zbMATH — the first resource for mathematics

Dimension of \(C^1\)-splines on type-6 tetrahedral partitions. (English) Zbl 1062.65014
Multivariate polynomial spline spaces are most important in the theory of approximation and its applications. Especially when the breakpoints/lines between the individual polynomial parts are not gridded, they are difficult to analyse, for instance with respect to their dimension. In many cases, even in just two dimensions, only upper and lower bounds are known so far (while in one dimension, the exact dimensions are known).
Here, three dimensional spline spaces are studied, for arbitrary polynomial degree. Formulae for the spaces’ dimension are found, as well as minimal determining sets (for particular cases). The partitions for the piecewise polynomials which are used are still regular (“type-6 tetrahedral partitions”) but they are not just an equispaced lattice.

65D07 Numerical computation using splines
65D17 Computer-aided design (modeling of curves and surfaces)
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
41A15 Spline approximation
Full Text: DOI
[1] Alfeld, P., A trivariate \(C^1\) clough – tocher interpolation scheme, Comput. aided geom. design, 1, 169-181, (1984) · Zbl 0566.65003
[2] Alfeld, P.; Piper, B.; Schumaker, L.L., An explicit basis for \(C^1\) quartic bivariate splines, SIAM J. numer. anal., 24, 891-911, (1987) · Zbl 0658.65008
[3] Alfeld, P.; Schumaker, L.L.; Sirvent, M., The dimension and existence of local bases for multivariate spline spaces, J. approx. theory, 70, 243-264, (1992) · Zbl 0761.41007
[4] Alfeld, P.; Schumaker, L.L.; Whiteley, W., The generic dimension of the space of \(C^1\) splines of degree \(d \geqslant 8\) on tetrahedral decompositions, SIAM J. numer. anal., 30, 889-920, (1993) · Zbl 0774.41012
[5] C.L. Bajaj, F. Bernardini, G. Xu, Automatic reconstruction of surfaces and scalar fields from 3D scans, in: AMS SIGGRAPH 1995, pp. 109-118.
[6] de Boor, C., A practical guide to splines, (1978), Springer New York · Zbl 0406.41003
[7] C. de Boor, B-form basics, in: G. Farin (Ed.), Geometric Modeling: Algorithms and New Trends, SIAM Publication, Philadelphia, pp. 131-148.
[8] H. Carr, T. Möller, J. Snoeyink, Simplicial subdivisions and sampling artifacts, in: Proceedings of IEEE Visualization 2001, pp. 99-106.
[9] C.K. Chui, Multivariate Splines, CBMS, SIAM, Philadelphia, 1988, 189p.
[10] Chui, C.K.; Wang, R.H., On smooth multivariate spline functions, Math. comp., 41, 131-142, (1983) · Zbl 0542.41008
[11] Davydov, O.; Zeilfelder, F., Scattered data Fitting by direct extension of local polynomials to bivariate splines, Adv. comput. math., 21, 3-4, 223-271, (2004) · Zbl 1065.41017
[12] Farin, G., Triangular bernstein-Bézier patches, Comput. aided geom. design, 3, 83-127, (1986)
[13] J. Haber, F. Zeilfelder, O. Davydov, H.-P. Seidel, Smooth approximation and rendering of large scattered data sets, in: T. Ertl, K. Joy, A. Varshney (Eds.), Proceedings of IEEE Visualization 2001, pp. 341-347, 571. · Zbl 1147.65010
[14] Hong, D., Spaces of bivariate spline functions over triangulation, Approx. theory appl., 7, 56-75, (1991) · Zbl 0756.41017
[15] Ibrahim, A.; Schumaker, L.L., Super spline spaces of smoothness r and degree d⩾3r+2, Constr. approx., 7, 401-423, (1991) · Zbl 0739.41011
[16] Lai, M.-J.; Le Méhauté, A., A new kind of trivariate \(C^1\) spline, preprint, Adv. comput. math., 21, 3-4, 273-292, (2004)
[17] G. Nürnberger, Approximation by Spline Functions, Springer, Berlin, 1989, 243p. · Zbl 0692.41017
[18] G. Nürnberger, C. Rössl, H.-P. Seidel, F. Zeilfelder, Quasi-interpolation by quadratic piecewise polynomials in three variables, J. Comput. Aided Geom. Design, to appear.
[19] Nürnberger, G.; Schumaker, L.L.; Zeilfelder, F., Lagrange interpolation by \(C^1\) cubic splines on triangulated quadrangulations, Adv. comput. math., 21, 3-4, 357-380, (2004) · Zbl 1053.41014
[20] Nürnberger, G.; Zeilfelder, F., Developments in bivariate spline interpolation, J. comput. appl. math., 121, 125-152, (2000) · Zbl 0960.41006
[21] Nürnberger, G.; Zeilfelder, F., Lagrange interpolation by bivariate \(C^1\)-splines with optimal approximation order, Adv. comput. math., 21, 3-4, 381-419, (2004) · Zbl 1064.41005
[22] H. Prautzsch, W. Boehm, M. Paluszny, Bézier and B-Spline Techniques, Springer, Berlin, 2002, 304p.
[23] C. Rössl, F. Zeilfelder, G. Nürnberger, H.-P. Seidel, Reconstruction of volume data with quadratic super splines, in: J.J. van Wijk, R.J. Moorhead, G. Turk (Eds.), IEEE Transactions on Visualization and Computer Graphics, Vol. 10(4), 2004, pp. 397-409.
[24] L.L. Schumaker, Spline Functions: Basic Theory, Wiley, New York, 1980, 553p. · Zbl 1123.41008
[25] Schumaker, L.L., On the dimension of piecewise polynomials in two variables, (), 396-412
[26] Schumaker, L.L., Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky mountain J. math., 14, 251-264, (1984) · Zbl 0601.41034
[27] Schumaker, L.L., Dual bases for spline spaces on cells, Computer aided geom. design, 5, 277-284, (1988) · Zbl 0652.41012
[28] Schumaker, L.L.; Sorokina, T., Quintic spline interpolation on type-4 tetrahedral partitions, Adv. comput. math., 21, 3-4, 421-444, (2004) · Zbl 1053.41015
[29] L.L. Schumaker, T. Sorokina, A trivariate box macro element, Constr. Approx., preprint. · Zbl 1077.41009
[30] T. Sorokina, F. Zeilfelder, Trivariate spline operators, private communication, 2004.
[31] Worsey, A.; Farin, G., An n-dimensional clough – tocher interpolant, Constr. approx., 3, 2, 99-110, (1987) · Zbl 0631.41003
[32] F. Zeilfelder, H.-P. Seidel, Splines over triangulations, in: G. Farin, J. Hoschek, M.-S. Kim (Eds.), Handbook on Computer Aided Design, pp. 701-722.
[33] Ženišek, A., Polynomial approximation on tetrahedrons in the finite element method, J. approx. theory, 7, 334-351, (1973) · Zbl 0279.41005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.