Alfeld, P. and Schumaker, L.: On the dimension of bivariate spline spaces of smoothness r and degree d 1. Numerische Mathematik 57, 651--661 (1990)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be handed over to boys or girls under the age of 1 for fear of injury from the sharp corner. Matching derivatives clearly do not always imply smoothness. Conversely, smoothness does not imply matching derivatives. The C of VC is parametrized by the two quadratic pieces, u; v 2 [0; 1] q 3 (u) [ 3 2 ] (1 u) 2 [ 0 2 ] 2(1 u)u [ 0 0 ] u 2 and q 4 (v) 0 0 ] 1 v) 2 [ 0 1 ] 2(1 v)v [ 3 1 ] v 2 : The C is visually (and geometrically) smooth at the common point q 3 (1) 0 0 ] since the two pieces have the same vertical tangent line but the derivatives do not agree: Dq ....

....boys or girls under the age of 1 for fear of injury from the sharp corner. Matching derivatives clearly do not always imply smoothness. Conversely, smoothness does not imply matching derivatives. The C of VC is parametrized by the two quadratic pieces, u; v 2 [0; 1] q 3 (u) 3 2 ] 1 u) 2 [ 0 2 ] 2(1 u)u [ 0 0 ] u 2 and q 4 (v) 0 0 ] 1 v) 2 [ 0 1 ] 2(1 v)v [ 3 1 ] v 2 : The C is visually (and geometrically) smooth at the common point q 3 (1) 0 0 ] since the two pieces have the same vertical tangent line but the derivatives do not agree: Dq 3 ) 1) 0 4 ] 6= ....

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Peter Alfeld and Larry L. Schumaker. On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1. Numerische Mathematik, 57(6/7):651--661, July 1990.

.... Alfeld et al. 12] such complex arguments do not have to be used if is a non degenerate triangulation (i.e. a triangulation that contains no degenerate edges, see Section 2) For this class of triangulations , the dimension of S r 3r 1 ( r 2, has been determined by Alfeld and Schumaker [8]. By Euler s formulas (see Section 2) the lower bound in (3) denoted by lb 1 3 for S 1 3 ( can be written as follows lb 1 3 = 3VB 2V I 1: 7) Therefore, the dimension of S 1 3 ( is larger than the number of vertices of . This is in contrast to the case of quadratic C 1 splines ....

P. Alfeld and L.L. Schumaker, On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1, Numer. Math. 57 (1990) 651-661.

....and using the fact that p (j Gamma1) 1 ) 0 for j = 1; m, we conclude that p j 0. We will apply Theorem 5 to the hexagonal triangulation 4H where n = 6 and e = 3. In this case we take M = f2; 3; 4; 5; 6; 1g. 6 p. alfeld and l. l. schumaker Ring T [1] T [2] T [3] T [4] r = 2m r = 2m 1 R 0 (v) 1 0 0 0 R 1 (v) 2 0 0 0 . R r (v) r 1 0 0 0 R r 1 (v) r 2 1 0 0 . R r m (v) r m 1 m 0 0 R r m 1 (v) r m 2 m 1 1 0 R r m 2 (v) r m 3 m 2 3 2 . R 2r (v) 2r 1 r 2m ....

....includes all of the points on that ring in triangle T [1] The number of such points is i 1 and is listed in the second column of the table. In addition, for each i = r 1; r k, Gamma also includes the last i Gamma r points on R i (v) in the triangles T [2] T [3] T [4] . The numbers of these points are shown in the third column of the table. If r = 2m, Gamma also includes the last 2(i Gamma m) Gamma 1 points on R r i (v) in the triangles T [5] for i = m 1; k. These points are shown in the fourth column of the table. Finally, if r = 2m 1, Gamma ....

P. Alfeld and L.L. Schumaker, On the dimension of bivariate splines spaces of smoothness r and degree d = 3r + 1, Numer. Math., 3 (1990), pp. 651-661.

....Note that the above proof also applies to the superspline spaces S r;ae d (4) whenever ae v 2r for some near singular vertex v. On the other hand, if d 4r 1 and ae v 2r for all vertices, then the basis constructed in [23] is both stable and LLI. x8. Remarks Remark 8.1. It is well known [5 8,22,23] that the dimension of spline spaces and superspline spaces (when ae v 2r) generally depends on the exact geometry of the triangulation, and in particular may change as certain near singular vertices become singular. Thus, it may seem surprising that it is possible to construct stable bases even ....

Alfeld, P. and L. L. Schumaker, On the dimension of bivariate splines spaces of smoothness r and degree d = 3r + 1, Numer. Math. 57 (1990), 651--661.

....with the boundary conditions. Note that d Gamma r Gamma 1 = r k. To find a set Gamma which determines S r d (4H ) on D d Gammar Gamma1 , we identify the domain points of s 2 S r d (4H ) lying in D d Gammar Gamma1 with the domain points of a spline Ring T [1] T [2] T [3] T [4] r = 2m r = 2m 1 R 0 (v) 1 0 0 0 R 1 (v) 2 0 0 0 . R r (v) r 1 0 0 0 R r 1 (v) r 2 1 0 0 . R r m (v) r m 1 m 0 0 R r m 1 (v) r m 2 m 1 1 0 R r m 2 (v) r m 3 m 2 3 2 . R 2r (v) 2r 1 r 2m ....

....includes all of the points on that ring in triangle T [1] The number of such points is i 1 and is listed in the second column of the table. In addition, for each i = r 1; r k, Gamma also includes the last i Gamma r points on R i (v) in the triangles T [2] T [3] T [4] . The numbers of these points are shown in the third column of the table. If r = 2m, Gamma also includes the last 2(i Gamma m) Gamma 1 points on R r i (v) in the triangles T [5] for i = m 1; k. These points are shown in the fourth column of the table. Finally, if r = 2m 1, ....

Alfeld, P. and L. L. Schumaker, On the dimension of bivariate splines spaces of smoothness r and degree d = 3r + 1, Numer. Math. 57 (1990), 651--661.

....faces. Then the dimension of H r 3r 1 (T ) is given by (2.13) or (2.14) depending on whether T is total or partial. Moreover, there exists a basis with local supports as in Theorem 4.4. Proof: A minimal determining set can be constructed by an obvious adaptation of the prescription given in [8] for the planar case. It is also of interest to consider certain generic decompositions, see [10] for the planar case. 15 Fig. 3. Division of domain points, d = 23, r = 5. Definition 6.2. A trihedral decomposition T is said to be generic with respect to r and d provided that for all sufficiently ....

Alfeld, P., and L. L. Schumaker, On the dimension of bivariate splines spaces of smoothness r and degree d = 3r + 1, Numer. Math. 57 (1990), 651--661.

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Alfeld, P. and Schumaker, L.: On the dimension of bivariate spline spaces of smoothness r and degree d 1. Numerische Mathematik 57, 651--661 (1990)

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P. Alfeld and L. Schumaker, On the dimension of bivariate spline spaces of smoothness r and degree d =3r+1,Numerische Mathematik 57 (1990), 651-661.

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P. Alfeld and L. Schumaker, On the dimension of bivariate spline spaces of smoothness r and degree d =3r+1,Numerische Mathematik 57 (1990), 651-661. 19

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Alfeld, P. and Schumaker, L., (1990) "On the Dimension of Bivariate Spline Spaces of Smoothness r and Degree d = 3r + 1", Numerische Mathematik, 57, 651-661.

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Alfeld, P., and L. L. Schumaker, On the dimension of bivariate spline spaces of smoothness r and degree d = 3r + 1, Numer. Math. 57 (1990), 651--661.

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