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Construction of cubic 3D spline surfaces by Lagrange interpolation at selected points. (English) Zbl 1043.65029

Lyche, Tom (ed.) et al., Curve and surface design: Saint Malo 2002. Fifth international conference on curves and surfaces, Saint-Malo, France, June 27 – July 3, 2002. Proceedings. Brentwood, TN: Nashboro Press (ISBN 0-9728482-0-7/hbk). Modern Methods in Mathematics, 245-254 (2003).
Summary: We develop the first method for constructing cubic 3D spline surfaces by using Lagrange interpolation. Given a (triangular) 3D surface \(\Omega_{\mathbf f}\), we consider splines defined w.r.t. coarse triangulations which interpolate \(\Omega_{\mathbf f}\) at selected points. We first construct a continuous cubic 3D spline surface which interpolates the surface \(\Omega_{\mathbf f}\). Then by using a coloring of the triangles we modify the Bernstein-Bézier net of the spline such that certain smoothness conditions are satisfied and the approximation order is preserved.
In this paper we describe our algorithm, while the non-standard proof of optimal approximation order of the spline is given in the following paper [ibid. 255–264 (2003; reviewed below)]. A complicated example with a large number of data points shows that 3D spline surfaces of high accuracy are obtained.
For the entire collection see [Zbl 1023.00024].

MSC:

65D17 Computer-aided design (modeling of curves and surfaces)
65D07 Numerical computation using splines

Citations:

Zbl 1043.65030
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