivariate $B$-splines, SIAM J. Numer. Anal. 17 (1980), no. 2, 179–191. MR 567267, DOI 10.1137/0717017
Wolfgang A. Dahmen and Charles A. Micchelli, On the linear independence of multivariate $B$-splines. I. Triangulations of simploids, SIAM J. Numer. Anal. 19 (1982), no. 5, 993–1012. MR 672573, DOI 10.1137/0719072
Wolfgang Dahmen, Charles A. Micchelli, and Hans-Peter Seidel, Blossoming begets $B$-spline bases built better by $B$-patches, Math. Comp. 59 (1992), no. 199, 97–115. MR 1134724, DOI 10.1090/S0025-5718-1992-1134724-1
Paul Dierckx, On calculating normalized Powell-Sabin $B$-splines, Comput. Aided Geom. Design 15 (1997), no. 1, 61–78. MR 1484258, DOI 10.1016/S0167-8396(97)81785-2
Nira Dyn and Tom Lyche, A Hermite subdivision scheme for the evaluation of the Powell-Sabin $12$-split element, Approximation theory IX, Vol. 2 (Nashville, TN, 1998) Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 33–38. MR 1743030
Thomas A. Grandine, The computational cost of simplex spline functions, SIAM J. Numer. Anal. 24 (1987), no. 4, 887–890. MR 899710, DOI 10.1137/0724057
Thomas A. Grandine, The stable evaluation of multivariate simplex splines, Math. Comp. 50 (1988), no. 181, 197–205. MR 917827, DOI 10.1090/S0025-5718-1988-0917827-2
Klaus Höllig, Multivariate splines, SIAM J. Numer. Anal. 19 (1982), no. 5, 1013–1031. MR 672574, DOI 10.1137/0719073
Ming-Jun Lai and Larry L. Schumaker, Spline functions on triangulations, Encyclopedia of Mathematics and its Applications, vol. 110, Cambridge University Press, Cambridge, 2007. MR 2355272, DOI 10.1017/CBO9780511721588
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.
Tom Lyche and Karl Scherer, On the $p$-norm condition number of the multivariate triangular Bernstein basis, J. Comput. Appl. Math. 119 (2000), no. 1-2, 259–273. Dedicated to Professor Larry L. Schumaker on the occasion of his 60th birthday. MR 1774222, DOI 10.1016/S0377-0427(00)00383-6
Jan Maes, Evelyne Vanraes, Paul Dierckx, and Adhemar Bultheel, On the stability of normalized Powell-Sabin B-splines, J. Comput. Appl. Math. 170 (2004), no. 1, 181–196. MR 2075830, DOI 10.1016/j.cam.2004.01.001
Charles A. Micchelli, On a numerically efficient method for computing multivariate $B$-splines, Multivariate approximation theory (Proc. Conf., Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 51, Birkhäuser Verlag, Basel-Boston, Mass., 1979, pp. 211–248. MR 560673
Marian Neamtu, What is the natural generalization of univariate splines to higher dimensions?, Mathematical methods for curves and surfaces (Oslo, 2000) Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2001, pp. 355–392. MR 1858970
Marian Neamtu, Delaunay configurations and multivariate splines: a generalization of a result of B. N. Delaunay, Trans. Amer. Math. Soc. 359 (2007), no. 7, 2993–3004. MR 2299443, DOI 10.1090/S0002-9947-07-03976-1
M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977), no. 4, 316–325. MR 483304, DOI 10.1145/355759.355761
Larry L. Schumaker and Tatyana Sorokina, Smooth macro-elements on Powell-Sabin-12 splits, Math. Comp. 75 (2006), no. 254, 711–726. MR 2196988, DOI 10.1090/S0025-5718-05-01813-2
Hans-Peter Seidel, Polar forms and triangular $B$-spline surfaces, Computing in Euclidean geometry, Lecture Notes Ser. Comput., vol. 1, World Sci. Publ., River Edge, NJ, 1992, pp. 235–286. MR 1239195, DOI 10.1142/9789814355858_{0}007
Hendrik Speleers, Paul Dierckx, and Stefan Vandewalle, On the $L_p$-stability of quasi-hierarchical Powell-Sabin B-splines, Approximation theory XII: San Antonio 2007, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2008, pp. 398–413. MR 2544025
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.
Hendrik Speleers, A normalized basis for quintic Powell-Sabin splines, Comput. Aided Geom. Design 27 (2010), no. 6, 438–457. MR 2657545, DOI 10.1016/j.cagd.2010.05.001