C 1 Quintic Spline Interpolation over Tetrahedral Partitions

We use C 1 quintic superspline functions to interpolate any given scattered data. The space of C 1 quintic superspline functions is introduced in Lai and LeMehaute'99] and is an improvement of the Alfeld scheme of 3D scattered data interpolation. We have implemented the spline space in MATLAB and tested for the accuracy of reproduction of all quintic polynomials. We present some numerical evidences that when the partition is reened, the spline interpolant converges to the function to be approximated. x1. Introduction There are a few trivariate splines spaces available for interpolation over a tetrahedral partition 4 of a polygonal domain in R 3. We would like to mention a direct polynomial interpolation by Zenisek in Zenisek'73]. This scheme requires piecewise polynomials of degree 9 which is globally C 1 over while C 4 around the vertices and C 2 around the edges of 4. Another scheme is the Alfeld scheme (cf. Alfeld'84]) which uses polynomials of degree 5 to construct spline functions over a 3D Clough-Tocher reenement of tetrahedral partition 4. The scheme produces spline interpolants which are globally C 1 over while C 2 around the vertices and C 1 around the edges of 4. A further generalization of the Clough-Tocher reenement enables Worsey and Farin in Worsey and Farin'87] to construct interpolation by C 1 cubic splines. Worsey and Piper constructed C 1 quadratic spline functions based on special tetrahe-dral partition in Worsey and Piper'88]. The present paper is concerned with the implementation of a new kind of trivariate splines introduced in Lai and LeMehaute'99]. The new scheme is like the Alfeld scheme. That is, it uses polynomi-als of degree 5 over the Alfeld version of Clough-Tocher reenement of 4 to construct spline interpolants globally C 1 over and locally C 2 around the vertices and C 1 around the edges of 4. The main diierence is that the Alfeld ISBN 0-8265-xxxx-x. All rights of reproduction in any form reserved.