Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

Request Permissions   Purchase Content 


On the dimension of spline spaces on planar T-meshes

Author: Bernard Mourrain
Journal: Math. Comp. 83 (2014), 847-871
MSC (2010): Primary 14Q20, 14Q99, 13P25; Secondary 68W30, 65D17, 65D07
Published electronically: July 12, 2013
MathSciNet review: 3143695
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We analyze the space $ \mathcal {S}_{m, m'}^{\mathbf {r}} (\mathcal {T})$ of bivariate functions that are piecewise polynomial of bi-degree $ \leqslant (m, m')$ and of smoothness $ \mathbf {r}$ along the interior edges of a planar T-mesh $ \mathcal {T}$. We give new combinatorial lower and upper bounds for the dimension of this space by exploiting homological techniques. We relate this dimension to the weight of the maximal interior segments of the T-mesh, defined for an ordering of these maximal interior segments. We show that the lower and upper bounds coincide, for high enough degrees or for hierarchical T-meshes which are regular enough. We give a rule of subdivision to construct hierarchical T-meshes for which these lower and upper bounds coincide. Finally, we illustrate these results by analyzing spline spaces of small degrees and smoothness.

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

  • [1] Dmitry Berdinsky, Min-jae Oh, Tae-wan Kim, and Bernard Mourrain.
    On the problem of instability in the dimension of a spline space over a T-mesh.
    Comput. Graph., 36(5):507-513, 2012.
  • [2] Louis J. Billera, Homology of smooth splines: generic triangulations and a conjecture of Strang, Trans. Amer. Math. Soc. 310 (1988), no. 1, 325-340. MR 965757 (89k:41010),
  • [3] A. Buffa, D. Cho, and G. Sangalli, Linear independence of the T-spline blending functions associated with some particular T-meshes, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 23-24, 1437-1445. MR 2630153 (2011d:65042),
  • [4] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR 1731415 (2000h:18022)
  • [5] Carl de Boor, A practical guide to splines, Revised edition, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York, 2001. MR 1900298 (2003f:41001)
  • [6] Jiansong Deng, Falai Chen, and Yuyu Feng, Dimensions of spline spaces over $ T$-meshes, J. Comput. Appl. Math. 194 (2006), no. 2, 267-283. MR 2239393 (2007a:41017),
  • [7] Jiansong Deng, Falai Chen, and Liangbing Jin, Dimensions of biquadratic spline spaces over T-meshes, J. Comput. Appl. Math. 238 (2013), 68-94. MR 2972590,
  • [8] Jiansong Deng, Falai Chen, Xin Li, Changqi Hu, Weihua Tong, Zhouwang Yang, and Yuyu Feng.
    Polynomial splines over hierarchical T-meshes.
    Graph. Models, 70(4):76-86, 2008.
  • [9] Tor Dokken, Tom Lyche, and Kjell-Fredrik Pettersen.
    Polynomial splines over locally refined box-partitions,
    Comput. Aided Geom. Design, 30(3):331-356, 2013. MR 3019748
  • [10] Richard Ehrenborg and Gian-Carlo Rota, Apolarity and canonical forms for homogeneous polynomials, European J. Combin. 14 (1993), no. 3, 157-181. MR 1215329 (94e:15062),
  • [11] Gerald Farin. Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, 5th Edition. Morgan Kaufmann, San Mateo, CA, 2001.
  • [12] Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
  • [13] Zhang-jin Huang, Jian-song Deng, Yu-yu Feng, and Fa-lai Chen, New proof of dimension formula of spline spaces over T-meshes via smoothing cofactors, J. Comput. Math. 24 (2006), no. 4, 501-514. MR 2243118 (2007a:41019)
  • [14] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 39-41, 4135-4195. MR 2152382 (2006a:65018),
  • [15] Joseph P. S. Kung and Gian-Carlo Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 1, 27-85. MR 722856 (85g:05002),
  • [16] Chong-Jun Li, Ren-Hong Wang, and Feng Zhang.
    Improvement on the dimensions of spline spaces on T-mesh.
    Journal of Information & Computational Science, 3(2):235-244, 2006.
  • [17] Xin Li and Falai Chen, On the instability in the dimension of splines spaces over T-meshes, Comput. Aided Geom. Design 28 (2011), no. 7, 420-426. MR 2836487 (2012m:65035),
  • [18] G. Salmon.
    Modern Higher Algebra.
    G.E. Stechert and Co., New York, 1885.
    Reprinted 1924.
  • [19] Hal Schenck and Mike Stillman, Local cohomology of bivariate splines, J. Pure Appl. Algebra 117/118 (1997), 535-548. Algorithms for algebra (Eindhoven, 1996). MR 1457854 (99d:13011),
  • [20] Thomas W. Sederberg, David L. Cardon, G. Thomas Finnigan, Nicholas S. North, Jianmin Zheng, and Tom Lyche.
    T-spline simplification and local refinement.
    ACM Trans. Graph., 23(3):276-283, 2004.
  • [21] Thomas W. Sederberg, Jianmin Zheng, Almaz Bakenov, and Ahmad Nasri.
    T-splines and T-nurccs.
    ACM Trans. Graph., 22(3):477-484, 2003.
  • [22] Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York, 1966. MR 0210112 (35 #1007)
  • [23] Ren-Hong Wang, Multivariate spline functions and their applications, Mathematics and its Applications, vol. 529, Kluwer Academic Publishers, Dordrecht, 2001. Translated from the 1994 Chinese original by Shao-Ming Wang and revised by the author. MR 1891792 (2003f:41002)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 14Q20, 14Q99, 13P25, 68W30, 65D17, 65D07

Retrieve articles in all journals with MSC (2010): 14Q20, 14Q99, 13P25, 68W30, 65D17, 65D07

Additional Information

Bernard Mourrain
Affiliation: Galaad, Inria Méditerranée, 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis, France

Received by editor(s): May 26, 2013
Received by editor(s) in revised form: December 23, 2011, and July 9, 2012
Published electronically: July 12, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society