Original articleVariational trivariate fitting using Worsey–Piper macro elements on tetrahedral partitions☆
Section snippets
Introduction and preliminaries
In the last years different variational methods based on the minimization of an “energy functional” have been used in CAD and CAGD in the field of the fitting and design of curves and surfaces. Different works in this research area have shown that these methods are useful and efficient (see [1], for instance). The energy functional is usually a sum of two terms: the first measures how well the resulting curve or surface approximates a set of scattered data which may be of Lagrange or Hermite
Berstein–Bézier representation of quadratic polynomials
Let be a nondegenerate tetrahedron, having vertices with cartesian coordinates (xi, yi, zi), for i = 1, …, 4. Let λ = (λ1, λ2, λ3, λ4) denote the barycentric coordinates of a point with respect to the tetrahedron ρ. Such barycentric coordinates can be determined as the unique solution of the system
Setting , then each polynomial p in has a unique representation
where
Formulation of the problem
Given a finite set of points in and a set of real values , we are looking for a WP-spline that approximates the points and minimizes an “energy functional” that we describe next. From the continuous injection of into , we can define the evaluation operator for all Let us suppose that
Given τ1 ∈ (0, + ∞) and τ2 ∈ (0, + ∞), we consider the functional defined on by
Observe
Convergence
Let us consider, for each , a finite set of points in and a given vector of real values . Let be a -unisolvent subset of Ω and let us suppose that
Then, there exist C > 0 and such that for all s ≥ s1 there exist satisfying
Lemma 3 Let us suppose that (4.1) is satisfied and let be any subset satisfying (4.2). Then, there
Estimation of the optimum smoothing parameters
We have also implemented a method providing optimum values of the parameters τ1 and τ2. Such a method was developed in [4] for the two-dimensional case. Next we briefly explain it: let us fix a uniform tetrahedral partition Δ of , its associated Worsey–Piper refinement Δ∗, the data point set coming from a function and be the solution of Problem 2 for given values of the smoothing parameters τ = (τ1, τ2). We will look for values of the parameters τ which minimize the
Graphical examples
In this section, we show the results of some numerical tests on the method proposed in this paper. To do this, let us consider a tetrahedral partition Δn, n ≥ 1 of the domain Ω = [0, 1] × [0, 1] × [0, 1] obtained as follows:
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Use n + 1 parallel planes in each of the three space dimensions and subdivide Ω into n3 subcubes where i, j, k = 0, ⋯ , n − 1 and .
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Subdivide each subcube into six tetrahedra (see Fig. 4).
By applying Algorithm 1 to Δn we obtain the corresponding
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Research supported by AI MA/08/182 and URAC-05.