zbMATH — the first resource for mathematics

Smooth interpolation to scattered data by bivariate piecewise polynomials of odd degree. (English) Zbl 0708.65012
An algorithm for constructing bivariate spline interpolants to scattered data is developed. Given scattered data points \(x^ 1,...,x^ n\) in \({\mathbb{R}}^ 2\) and a corresponding triangulation, the authors discuss the dimension of spaces of splines of degree k and smoothness m with respect to this triangulation, and the description of the smoothness properties with the aid of Bézier ordinates.
In the first step of the algorithm, a continuous spline p of degree k is constructed which interpolates given data at \(x^ 1,...,x^ n\) and satisfies certain differentiability conditions at these points. In a second step, the interpolant p is perturbed such that the resulting interpolating spline s is m-times continuously differentiable. This approach guarantees that s is as close as possible to p and that polynomials of degree m are reproduced. For a given smoothness m of the spline interpolants, the method keeps their degree k as low as possible.
A similar version was already given by T. A. Grandine [Comput. Aided Geom. Des. 4, 307-319 (1987; Zbl 0637.65008)] for the case of differentiable cubic splines. Finally, the authors describe several numerical examples for quintic splines of smoothness two.
Reviewer: G.Nürnberger

65D07 Numerical computation using splines
65D05 Numerical interpolation
41A15 Spline approximation
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
nag; NAG; symrcm
Full Text: DOI
[1] Alfeld, P., (), 1-23, June 25-28, 1985
[2] Alfeld, P., Derivative generation from multivariate scattered data by functional minimization, Computer aided geometric design, 2, 281-296, (1985) · Zbl 0585.65014
[3] Alfeld, P., (), 149-159
[4] Alfeld, P.; Barnhill, R.E., A transfinite C2 interpolant over triangles, Rocky mountain J. math., 14, 17-40, (1984) · Zbl 0558.65002
[5] Alfeld, P.; Piper, B.; Schumaker, L., An explicit basis for C1 quartic bivariate splines, SIAM J. numer. anal., 24, 891-911, (1987) · Zbl 0658.65008
[6] Alfeld, P.; Schumaker, L., The dimension of bivariate spline spaces of smoothness r for degree d ⩾ 4r + 1, Constr. approx., 3, 189-197, (1987) · Zbl 0646.41008
[7] Barnhill, R.E.; Farin, G., C1 quintic interpolation over triangles: two explicit representations, Int. J. num. methods in engineering, 17, 1763-1778, (1981) · Zbl 0477.65009
[8] de Boor, C., (), 131-148
[9] Chui, C.K.; Lai, M.J., (), 84-115
[10] Dahmen, W., Bernstein-Bézier representation of polynomial surfaces, Proc. ACM SIGGRAPH, (1986), Aug. 18-22, 1986, Dallas · Zbl 0606.41009
[11] Farin, G., Triangular bernstein-Bézier patches, Computer aided geometric design, 3, 83-127, (1986)
[12] Franke, R., Scattered data interpolation: tests of some methods, Math. comp., 38, 181-200, (1982) · Zbl 0476.65005
[13] George, A.; Liu, J.W., Computer solution of large sparse positive definite systems, (1981), Prentice Hall Englewood Cliffs, NJ · Zbl 0516.65010
[14] Gmelig Meyling, R.H.J., Approximation by piecewise cubic C1-splines on arbitrary triangulations, Numer. math., 51, 65-85, (1987) · Zbl 0595.41010
[15] Gmelig Meyling, R.H.J., On interpolation by bivariate quintic splines of class C2, Proc. conf. constructive theory of functions, 152-161, (1987), Varna, Bulgaria · Zbl 0617.41025
[16] Gmelig Meyling, R.H.J.; Pfluger, P.R., (), 180-190
[17] Golub, G.H.; Van Loan, C.F., Matrix computations, (1983), North Oxford Academic London · Zbl 0559.65011
[18] Grandine, T.A., An iterative method for computing multivariate C1 piecewise polynomial interpolants, Computer aided geometric design, 4, 307-320, (1987) · Zbl 0637.65008
[19] Lawson, C.L., (), 161-194
[20] Lawson, C.L., C1 surface interpolation for scattered data on a sphere, Rocky mountain J. math., 14, 177-202, (1984) · Zbl 0579.65008
[21] Morgan, J.; Scott, R., A nodal basis for C1-piecewise polynomials of degree n ⩾ 5, Math. comp., 29, 736-740, (1975) · Zbl 0307.65074
[22] Morgan, J.; Scott, R., The dimension of piecewise polynomials, Unpublished manuscript, (1977)
[23] Nadler, E., Hermite interpolation by C1 piecewise polynomials on triangulations, IBM report RC 12507, (1987), (#56249)
[24] FORTRAN library manual, Mark 11, (1984), Numerical Algorithms Group Oxford
[25] Reid, J.K., (), 231-254
[26] Rescorla, K.L., Cardinal interpolation: a bivariate polynomial example, Computer aided geometric design, 3, 313-321, (1986) · Zbl 0628.65006
[27] Schumaker, L.L., (), 151-197
[28] Schumaker, L.L., Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky mountain J. math., 14, 251-264, (1984) · Zbl 0601.41034
[29] Wang, R.H.; Chou, Y.S.; Su, L.Y., (), 71-83
[30] Whelan, T., A representation of a C2 interpolant over triangles, Computer aided geometric design, 3, 53-66, (1986) · Zbl 0626.65006
[31] Ẑenišek, A., Interpolation polynomials on the triangle, Numer. math., 15, 283-296, (1970) · Zbl 0216.38901
[32] Ẑenišek, A., Hermite interpolation on simplices in the finite element method, Proc. equadiff. III, 271-277, (1973), Brno
[33] Ẑenišek, A., Polynomial approximation on tetrahedrons in the finite element method, J. appr. 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.