Tamtam Proceedings - lamsin

Smoothing and compressing surfaces 3593. ConclusionStarting from the results given in [2] and [3], we have also proposed in [4] a methodwhich allows to compute recursively the tensor product Hermite interpolant f k,l . Thismethod is similar to the one studied in this paper, but it seems that the computationalcost for evaluating f k,l with the later one is better. Indeed, it is easy to verify that thenumber of coefficients needed for computing the detailed functions in the decompositionof f k,l using the method introduced in [4] is 4nm(k + l + 1), while we need only2n(m + 1)l + 2m(n + 1)k + 4nm if we use the method developed here. Numericalexperiments have still to be done in order to give other further information concerning thecomparison of these two methods.Obviously, the above decomposition of f k,l has several advantages. It allows to computethis interpolant step by step using basis functions which they have simple expressions.But this method is adapted only for tensor product Hermite interpolants. Hence, itis interesting to study the decomposition of any bivariate Hermite interpolant. In regard tothis topic, we have proposed in [5] a method allowing to build recursively bivariate Hermitepolynomial interpolants defined on triangles, and in [6] a Hierarchical computationof particular Hermite spline interpolants of class C k on IR 2 . The development of otherapplications of these results is still under investigation.4. References[1] D. HONG, L. L. SCHUMAKER, “Surface Compression Using A Space of C 1 Cubic SplinesWith A Hierarchical Basis.”, Computing 72 (2004), pp. 79-92.[2] A. MAZROUI, D. SBIBIH, AND A. TIJINI, “A recursive construction of Hermite interpolantsand applications.”, To appear in J. Comput. Appl. math.[3] A. MAZROUI, D. SBIBIH, AND A. TIJINI, “A simple method for smoothing functions andcompressing Hermite data.”, To appear in Adv. Comput. Math.[4] A. MAZROUI, D. SBIBIH, AND A. TIJINI, “A recursive method for the construction of tensorproduct Hermite interpolants.”, Curve and Surface Design, Saint-Malo 2002, T. Lyche, M.-L.Mazure, and L. L. Shcumaker, (eds), Nashboro Press, Brentwood, 2003, pp. 303-314.[5] A. MAZROUI, D. SBIBIH, AND A. TIJINI, “AHierarchical computation of bivariate Hermiteinterpolants.”, Curve and Surface Design, Saint-Malo 2002, T. Lyche, M.-L. Mazure, and L. L.Shcumaker, (eds), Nashboro Press, Brentwood, 2003, pp. 315-324.[6] A. MAZROUI, D. SBIBIH, AND A. TIJINI, “Recursive Computation of Bivariate HermiteSpline Interpolants.”, Submitted.TAMTAM –Tunis– 2005