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Scattered data interpolation and approximation using bivariate \(C^{1}\) piecewise cubic polynomials. (English) Zbl 0873.65011
Summary: We show that if the scattered data over a polygonal domain can be quadrangulated, then the space of bivariate \(C^{1}\) piecewise cubic polynomial functions on a triangulation obtained from the quadrangulation has the full approximation order. We point out that our method is more efficient than the Clough-Tocher scheme.

MSC:
65D17 Computer-aided design (modeling of curves and surfaces)
68U07 Computer science aspects of computer-aided design
65D07 Numerical computation using splines
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[1] Alfeld, P., A bivariate C2 clough-tocher scheme, Computer aided geometric design, 1, 257-267, (1984) · Zbl 0597.65005
[2] Alfeld, P.; Piper, B.; Schumaker, L.L., An explicit basis for C1 quartic bivariate splines, SIAM J. numer. anal., 24, 891-911, (1987) · Zbl 0658.65008
[3] Bramble, J.H.; Hilbert, S.R., Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. math., 16, 362-369, (1971) · Zbl 0214.41405
[4] Chui, C.K.; Lai, M.-J., On bivariate super vertex splines, Constr. approx., 6, 399-419, (1990) · Zbl 0726.41012
[5] Ciarlet, P.G., Sur l’élement de clough et tocher, RAIRO anal. numér., R2, 19-27, (1974) · Zbl 0306.65070
[6] de Boor, C., B-form basics, (), 131-148
[7] de Boor, C.; Höllig, K., Approximation order from bivariate C1-cubics: a counterexample, (), 649-655 · Zbl 0545.41017
[8] de Boor, C.; Höllig, K., Approximation power of smooth bivariate pp functions, Math. Z., 197, 343-363, (1988) · Zbl 0616.41010
[9] Farin, G., Triangular Bernstein-Bézier patches, Computer aided geometric design, 3, 83-127, (1986)
[10] Gao, J., A scheme of C2 interpolation over triangulations, (1993), manuscript
[11] Gmelig Meyling, R.H.J., Approximation by cubic C1-splines on arbitrary triangulation, Numer. math., 51, 65-85, (1987) · Zbl 0595.41010
[12] Gmelig Meyling, R.H.J.; Pfluger, P.R., Smooth interpolation to scattered data by bivariate piecewise polynomials of odd degree, Computer aided geometric design, 7, 439-458, (1990) · Zbl 0708.65012
[13] Grandine, T.A., An iterative method for computing multivariate C1 piecewise polynomial interpolants, Computer aided geometric design, 4, 307-320, (1987) · Zbl 0637.65008
[14] Lai, M.J., Approximation order from bivariate C1-cubics on a four-directional mesh is full, Computer aided geometric design, 11, 215-223, (1994) · Zbl 0792.41023
[15] Lai, M.J.; Schumaker, L.L., Scattered data interpolation using C2 supersplines of degree six, SIAM numer. anal., (1995), to appear
[16] Schumaker, L.L., Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky mountain J. math., 14, 251-264, (1984) · Zbl 0601.41034
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