Alfeld, P. Piper, B., and Schumaker, L., (1987) " Minimally Supported Bases for Spaces of Bivariate Piecewise Polynomials of Smoothness r for Degree d ? 4r + 1", Computer Aided Geometric Design, 4, 105 - 123.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....construct an explicit basis for SP m (T ) Lemma 1.3 (Alfeld Schumaker [3] For m 4r 1 and any triangulation T , the dimension of SP m (T ) 3v (2m 1)e f 2(m Gamma 1)ff fi where fi = the number of rectangles triangulated with cross diagonals. Alfeld, Piper Schumaker [2] also construct explicit bases for SP m (T ) for m 4r 1 and for SP 4 [1] Algorithms for the explicit construction and graphics display of C algebraic splines over ST are given in [7, 8] Main Results The main results of this paper are as follows. In section 3 we prove that the dimension ....

Alfeld, P. Piper, B., and Schumaker, L., (1987) " Minimally Supported Bases for Spaces of Bivariate Piecewise Polynomials of Smoothness r for Degree d ? 4r + 1", Computer Aided Geometric Design, 4, 105 - 123.

....coecients in the above Bernstein B ezier representation. It is easy to see, that the cardinality of an admissible set for S r q ( is equal to the dimension of this space. Therefore, each admissible set for S r q ( can be identi ed with a minimal determining set for S r q ( in the sense of [2, 3, 30], and vice versa. The above Bernstein B ezier form can be used to express smoothness conditions of polynomial pieces on adjacent triangles T 1 ; T 2 with vertices v 1 ; v 2 ; v 3 , respectively v 1 ; v 2 ; v 4 . The following result was given by Farin [26] and de Boor [5] Theorem 2.6. Let s be a ....

....It suces to consider the polynomial pieces on the triangles of P which have e ; 1 respectively e ; 2 as an edge. Using Theorem 2.6. it is easy to see that the Bernstein B ezier coecients of the rst three rows w.r.t. e ; e ; 3 and the disc of order 2 around v (for the de nition see [3]) are uniquely determined. By using Theorem 2.6. we obtain a system of linear equations for the remaining Bernstein B ezier coecients on the disc of order 3 around v w.r.t. the triangle with the edges e ; 1 ; e ; 2 . By analyzing this system, we get that the Bernstein B ezier coecients of ....

P. Alfeld, B. Piper and L.L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d  4r + 1, Comp. Aided Geom. Design 4 (1987), 105-123.

....6) Figure 3.3: A cell. The results of Morgan and Scott were extended to spline spaces S r q ( q 4r 1. These extensions are based on the results of Schumaker [151] for spline spaces on cells coupled with the methods developed by Alfeld and Schumaker [7] and Alfeld, Piper, and Schumaker [11] (see also Carncier and Pe na [29] For a generalization to trivariate splines of degree at least 8r 1 on tetrahedral partitions, see Alfeld, Schumaker, and Sirvent [13] In these papers, B ezier Bernstein techniques were used and the concept of minimal determining sets was 7 introduced. ....

P. Alfeld, B. Piper, and L.L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d  4r + 1, Comp. Aided Geom. Design 4 (1987) 105-123.

....the basis functions constructed are minimally supported. We say Omega is a standard cell if it is triangulated with precisely one interior vertex v such that every boundary vertex is connected to v by a interior edge. We state the definition of minimally supported basis as following(see [2]) Definition 4. 1 A basis of S k; Delta is called a minimally supported basis, if the support of each spline in the basis is a subset of a standard cell. As before, we use P = P k ( Delta) to denote the set of all B net points on partition Delta, and for s 2 S 0 k; Delta , b s is its B net ....

Alfeld, P., B. Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree k 4r + 1, Computer Aided Geometric Design 4(1987), 105-123.

....coefficients are bounded by K max 2M jc j, where K is a constant dependening only on d and the smallest angle 4 in 4. If M is a MDS for S on all of D d;4 , we simply call it a minimal determining set. The algorithms presented here for constructing bases is based on the following well known (cf. [6]) result: 2 Algorithm 2.1. Suppose M is a MDS for S. For each 2 M, construct the unique spline B 2 S satisfying j B = ffi ;j ; all j 2 M: 2:1) Then the set fB g 2M is a basis for S. We call it the dual basis corresponding to M. Discussion: To construct the spline B , choose c = 1, ....

Alfeld, P., B. Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d 4r + 1, Comput. Aided Geom. Design 4 (1987), 105--123.

....; i; j = 1; 2; 3; 4; s j = 0; for all 2 N4 n R 2 (v) Clearly, supp s j = T j Gamma1 [ T j [ T j 1 ; and we can write (see [2,9] s j = X i2I j c [j] i B i ; where I j : fi : supp B i ae supp s j g. We now consider the spline s = Gammas 1 s 2 Gamma s 3 s 4 = Gamma X i2I1 c [1] i B i X i2I2 c [2] i B i Gamma X i2I3 c [3] i B i X i2I4 c [4] i B i = X i2I1[I2[I3[I4 a i B i : Using the smoothness conditions (9) it is easy to see that v i;p s = 0; v i;m s = Gamma1) i ; i = 1; 2; 3; 4; s = 0; for all 2 N4 n R 2 (v) and thus ksk 1 K ....

....we have to choose i 0 more carefully. The basis constructed in Algorithm 4 is not locally linearly independent. To get an LLI basis, step 4) has to be modified in a different way, see [4] Remark 4. Star supported bases were constructed for general spline spaces srd S r d (4) for d 4r 1 in [1], and for d 3r 2 in [10,11] The constructions were based on Bernstein B ezier techniques, and are not stable for triangulations that contain near degenerate edges and or near singular vertices. Remark 5. In [7] we use Bernstein B ezier techniques to construct stable local general bases for ....

Alfeld, P., B. Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d 4r + 1, Comput. Aided Geom. Design 4 (1987), 105--123.

.... They are also of interest since an LLI basis B for S r d (4) is a least supported basis in the sense that it is optimal with respect to the size of the supports of the B i , see [7] Locally supported bases have been constructed for the spline spaces S r d (4) and their superspline subspaces in [3,4,16,17,20,24], but they are mostly not LLI, 1) Mathematical Institute, Justus Liebig University, D 35392 Giessen, Germany, oleg.davydov math.uni giessen.de 2) Department of Mathematics, Vanderbilt University, Nashville, TN 37240, s mars.cas.vanderbilt.edu. Supported by the National Science Foundation under ....

....in Sect. 6 and Sect. 7, respectively. Sect. 8 is devoted to a few remarks. x2. The space S 0 d (4) For the sake of completeness and in order to set some notation, we briefly describe the situation for the spline space S 0 d (4) We make use of standard Bernstein B ezier methods as in [3,4,16,17]. Given a triangle T = hv 1 ; v 2 ; v 3 i, the points T ijk : iv 1 jv 2 kv 3 d ; i j k = d; are called the domain points. Each polynomial of degree d can be written in the Bernstein B ezier (B ) form p = X i j k=d c ijk B d ijk ; where B d ijk are the Bernstein polynomials of ....

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Alfeld, P., B. Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d 4r + 1, Comput. Aided Geom. Design 4 (1987), 105--123.

....in the third step, we find that the coefficient associated with the point at v (marked with an open circle) is also zero. The following restatement of Lemma 3.3 of [7] was used in the proof of Theorem 2.1, and will also be used again later. Lemma 2.2. Let T [1] hv 0 ; v 1 ; v 2 i and T [2] = hv 0 ; v 2 ; v 3 i be two triangles sharing the common edge e : hv 0 ; v 2 i. Suppose p 1 ; p 2 are polynomials of degree d on T [1] T [2] which join together with C k smoothness across the edge e for some 0 k d. Given k j d, suppose that all coefficients of p 1 and p 2 in the ....

....of Lemma 3.3 of [7] was used in the proof of Theorem 2.1, and will also be used again later. Lemma 2.2. Let T [1] hv 0 ; v 1 ; v 2 i and T [2] hv 0 ; v 2 ; v 3 i be two triangles sharing the common edge e : hv 0 ; v 2 i. Suppose p 1 ; p 2 are polynomials of degree d on T [1] T [2] which join together with C k smoothness across the edge e for some 0 k d. Given k j d, suppose that all coefficients of p 1 and p 2 in the set D j Gamma1 (v 0 ) are zero, and define c i =c [1] d Gammaj;i;j Gammai c Gammai =c [2] d Gammaj;j Gammai;i i = 0; j: Suppose ....

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Alfeld, P., B. Piper, Schumaker, and L. L., Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d 4r+1, Comput. Aided Geom. Design 4 (1987), 105--123.

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Alfeld, P., B. Piper, and L. L. Schumaker, Minimally supported bases for spaces of bivariate piecewise polynomials of smoothness r and degree d 4r + 1, Comput. Aided Geom. Design 4 (1987), 105--123.

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