Recursive computation of Hermite spherical spline interpolants

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1 Journal of Computational and Applied Mathematics 213 (2008) Recursive computation of Hermite spherical spline interpolants A. Lamnii, H. Mraoui, D. Sbibih Laboratoire MATSI, Université Mohammed I, Ecole Supérieure de Technologie, Oujda, Morocco Received 25 May 2006; received in revised form 12 January 2007 Abstract Let u be a function defined on a spherical triangulation Δ of the unit sphere S. In this paper, we study a recursive method for the construction of a Hermite spline interpolant u k of class C k and degree 4k + 1onS, defined by some data scheme D k (u). We show that when the data sets D r (u) are nested, i.e., D r 1 (u) D r (u), 1 r k, the spline function u k can be decomposed as a sum of k + 1 simple elements. This decomposition leads to the construction of a new and interesting basis of a space of Hermite spherical splines. The theoretical results are illustrated by some numerical examples Elsevier B.V. All rights reserved. MSC: 41A05; 41A15; 43A90; 65D05; 65D07; 65D10 Keywords: Spherical splines; Hermite interpolation; Recursive computation; Decomposition 1. Introduction The well-known methods for building the classical univariate or bivariate Hermite spline interpolants are based on the Hermite fundamental functions. But, the lack of recursive formulae for computing these basis functions makes this construction rather complicated. In order to overcome this difficulty, a simple method allowing to compute recursively a univariate Hermite spline interpolant of class C k and degree 2k + 1 of a function f defined on an interval [a,b] was proposed in [9] (see also [10]). More precisely, if f k is such an interpolant, then it can be decomposed in the form f k = f 0 + g 1 + +g k, where f 0 is the piecewise linear interpolant of f, and g r,1 r k, are particular splines of C r 1 and degree 2r + 1 that satisfy interesting properties. The simplicity and the multiresolution structure of this decomposition make it attractive for applications, such as computing integrals, smoothing curves and compressing data. For more details on these subjects, see [9,10]. In view of the importance and the originality of this method, it is natural to extend it to several variables. One obvious way to do this is to use the tensor product. With regard to this extension, a recursive construction for tensor product Hermite interpolants was described in [7]. In[11] (see also [8], a method allowing to build recursively bivariate Hermite spline interpolants of class C k on R 2 was proposed. In this paper, we deal with a hierarchical computation of particular C k Hermite spherical spline interpolants. Assume that S is the unit sphere, and V ={v i } n is a set of scattered points located on S. Let us denote by Δ a spherical triangulation of S whose set of vertices is V. For a regular function u, defined on S, we denote by D k (u) the Research supported in part by PROTARS III, D11/18. Corresponding author. addresses: alamnii@gmail.com (A. Lamnii), hamid_mraoui@yahoo.fr (H. Mraoui), sbibih@yahoo.fr (D. Sbibih) /$ - see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.cam

2 440 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) set of data formed by the values and the derivatives of u at all the vertices v i of Δ and at other points lying in S. Let D k,t be the data set in D k (u) restricted to a spherical triangle T of Δ. We show that there exists a unique spherical Bernstein Bézier (SBB) polynomial u k,t of degree 4k + 1 defined on T which interpolates the data D k,t (u). Setting u k = T Δ u k,t, then u k is the unique spherical spline of smoothness k and degree 4k + 1 which interpolates D k (u). Methods of this type are called macro-element methods. Our aim in this paper is to define a recursive formula allowing to compute u k step by step if some conditions are satisfied. In order to do this, assume that the sets D r,t (u), 0 r k, are nested, i.e., D 0,T D 1,T D k,t for all T Δ. (1.1) Then, the Hermite spherical spline u k can be written in the form u k = u 0 + d 1 + +d k, where u 0 is the interpolant to u of class C 0 and degree 1 and d r,1 r k, is a spherical spline of class C r 1 and degree 4r + 1onS. Ifwe put d r,t = d r T, then we have u k,t = u 0,T + d 1,T + +d k,t,sou 0 = T Δ u 0,T and d r = T Δ d r,t,1 r k. Moreover, each d r,t, which is an homogeneous Bernstein Bézier polynomial, is completely determined by the data D k,t (u u r ),1 r k. The multiresolution structure of this decomposition means that u 0 may be considered as a coarse approximation of u k, and d r,0 r k, are correction terms or detail functions. This representation of u k gives rise to a family that generates the space of spherical splines of smoothness k and degree 4k + 1, and to a new basis for the space B 4k+1 (T ), T Δ, of homogeneous Bernstein Bézier polynomials of degree 4k + 1. As a consequence of (1.1), we will see later that the new bases for the spaces B 4r+1 (T ), 0 r k, are hierarchical. Then, they can be used as tools for solving several mathematic problems like those studied in [4]. The paper is organized as follows. In Section 2 we give some preliminary results on homogeneous Bernstein Bézier polynomials and spherical splines. Section 3 is devoted to local interpolation method based on C k macro-elements of degree 4k + 1. In Section 4 we define a recursive computation of local Hermite polynomials u k,t B 4k+1 (T ), T Δ, when their corresponding data schemes D k,t (u) are nested. Then we deduce a decomposition of the spherical spline u k of class C k and degree 4k + 1onS. As a consequence of this method, we obtain a new and interesting basis for B 4k+1 (T ). Finally, in order to illustrate our results, we give in Section 5 some numerical examples. 2. Preliminary results In this section, we present the connection between the functions defined on S and homogeneous trivariate functions, and we introduce some definitions. A trivariate function F is said to be positively homogeneous of degree t R provided that for every real number a>0, F(av)= a t F(v), v R 3 \{0}. Lemma 1 (see Alfeld et al. [3]). Given a function f defined on S and let t R. Then ( ) v F t (v) = v t f v is the unique homogeneous extension of f of degree t to all of R 3 \{0}, i.e., F t S = f, and F t is homogeneous of degree t. Let g be a given unit vector. Then, as in [3], we define the directional derivative D g of f at a point v S by D g f(v)= D g F(v)= g T F(v), where F is some homogeneous extension of f and F is the gradient of the trivariate function F. While a polynomial of degree d has a natural homogeneous extension to R 3, a general function f on S has infinitely many different extensions. The value of its derivative may depend on which extension that we take (for more details see [3]). Let P d be the space of trivariate polynomials of total degree at most d, and let H d = P d S be its restriction to the sphere S. A trivariate polynomial p is called homogeneous of degree d if p(λx, λy, λz) = λ d p(x,y,z) for all λ R, and harmonic if Δp = 0, where Δ is the Laplace operator defined by Δf = (D 2 x + D2 y + D2 z )f.

3 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) Definition 2 (see Fasshauer and Schumaker [5]). The linear space H d ={p S : p P d and p is homogeneous of degree d and harmonic} is called the space of spherical harmonics of exact degree d. Let there be given a spherical triangle T. The associated spherical Bernstein basis functions of degree d are defined by Bij d d! k (v) = i!j!k! bi 1 (v)bj 2 (v)bk 3 (v), i + j + k = d, ( ) where b 1 (v), b 2 (v), b 3 (v) are spherical barycentric coordinates of v relative to T. These d+2 2 functions are linearly independent [5], and form a basis for the space denoted, in what follows, by B d. Each p B d is called an SBB polynomial. It is clear that p can be written in the form p = i+j+k=d c ij kbij d k and it is uniquely determined by its B coefficients c ij k. It is well known (see [5]) that Bij d k are actually linear combinations of spherical harmonics. Proposition 3 (see Fasshauer and Schumaker [5]). For all d 1, we have { H0 H 2 H 2k if d = 2k, B d = H 1 H 3 H 2k+1 if d = 2k + 1. From the above proposition, it is simple to see that B d 1 / B d but B d 2 B d. Let Δ ={T i } N be a triangulation of the unit sphere S. Given integers r and d, we define the space of spherical splines (Δ) by S r d S r d (Δ) ={s Cr (S) : s Ti B d,i= 1,...,N}. If s Sd r (Δ), then its pieces are SBB polynomials whose exact degree is even if d is even, and odd if d is odd. As documented in the recent literature [6,13], the spherical Bézier splines are extensively used in the computer design area. We say that a macro-element has smoothness C k provided that if the element is used to construct an interpolating spline locally on each spherical triangle of Δ, then the resulting piecewise function is C k continuous globally. A quintic C 1 macro-element have been constructed in the literature, see [5]. In the next section, we give some theoretical results for the construction of the C k macro-elements. 3. A C k macro-element Using the macro-element method, we can now construct an interpolating spherical spline of class C k and degree 4k + 1. Let v 1, v 2 and v 3 be the vertices of T, and, for convenience, let v 4 = v 1 and v 5 = v 2. For each i = 1, 2, 3, let M i,s be some arbitrary distinct points on the edges from v i to v i+1 not equal to its vertices v i, and B an anterior point of T. To define some useful derivatives associated with T, let g i,j be a tangent vector to S at v i contained in the plane passing through v i,v j and the origin, not parallel with v i,ifi, j = 1, 2, 3, i = j, and, for convenience, g i,0 = g i,3 and g i,4 = g i,1. In addition, let h i be a vector tangent to S at M i,s which is not contained in the plane passing through v i,v i+1 and the origin, and let ν be a tangent vector to S at B.

4 442 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) v 3 v 3 v 2 v 2 v 1 v 1 o o Fig. 1. The C k macro-element for k = 3 and 4, respectively. For a regular function u defined on S, we denote by D k,t its interpolating data set restricted to a triangle T = v 1 v 2 v 3 of Δ, and by H k,t the associated Hermite interpolation problem defined as follows, see [11]: Find u k,t B 4k+1 (T ) such that D α u k,t (v i ) = D α u(v i ), α 2k, 1 i 3 for k 0, H k,t = Dh r i u k,t (M i,s ) = Dh r i u k (M i,s ), 1 s r k, 1 i 3 for k 1, D γ u k,t (B) = Dνu γ k,t (B) = Dνu(B), γ γ k 2 for k 2, where D α u(v i ) = D α 1 g i,i+1 D α 2 g i,i 1 u(v i ), α =α 1 + α 2. Definition 4. Let D k (u) = T D k,t. We say that D k (u) is an S4k+1 k (Δ)-unisolvent data scheme if for all T Δ, the problem H k,t has a unique solution u k,t B 4k+1 (T ) and the function u k is such that u k T = u k,t is of class C k on S. Lemma 5. The data set D k,t (u) uniquely determines an SBB polynomial u k,t of degree 4k + 1 on T. Proof. The proof is similar to the proof of the bivariate case (see [14]). Indeed, assume that u k,t is written in its SBB form, and the corresponding Bézier coefficients are numbered as in Fig. 1. It is simple to verify that ( ) 4k + 1 dim B 4k+1 (T ) = card(d k,t (u)) = = 8k k Then, showing that D k,t (u) is a determining set for B 4k+1 (T ) is equivalent to showing that D k,t (u) uniquely determines all B coefficients of u k,t. Indeed, the C 2k smoothness at v 1 implies that the data set {D α u(v 1 ), α 2k} uniquely determines the (2k + 1)(k + 1) coefficients corresponding to domain points marked with closest to vertex v 1 (see Fig. 1). Once the above coefficients are computed, for r =1,...,kthe coefficients (c 2k,2k r+1,r,...,c 2k r+1,2k,r ) corresponding to domain points marked with (square) closest to vertex v 1 are uniquely determined by {D r h 1 u(m 1,s ), s=1,...,r},

5 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) and can be computed from a nonsingular linear system of r equations, where the corresponding matrix depends only on spherical barycentric coordinates. The situation at v 2 and v 3 is analogous. Moreover, it is easy to see that {D γ u k,t (B), γ k 2} uniquely determines the k(k 1)/2 coefficients corresponding to domain points marked with (up triangle). Thus, a total of 3(2k + 1)(k + 1) + 3k(k + 1)/2 + k(k 1)/2 = 8k k + 3 coefficients are already determined, and this completes the proof. Theorem 6. For k N, the data set D k (u) is S k 4k+1 (Δ)-unisolvent. Proof. From Lemma 5, the given data set D k,t (u) uniquely defines an SBB polynomial of degree 4k + 1 on each triangle T of Δ. We now show that these polynomials join together smoothly to form a spline in S4k+1 k (Δ). The argument is the same as in the planar case. Suppose that T and T are two triangles in Δ which share an edge e joining the vertices v 1 and v 2 u k,t and u k, T are both circular Bernstein Bézier polynomials (CBB polynomials) (see [1]) of degree 4k + 1 on e. Since the coefficients of both are computed from common data at points on e, it follows that u k,t = u k, T, and we conclude that u k,t and u join continuously across e. Now,letp k, T r = Dh r 1 u k,t and p r = Dh r 1 u k, T for r = 0,...,k, then p r and p r reduce to CBB polynomials on e of degree 4k + 1 r satisfying the following interpolation conditions: { D α g12 p r (v i ) = Dg α 12 p r (v i ) = Dg α 12 u(v i ), i = 1, 2 for 0 α 2k r, D r h 1 p r (M 1,s ) = D r h 1 p r (M 1,s ) = D r h 1 u(m 1,s ) for 1 s r. We have 2(2k r 1) data at vertices v 1 and v 2, and r data at points M 1,s, then we have 4k r + 2 data. Moreover, these data are linearly independent. Since the dimension of the space of CBB polynomials on e of degree 4k + 1 r is equal to 4k r + 2, we conclude that p r = p r,0 r k. This establishes the C k continuity between u k,t and u. k, T The same argument works for every edge and the claim follows. Using standard arguments (see e.g., [12]) we can establish an optimal order error bound for functions in the classical Sobolev spaces W 4k+2 (S). Let Δ be the mesh size of Δ, i.e., the diameter of the largest triangle in Δ. Theorem 7. There exists a constant C depending only on k and the smallest angle in Δ such that for every u C 4k+2 (S), u u k S C Δ 4k+2 u 4k+2,S, (3.1) where u 4k+2,S = α =4k+2 Dα u S and we write S for the infinity norm. Proof. Since the proof is similar to the proof of Theorem 5.3 in [12], we can be brief. Fix T Δ, let u W 4k+2. From [12, Theorem 4.2], there exists a spherical polynomial q B 4k+1 (T ) such that u q ΩT u q 4k+2,ΩT C 1 Ω 4k+2 u 4k+2,ΩT, (3.2) where Ω T is the union of the triangles in the star(t) (for the definition of star(t) see [12]). Let I k,t be the Hermite interpolation operator defined for a function u by I k,t u = u k,t B 4k+1 (T ). Since I k,t is exact on B 4k+1 (T ), i.e., I k,t p = p for all p B 4k+1 (T ), we deduce that u u k T = u I k,t u T u q T + I k,t (u q) T. It suffices to estimate the second quantity. Applying the Markov inequality (see [12]) to each of the polynomials I k,t (u q), T Δ, we deduce the existence of a constant C depending only on the minimum angle in the triangulation of Ω T and k such that I k,t (u q) T C Ω T 4k+2 u 4k+2,ΩT. (3.3) Finally, to get (3.1) we take the maximum over all T Δ. Let B k,t ={ α i,k, ψr i,s,k, ψγ k, α 2k, 1 s r k, γ k 2, 1 i 3} be the Hermite basis for B 4k+1(T ) associated with the problem H k,t, where the elements of B k,t are entirely determined by the following interpolation

6 444 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) conditions: D β i,k α (v l) = δ i,l δ α,β, β 2k, 1 l 3 for k 0, D r 1 h l α i,k (M l,s 1 ) = 0, 1 s 1 r 1 k, 1 l 3 for k 1, D γ 1 α i,k (B) = 0, γ 1 k 2 for k 2, D β ψ r i,s,k (v l) = 0, β 2k, 1 l 3 for k 0, D r 1 h l ψ r i,s,k (M l,s 1 ) = δ i,l δ r,r1 δ s,s1, 1 s 1 r 1 k, 1 l 3 for k 1, D γ 1ψ r i,s,k (B) = 0, γ 1 k 2 for k 2, D β ψ γ k (v l) = 0, β 2k, 1 l 3 for k 0, D r 1 h l ψ γ k (M l,s 1 ) = 0, 1 s 1 r 1 k, 1 l 3 for k 1, D γ 1ψ γ k (B) = δ γ,γ 1, γ 1 k 2 for k 2, where δ is the Kronecker delta. Then, by using the above basis, the polynomial u k,t, solution of the problem H k,t, can be written in the form 3 u 0,T = u(v i ) (0,0) i,0, u 1,T = and for k 2, u k,t = 3 α 2 3 α 2k (3.4) (3.5) (3.6) 3 D α u(v i ) α i,1 + D hi u(m i,1 )ψ 1 i,1,1, (3.7) D α u(v i ) α i,k s r k D r h i u(m i,s )ψ r i,s,k + γ k 2 D γ u(b)ψ γ k. (3.8) As mentioned in the Introduction, the computation of the spline u k is equivalent to that of its restriction u k,t on each triangle T of Δ. Then, we reserve almost the next section for the study of u k,t, T Δ. 4. Recursive computation of Hermite spherical spline interpolants of class C k The lack of recursive formulae for computing the basis elements of B k,t makes the use of expression (3.7) and (3.8) rather complicated. To remedy this problem, we have established a decomposition of u k,t. Indeed, as B 4k 3 (T ) B 4k+1 (T ) for k 1, we deduce that u k,t =u k 1,T +d k,t, where d k,t is a particular polynomial in B 4k+1 (T ). This decomposition is derived from connections between some elements of B k,t and B k 1,T. Hence, we first prove the following result. Lemma 8. For 1 i 3, we have α i,k = α i,k 1 α i,k, α 2k 2 for k 1, and where ψ r i,s,k = ψr i,s,k 1 ψr i,s,k, ψ γ k = ψγ k 1 ψγ k, (0,0) i,1 = 3 2 l=1 β =1 1 s r k 1 for k 2 γ k 3 for k 3, D β (0,0) i,0 (v l ) β l,1 + 3 l=1 D hl (0,0) i,0 (M l,1 )ψ 1 l,1,1

7 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) and α i,k = ψ r i,s,k = ψ γ k = 3 2k l=1 β =2k γ 1 =k 2 2k l=1 β =2k γ 1 =k 2 2k l=1 β =2k 1 + γ 1 =k 2 D β α i,k 1 (v l) β l,k + 3 D γ 1 α i,k 1 (B)ψγ 1 k l=1 s 1 =1 for k 2, D β ψ r i,s,k 1 (v l) β l,k + 3 D γ 1ψ r i,s,k 1 (B)ψγ 1 k, D β ψ γ k 1 (v l) β l,k + 3 D γ 1ψ γ k 1 (B)ψγ 1 k. k Dh k l α i,k 1 (M l,s 1 )ψ k l,s 1,k l=1 s 1 =1 l=1 s 1 =1 k Dh k l ψ r i,s,k 1 (M l,s 1 )ψ k l,s 1,k k Dh k l ψ γ k 1 (M l,s 1 )ψ k l,s 1,k Proof. Let T be a triangle of Δ and I k,t be the Hermite interpolation operator defined for a function u by I k,t u=u k,t B 4k+1 (T ).AsI k,t is exact on B 4k+1 (T ), i.e., I k,t p = p for all p B 4k+1 (T ), we deduce that I k,t α i,k 1 = α i,k 1.In other words, we have α i,k 1 = 3 l=1 β 2k + γ 1 k 2 D β α i,k 1 (v l) β l,k + 3 D γ 1 α i,k 1 (B)ψγ 1 k. l=1 1 s 1 r 1 k D r 1 h l α i,k 1 (M l,s 1 )ψ r 1 l,s1,k On the other hand, from (3.4) (3.6) we deduce that for all α 2(k 1), 3 D β α i,k 1 (v l) β l,k = α i,k l=1 β 2k 2 and 3 l=1 1 s 1 r 1 k 1 D r 1 h l α i,k 1 (M l,s 1 )ψ r 1 l,s1,k = 0, where after, we get the first equality. Using a similar technique, one can establish the other equalities. Theorem 9. Assume that the data schemes corresponding to the problems H k 1,T and H k,t satisfy D k 1,T D k,t for k 1. Then, the polynomial u k,t can be decomposed in the form u k,t = u k 1,T + d k,t, k 1, where d 1,T = D α (u u 0,T )(v i ) α i,1 + D hi (u u 0,T )(M i,1 )ψ 1 i,1,1, α =1

8 446 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) and for k 2, d k,t = 3 2k α =2k 1 + γ =k 2 D α (u u k 1,T )(v i ) α i,k + 3 D γ (u u k 1,T )(B)ψ γ k. k Dh k i (u u k 1,T )(M i,s )ψ k i,s,k s=1 Proof. If we put ũ k,t = u k 1,T + d k,t, then one can check that the polynomial ũ k,t of degree 4k + 1 satisfies the Hermite interpolation conditions in H k,t. As the Hermite polynomial is unique, we deduce that ũ k,t = u k,t. Remark 10. From the above expression of d k,t, we deduce that its corresponding data set is D k (u u k 1,T ). Corollary 11. For a triangle T Δ, assume that D 0,T D 1,T D k,t. Then, the polynomial u k,t can be decomposed in the form u k,t = u 0,T + d 1,T + +d k,t, k 1, where d 1,T = Ci,1 α α i,1 + C i,1 1 ψ1 i,1,1, and for k 2, α =1 3 d n,t = 2n C α i,n α i,n + 3 α =2n 1 n C i,t n ψn i,t,n + Ĉnψ γ γ n t=1 γ =n 2 for 2 n k. u 0,T is the unique solution of the Hermite problem H 0,T, and the coefficients Ci,n α = Dα u(v i ) D α u n 1,T (v i ), C i,t n = Dn h i u(m i,t ) Dh n i u n 1,T (M i,t ), Ĉn γ = D γ u(b) D γ u n 1,T (B) can be computed recursively as follows: C α i,1 = Dα u(v i ) D α u 0 (v i ) if α =1 and 1 i 3, C α i,1 = Dα u(v i ) if α =2 and 1 i 3, C 1 i,1 = D h i u(m i,1 ) D hi u 0 (M i,1 ) if 1 i 3, Ĉ γ 1 = 0 for all γ, and for n 2 and α =2n 1 or α =2n, we have Ci,n α = n 1 3 2t Dα u(v i ) C β l,t Dα β l,t (v i) + + θ =t 2 t=1 l=1 β =2t 1 Ĉt θ Dα ψ θ t (v i), t C l,m t Dα ψ t l,m,t (v i) m=1

9 and 1 t k and 1 i 3, we also have C i,t n = n 1 3 Dn h i u(m i,t ) + t m=1 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) t=1 l=1 2t β =2t 1 ] C l,m t Dn h i ψ t l,m,t (M i,t) + and for γ =n 2, we have Ĉn γ = D γ u(b) D γ n 1 3 u 0 (B) + θ =t 2 t=1 Ĉt θ Dγ ψ θ t (B). l=1 C β l,t Dn h i β l,t (M i,t) θ =t 2 2t β =2t 1 Ĉt θ Dn h i ψ θ t (M i,t), C β l,t Dγ β l,t (B) + t m=1 C l,m t Dγ ψ t l,m,t (B) Proof. It is derived from Theorem 9. Now, we give another main result of this paper. Theorem 12. The family B k,t,k 1, defined by B 1,T ={ (0,0) i,0, α i,1, ψ1 i,1,1, 1 α 2 and 1 i 3} B k,t ={ (0,0) i,0, α i,n, ψn i,t,n, ψγ n, 2n 1 α 2n, γ =n 2, 1 t n k and 1 i 3}, k 2 forms a basis for the space B 4k+1 (T ). Moreover, B k,t, k N, are hierarchical. Proof. Let p B 4k+1 (T ), T Δ. Since the Hermite interpolation operator I k,t is exact on B 4k+1 (T ), we deduce that p = I k,t p = u k,t = u 0,T + d 1,T + +d k,t, where u 0,T = 3 p(v i ) (0,0) i,0 is the unique solution of the Hermite interpolation problem H 0,T, and d n,t,1 n k, are polynomials in B 4n+1 (T ) and are given by d n,t = 3 2n α =2n 1 D α (p u n 1,T )(v i ) α i,n D γ (p u n 1,T )(B)ψ γ n. γ =n 2 n Dh n i (u u n 1,T )(M i,t )ψ n i,t,n t=1 Then, B k,t generates the space B 4k+1 (T ). Since card( B k,t ) = dim(b 4k+1 (T )), we deduce that B k,t is a basis for the space B 4k+1 (T ). On the other hand, if we put B 0,T = B 0,T, it is simple to check that B k,t = B k 1,T B k,t, where B 1,T ={ α i,1, ψ1 i,1,1, 1 α 2 and 1 i 3}, and for k 2, B k,t ={ α i,k, ψk i,s,k, ψγ k, 2k 1 α 2k, 1 s k, γ =k 2 and 1 i 3}. Then we have B k 1,T B k,t.

10 448 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) Now, as a consequence of the above results, we establish the following decomposition of the spline interplant u k solution of the problem H k. Theorem 13. Let D k (u) be an S k 4k+1 (Δ)-unisolvent and H k = T Δ H k,t its corresponding Hermite interpolation problem. Then, the solution u k of H k can be decomposed in the form u k = u 0 + d 1 + +d k, where u 0 = T Δ u 0,T and d n = T Δ d n,t, 1 n k. Proof. It is derived from Corollary 11 and the fact that D k (u) = T Δ D k,t (u). Remark 14. The comparison of the two bases B k,t and B k,t of the space B 4k+1 (T ) leads to the following observations: (i) The hierarchical structure of the bases B k,t, k N, can be used for several practices in numerical analysis like compressing data and surfaces. (ii) If we denote by T α,k, T r,s,k and T γ,k the number of B coefficients of α i,k, ψr i,s,k and ψγ k, respectively, that are not necessarily equal to zeros, then by straightforward computation we get ( ) ( ) ( ) 4k + 3 2k + 2 α +1 T α,k = and ( ) 4k + 3 T r,s,k = T γ,k = 3 2 ( 2k ). These B coefficients are solutions of linear systems of size T α,k T r,s,k or T γ,k that are derived from Hermite interpolation problems given by (3.4) (3.6). Then, it is easy to verify that the total number of B coefficients needed for the determination of B k,t in B 4k+1 (T ) is given by Σ k = 3 α 2k T α,k + 3 k s=1 r=s k T r,s,k + γ k 2 = k k2 + 50k k 4, while the number of B coefficients for the determination of the new basis B k,t is given by σ k = 3T (0,0),0 + k 3 t=1 2t α =2t 1 T α,t + 3 T γ,k t T t,s,t + s=1 = k + 30k k3 + 8k 4. In Table 1, wegiveσ k and σ k for the first values of k. γ =t 2 T γ,t Table 1 k Σ k ,208 41,028 72, , , ,638 σ k ,017 29,480 47,787 73, ,493

11 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) (iii) According to (ii), the computation of a polynomial u k,t B 4k+1 (T ) at several points needs a lot of operations. As in practice this computation is required for several points of T, it is useful to work with the new basis which allows to reduce extensively the number of operations (see Table 1). In conclusion, we remark that the basis B k,t presents several practical advantages. Some are illustrated in this paper, but the development of others is still under investigation. In order to illustrate our results, we give in the next section some numerical examples. 5. Numerical examples In this section, we give two examples which illustrate the theoretical results. Let Δ 1, Δ 2,...be a sequence of regular spherical triangulations of S such that the number of vertices (resp. triangles) of Δ l is 2 2l +2 (resp. 2 2l+1 ). The sequence (Δ l ) l 1 is created as follows: Δ 1 is the Delauny triangulation associated with 6 vertices of a regular octahedron, i.e., with the points ±e i,i= 1, 2, 3, where e i are the Cartesian coordinate vectors. So, this triangulation consists of the eight quadrantal spherical triangles. Then, for each l 2 we compute the vertices of Δ l from those of Δ l 1 by adding the midpoints of each edge of Δ l 1 to the vertices of Δ l 1. This amounts to splitting each triangle of Δ l 1 into four subtriangles in a standard way, and in fact each of these triangulations is a Delauny triangulation of its vertex set (Figs. 2 and 3) Example 1 Using the results given in the preceding section, we describe in this example the recursive computation of the Hermite interpolant u 2 S 2 9 (Δ 3) to the function u defined on S by where and u(x,y,z)= 3 (g i (x,y,z)) 1/2, (5.1) ( ) x 2 ( ) y 2 ( ) z 2 g i (x,y,z)= + + α i α i+1 α i+2 (α 1,...,α 5 ) = (5, 1, 2, 5, 1). The decomposition of u 2 is such that u 2 = u 0 + d 1 + d 2, where the details d 1 and d 2 are computed from the formulae given in Corollary 11. To assist in understanding the behavior of the decomposition method, we visually examined the Fig. 2. Regular triangulations of the sphere S.

12 450 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) Fig. 3. Regular triangulations of S(u) using Δ 3. a b c d e f Fig. 4. Decomposition of u 2. (a) Graph of u 0, (b) graph of d 1, (c) graph of u 1, (d) graph of d 2, (e) graph of u 2 and (f) graph of u. surfaces S(u) ={u(v)v : v S}, corresponding to our test function u, its various interpolants u 0, u 1, u 2 and the detail functions d 1, d 2 in Fig. 4.

13 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) Example 2 Alfeld et al. have studied in [3] the Hermite interpolant of class C 1 and degree 5 of the function f defined on S by f(x,y,z) = 1 + x 8 + e 2y3 + e 2z2 + 10xyz. (5.2) By comparing their method with the one proposed in this paper for computing the above interpolant, we remark that the obtained results are the same, see Table 2, but our method allows to reduce the computational cost, see Table 1. Moreover, we give for this function the recursive computation of the associated Hermite interpolant f 2 S 2 9 (Δ 3). More specifically, we have the decomposition f 2 = f 0 + d 1 + d 2, where d 1 and d 2 are computed from the formulae listed in Corollary 11 (Fig. 5). To assist in understanding the behavior of the decomposition method, we visually examined the surfaces S(f ) ={f(v)v : v S}, corresponding to our test function f, its various interpolants f 0, f 1, f 2 and the detail functions d 1, d 2 in Fig. 6. Remark 15. It is simple to verify that the above function f is not homogeneous of order 9. So, according to the results given in Section 2, we cannot expect order 10 convergence for k = 2 if we compute second, third and fourth derivatives of f directly from expression (5.2). Indeed, as shown in Table 2, if we do compute derivatives in this way, we seem to be getting order 3 convergence (see the second column of Table 2). The convergence shown in the first column of Table 2 corresponds to computing the derivatives from the order 9 homogeneous extension of f (cl. Lemma 1). The error and error given in Table 2 are relative errors and they are defined by error = max v V f k (v) f(v) max v V f(v), error = max v V fk (v) f(v), max v V f(v) where f k is the Hermite interpolant of class Ck and degree 4k + 1 using the derivatives which are computed from (5.2). Table 2 Approximation errors Hermite interpolant of Hermite interpolant of Hermite interpolant of class C 2 and degree 9 class C 1 and degree 5 class C 1 and degree 5 [3] error error error error error error Δ ( 4) ( 2) ( 2) 2.91 ( 1) ( 2) ( 1) Δ ( 5) ( 2) ( 3) ( 1) ( 3) ( 1) Δ ( 6) ( 3) ( 4) ( 2) ( 4) ( 2) Δ ( 8) ( 3) ( 5) ( 3) ( 5) ( 3) Δ ( 10) ( 4) ( 7) ( 3) ( 7) ( 3) Fig. 5. Regular triangulations of S(f ) using Δ 3.

14 452 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) Fig. 6. Decomposition of f 2. (a) Graph of f 0, (b) graph of d 1, (c) graph of f 1, (d) graph of d 2, (e) graph of f 2 and (f) graph of f. References [1] P. Alfeld, M. Neamtu, L.L. Schumaker, Circular Bernstein Bézier polynomials, in: M. Daehlen, T. Lyche, L.L. Schumaker (Eds.), Mathematical methods in CAGD, Vanderbilt University, [3] P. Alfeld, M. Neamtu, L.L. Schumaker, Fitting scattered data on sphere-like surfaces using spherical splines, J. Comput. Appl. Math. 73 (1996) [4] G.E. Fasshauer, J. Jerome, Multistep approximation algorithms: improved convergence rates through postconditioning with smoothing kernels, Adv. in Comput. Math. 10 (1999) [5] G.E. Fasshauer, L.L. Schumaker, Scattered data fitting on the sphere, in: M. Daehlen, T. Lyche, L.L. Schumaker (Eds.), Mathematical Methods for Curves and Surfaces II, Vanderbilt University Press, 1998, pp [6] M. Hofer, H. Pottmann, Energy minimizing splines in manifolds, ACM Trans. on Graphics 23 (2004) [7] A. Mazroui, D. Sbibih, A. Tijini, A recursive method for construction of tensor product Hermite interpolants, in: T. Lyche, M.L. Mazure, L.L. Schumaker (Eds.), Curves and Surfaces Design, Saint Malo 2002, Nashboro Press, Brentwood, 2003, pp [8] A. Mazroui, D. Sbibih, A. Tijini, Hierarchical computation of bivariate Hermite interpolants, in: T. Lyche, M.L. Mazure, L.L. Schumaker (Eds.), Curves and Surfaces Design, Saint Malo 2002, Nashboro Press, Brentwood, 2003, pp [9] A. Mazroui, D. Sbibih, A. Tijini, A recursive construction of Hermite interpolants and applications, J. Comput. Appl. Math. 183 (2005) [10] A. Mazroui, D. Sbibih, A. Tijini, A simple method for smoothing function and compressing Hermite data, Adv. in Comput. Math 23 (2005) [11] A. Mazroui, D. Sbibih, A. Tijini, Recursive computation of bivariate Hermite spline interpolants, Appl. Numer. Math., in press. [12] M. Neamtu, L.L. Schumaker, On the approximation order of splines on the spherical triangulations, Adv. in Comput. Math 21 (2004) 3 20.

15 A. Lamnii et al. / Journal of Computational and Applied Mathematics 213 (2008) [13] T. Popiel, L. Noakes, C 2 spherical Bézier splines, Comput. Aided Geom. Design 23 (2006) [14] A. Zeni seck, A general theorem on triangular finite C m elements, RAIRO Anal. Numer. 2 (1974) Further Reading [2] P. Alfeld, M. Neamtu, L.L. Schumaker, Bernstein Bézier polynomials on sphere and sphere-like surface, Comput. Aided Geom. Design 13 (1996)