Original article
Variational trivariate fitting using Worsey–Piper macro elements on tetrahedral partitions

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Abstract

In this paper we present a method to obtain a trivariate spline constructed over the Worsey–Piper split corresponding to a tetrahedron. Such spline approximates a set of Lagrangian scattered data by minimizing an “energy functional” which also controls the smoothness of the spline. We give a convergence result and we show some graphical examples.

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 ρ=t(v1,v2,v3,v4) be a nondegenerate tetrahedron, having vertices vi with cartesian coordinates (xi, yi, zi), for i = 1, …, 4. Let λ = (λ1, λ2, λ3, λ4) denote the barycentric coordinates of a point (x,y,z)R3 with respect to the tetrahedron ρ. Such barycentric coordinates can be determined as the unique solution of the system

x1x2x3x4y1y2y3y4z1z2z3z41111λ1λ2λ3λ4=xyz1.

Setting i=(i1,i2,i3,i4)N4, then each polynomial p in P2(ρ) has a unique representation

p(x,y,z)=b(λ)=|i|=2biBi(2)(λ),where {Bi(2)(λ

Formulation of the problem

Given a finite set of points D={ai}i=1k in Ω¯ and a set of real values β={βi}i=1k, we are looking for a WP-spline that approximates the points {(ai,βi)}i=1kR4 and minimizes an “energy functional” that we describe next. From the continuous injection of H2(Ω) into C0(Ω¯), we can define the evaluation operator ρ(v):=(v(ai))i=1kRk for all vH2(Ω). Let us suppose thatKer(ρ)P1(Ω)={0}.

Given τ1  (0, + ∞) and τ2  (0, + ∞), we consider the functional defined on H2(Ω) by

J(v)=<ρ(v)β>k2+m=12τm|v|m2.

Observe

Convergence

Let us consider, for each sN, a finite set Ds={ais}i=1k=k(s) of points in Ω¯ and a given vector of real values βs={βis}i=1k. Let D0={a10,a20,a30,a40} be a P1-unisolvent subset of Ω and let us suppose thatsupxΩminaDs<xa>3=O1s,s+.

Then, there exist C > 0 and s1N such that for all s  s1 there exist {a1s,a2s,a3s,a4s}Ds satisfying<ai0ais>3Csfor alli=1,,4and for allss1.

Lemma 3

Let us suppose that (4.1) is satisfied and let As={a1s,a2s,a3s,a4s}Ds 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 Ω¯, its associated Worsey–Piper refinement Δ, the data point set DsΩ¯ coming from a function gC0(Ω) and στS21(Ω,Δ) 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:

  • Use n + 1 parallel planes in each of the three space dimensions and subdivide Ω into n3 subcubes [i,i+1]h¯×[j,j+1]h¯×[k,k+1]h¯ where i, j, k = 0, ⋯ , n  1 and h¯=1/n.

  • Subdivide each subcube into six tetrahedra (see Fig. 4).

By applying Algorithm 1 to Δn we obtain the corresponding

References (22)

  • D. Girard

    Practical optimal regularization of large linear systems, M2AN

    Math. Model. Numer. Anal.

    (1986)
  • Cited by (0)

    Research supported by AI MA/08/182 and URAC-05.

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