On the Dimension of the Bivariate Spline Space S 1 3( )

Transcription

1 On the Dimension of the Bivariate Spline Space S 1 3( ) Gašper Jaklič Institute of Mathematics, Physics and Mechanics University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Gasper.Jaklic@fmf.uni-lj.si July 8, 2004 Abstract In the paper the problem of determining the dimension of the bivariate spline space S 1 3( ) is considered. It is shown that for a large class of triangulations the dimension of the spline space is equal to the Schumaker s lower bound. An algorithm is presented, which decides if a given triangulation belongs to such a class. Keywords: dimension, spline space, triangulation. C. R. Categories: G Introduction The space of bivariate splines has a complex structure. Even such basic problems as determining its dimension and construction of its basis are still unsolved in general. The main problem is the fact that the dimension depends not only on the topology (graph of the triangulation), but also on the geometry (exact vertex positions) of the triangulation. The space of cubic C 1 splines S 1 3( ) is of particular interest, since it is the smallest space that allows interpolation at vertices of a triangulation. For numerical computations, it is essential to know the dimension of S 1 3( ) in advance. The interpolation or approximation problem can then be attacked with a proper number of free parameters, avoiding a numerical guess of the dimension. Let Ω R 2 be a closed, simply connected polygonal region and let := {Ω i } N i=1, Ω = 1 N i=1 Ω i

2 denote a regular triangulation of Ω with N := N( ) := triangles. A triangulation is regular, if two triangles Ω i, Ω j can have only one vertex or the whole edge in common. Let Π n (R 2 ) denote the space of bivariate polynomials of total degree n. Let S r n( ) := {f C r (Ω); f Ωi Π n (R 2 ), i = 1, 2,..., N} be the spline space over the triangulation. The dimension of S3( ) 1 has been determined for special triangulations only (triangulations of type 1 and 2 ([1, 2]), nested polygon triangulations([3]), etc.). The lower and the upper bound on the dimension are given in [1, 2]. Some improvements on the upper bound have been obtained in [4, 5]. It has been conjectured, that the dimension is equal to the Schumaker s lower bound dim S 1 3( ) 3V B ( ) + 2V I ( ) + σ( ) + 1, (1) where V B ( ) denotes the number of boundary vertices, V I ( ) the number of internal vertices, and σ( ) = V I ( ) i=1 σ i, { 1, if vertex is singular, σ i = 0, otherwise. A vertex is singular if it is of degree exactly 4 and is lying on an intersection of two lines (Fig. 1). A triangulation is called a cell, if it has exactly one inner vertex. A vertex is called degenerate, if two of its edges are collinear. Figure 1: A singular vertex T. There are different methods to study the dimension problem: the smoothness conditions can be expressed in standard polynomial or Bernstein-Bézier basis, minimal determining set approach, and blossoming approach. The dimension problem is clearly linear, and any approach leads to symbolic linear relations that have to be studied. Here, the blossoming approach is followed. The problem of determining the dimension is transformed into the problem of computing ranks of particular symbolic matrices that depend on the triangulation. A particularly simple consequence of the results obtained reveals the following corollary. 2

3 Corollary 1.1. Let the degrees of inner vertices of the triangulation be at most 6 and let the degrees of vertices on the outer face be at most 5. Let there be no collinearities of the edges at the inner vertices of. Then the dimension of the spline space S 1 3( ) is equal to the Schumaker s lower bound (1). The main result of the paper, Theorem 2.2, is stronger, and determines the dimension of the spline space over larger class of triangulations. It also provides a basis for a dimension count algorithm. The paper is organized as follows. In section 2 the main results are outlined and in section 3 the dimension count algorithm is presented. In section 4 the proofs are given. We conclude the paper with remarks and an example. 2 The reduction step We will follow the approach in [6], and express the smoothness conditions in the blossoming form ([6], Thm. 2.1). The smoothness conditions are prescribed over all inner edges of the triangulation. If all the conditions are combined together, a system of linear equations is obtained and its matrix plays the key role in the dimension problem. The rank of this matrix gives the number of independent conditions. The cubic case is simplified to where dim S 1 3( ) = 7 N + 3 rank M, (2) [ ] M11 M M := M( ) := 12, 0 M 22 and M km are block matrices with E I block rows and N block columns. Each block row l corresponds to the smoothness conditions over the edge between the neighbouring triangles Ω i, Ω j. In every block row there are 2 nonzero blocks Q km,li and Q km,lj = Q km,li, and the matrices Q km,lj are circular matrices of size 2 (m + 2). Figure 2: Neighbouring triangles and the notation. Let v l = (α l, β l ) denote a normalized directional vector of the edge e l between triangles Ω i and Ω j, and let t l = (c l, d l ) denote some point on the edge e l (Fig. 2). Further let v i v j := (α i β j α j β i ) 3

4 denote the planar vector product. It is equal to 0 iff the vectors v i and v j are collinear. Throughout the paper it will be assumed that the triangulation is in general position, i.e. α i 0, β i 0, for all i. This can be simply achieved by using a proper rotation. In the block matrix M 11, the blocks read [ ] αl β Q 11,li = Q 11,lj = l 0, 0 α l β l and in M 22 [ ] α 2 Q 22,li = Q 22,lj = l 2α l β l βl αl 2 2α l β l βl 2. In the block M 12 there are not only directions, but also some vertices of the triangulation are involved. It is possible to simplify some of the blocks without changing the rank of M by choosing the points t l := (c l, d l ) at the inner vertices of the triangulation, and choosing some arbitrary additional points z k := (x k, y k ) R 2, k = 1, 2,..., N ([6], Lemma 3.1). The block Q 12,li is of the form αl (c l x i) α l (d l y i) + β l (c l x i) β l (d l y i) 0, 0 α l (c l x i ) α l (d l y i ) + β l (c l x i ) β l (d l y i ) (3) and the block Q 12,lj reads αl (c l x j ) α l (d l y j ) + β l (c l x j ) β l (d l y j ) 0 0 α l (c l x j ) α l (d l y j ) + β l (c l x j ), β l (d l y j ) (4) so the choice z k = t l, k {i, j} reduces (3) or (4) to zero block. Of course, not all blocks can be simplified in this way. It is very unlikely that the rank of a large symbolic matrix M R 4E I 7N could be determined in general. So it is perhaps the best idea to look for sufficient conditions that allow the problem to be reduced to a smaller one, and carry such step onto the smaller problem. Figure 3: A triangulation = 1 \ 1. 4

5 Here is one of the possible approaches. Pick a vertex T 0 on the outer face of the triangulation and let 1 denote a simply connected subtriangulation of that includes all the triangles, attached to the vertex T 0, and let \ 1 denote the rest of the triangulation (Fig. 3). The matrix M could be written as M = M( ) = M( 1) 0 M( 1, ) M(, 1 ). (5) 0 M( \ 1 ) The matrices M( 1, ), M(, 1 ) represent the common part of the smoothness conditions between 1 and \ 1, M( 1 ) represents the conditions inside 1 and M( \ 1 ) the conditions inside \ 1. Note that has at least 3 boundary edges, therefore 3N 2E I + 3. This implies that the matrix M has at most as many rows as columns If a submatrix 4E I 6(N 1) 7N. M s ( 1, ) := [ ] M( 1 ) R r c M( 1, ) satisfies the condition r c and the matrix M( 1, ) is of the full rank, then rank M( ) = rank M s ( 1, ) + rank M( \ 1 ). (6) Let now by inductive supposition the dimension of the spline space over \ 1 be equal to the lower bound dim S 1 3( \ 1 ) = 3V B ( \ 1 ) + 2V I ( \ 1 ) + σ( \ 1 ) + 1. Then (2) and (6) imply dim S 1 3( ) = 7 1 rank M s ( 1, ) + 3V B ( \ 1 )+ Therefore, if one is able to prove the inequality 2V I ( \ 1 ) + σ( \ 1 ) + 1. rank M s ( 1, ) (V B ( \ 1 ) V B ( )) + 2 (V I ( \ 1 ) V I ( )) + (σ( \ 1 ) σ( )), (7) the dimension dim S3( ) 1 will be equal to the lower bound too. At the last reduction step one is left with one triangle, i.e., N = 1, V B = 3, V I = 0, σ = 0, and 3V B + 2V I + σ + 1 = 10 = 7N + 3, so the induction hypothesis is satisfied in the beginning. Let B denote the intersection of 1 and \ 1. The subtriangulation 1 is proper, if it is simply connected, has a vertex T 0 on the outer face of and 5

6 contains all the triangles in with the vertex T 0, no triangle in 1 has two edges on B, there are no consecutive pairs of collinear edges at the vertices of degree 3 in 1 on B, and every singular vertex in 1 lies in the interior of 1. The following theorem describes the application of the reduction step in order to determine dim S 1 3( ). Theorem 2.1. Suppose that 1 is a proper subtriangulation of that satisfies V B ( 1 ) 2V I ( 1 ) + 6. If rank M s ( 1, ) = r σ( 1 ) and dim S 1 3( \ 1 ) is equal to the lower bound, then the dimension dim S 1 3( ) is equal to the lower bound (1) too. We call a sequence of reduction steps that start in a vertex on the outer face of, and remove proper subtriangulations 1 at the boundary till the end, satisfying the conditions of Theorem 2.1, a reduction path. There can be more than one reduction path for a given triangulation. Corollary 2.1. If there exists a reduction path for the triangulation, then the dimension dim S 1 3( ) is equal to the lower bound (1). Here is the main result of the paper. It provides simple conditions on the triangulation, where Theorem 2.1 can be applied, and will be the core of the algorithm, described in section 3. Theorem 2.2. Let 1 be a proper subtriangulation of with a vertex T 0 of degree s + 1 on the outer face of. If 1) s 4, 1 = s, (Fig. 4), or 2) s = 2, 3, 1 = s + 2, and 1 includes a cell (Fig. 5, Fig. 6), and dim S 1 3( \ 1 ) is equal to the lower bound, then the dimension dim S 1 3( ) is equal to the lower bound (1) too. Particularly simple sufficient conditions on the triangulation, where the reductions of Theorem 2.2 can be applied, are given in Corollary 1.1. Remark: Euler s formula implies that the average vertex degree in a triangulation is less than 6, and at least (V I + V B )/2 vertices are of degrees 8. Since the reduction steps decrease some vertex degrees, a reduction path exists for a large class of triangulations. Every triangulation includes a vertex of degree < 5, but unfortunately not necessary on the outer face, so Theorem 2.2 can not be used for the triangulations that have all the vertices on the outer face of high degrees. 3 An algorithm for the reduction of the triangulation By using Theorem 2.2 we can give an algorithm that determines if the triangulation belongs to the class of triangulations, where the dimension dim S 1 3( ) can be obtained by sequential reductions of the triangulation. 6

7 We are given a triangulation. First, it is rotated to a general position, such that no edge lies on the coordinate axes. Then the reduction step can be used on every boundary vertex that satisfies one of the conditions of Theorem 2.2. An algorithm for the reduction of the triangulation // T - a given triangulation // deg(v) - degree of the vertex v list<vertex> L = {list of all vertices of degree <=5 on the outer face of the triangulation T}; while (!L.empty()) { if (T=cell of degree 4 or T=triangle) break(success); check for cycling in L and continue/break(failure); v=l.pop(); check for collinearities between neighbours of v on the cut; check that all edges on the cut are inner edges; if (no collinearities and all edges are inner edges) { T.delete_vertex(v); if any neighbour(v) has new degree <=5, add it to list L; } else if (collinearity at the inner vertex z, deg(z)=4, deg(v)<5, and all edges on the cut are inner edges){ T.delete_vertex(v); T.delete_vertex(z); add neighbours on the cut with degree <=5 to L; } else { L.append(v); } } if (success) {//no cycling print("dimension equals Schumaker s lower bound"); } else {//cycling print("algorithm can not be used"); } } If all the vertices in the list L were checked, and no reduction could be applied, the list L remains the same, cycling occurs and the algorithm stops. In such a case (because of too large degrees of the boundary vertices or because of the collinearities) this method can not be applied for the study of the dimension. If the while loop successfully terminates, a reduction path was found (the sequence of deleted vertices v), and the dimension dim S 1 3( ) is equal to the Schumaker s lower bound (1). The answer of the algorithm is quite clearly 7

8 independent of the enumeration of the vertices, i.e., on a particular sequence of reductions. The algorithm was implemented in C++, using LEDA (Library of Efficient Data structures and Algorithms), [7]. An example of the steps of the algorithm is shown in section 5. 4 Proofs Proof (Theorem 2.1): Let us recall observations (2), (5) and (6), and let us conclude the proof of Theorem 2.1. The matrix M( 1, ) is of the full rank, since the assumptions imply that it is block upper triangular with blocks of the form Q 11,li and Q 22,li on the diagonal. Euler s formula for a triangulation implies The condition r c reads and by (8) the condition E B = V B, E I = 3 V I + V B 3, N = 2 V I + V B 2. (8) 4 (E I ( 1 ) + (E B ( 1 ) 2)) 7 N( 1 ), V B ( 1 ) 2 V I ( 1 ) + 6, or N( 1 ) 2V B ( 1 ) 8 is obtained. The assumptions on the triangulation 1 are necessary. Some of them follow from the idea of the reduction and from the structure of the matrix M, the others depend on the geometry of the triangulation. Let V (B) denote the number of vertices on B. Since and by assumption V B ( \ 1 ) = V B ( ) + V (B) 3, V I ( \ 1 ) = V I ( ) V I ( 1 ) V (B) + 2, σ( ) = σ( 1 ) + σ( \ 1 ), V (B) = V B ( 1 ) 1, (7) is satisfied. If there is a collinearity of a consecutive pair of edges at the vertex of degree 3 in 1 on B or if there is a triangle in 1 with 2 edges on B, it is easy to see, that such configuration causes the diminishment of the rank of M s ( 1, ), therefore the reduction step can not be applied, since the inequality (7) is not satisfied. Proof (Theorem 2.2): The proof of Theorem 2.2 requires the calculation of ranks of large symbolic matrices. It would be nice to use implemented algorithms in symbolic packages, such as Mathematica or Maple, but the problems are too large. In order to calculate the ranks other known algorithms, based on the special structure of the 8

9 matrices, were implemented: Gaussian elimination, Laplacian decomposition ([8]), Chio s pivotal condensation ([9]), and the methods for calculation of large determinants ([10]). The most successful turned out to be the method of the calculation of special minors, factorization of polynomials and careful use of Gaussian elimination. Here is a general idea: to prove that a given matrix is of the full rank r, it is enough to find a suitable minor of size r. Factorization of such minor proves that the matrix is of the rank r in general, and gives conditions on, in which the rank can be lower. In order to calculate the rank (in degenerate cases) we find a subminor that can never be zero. The change of the rank in the submatrix M( 1 ) is shown by using Gaussian elimination. Since the matrices have polynomial entries, all minors are multivariate polynomials, and the results can be easily verified by evaluating the determinant in sufficient number of points and using the theory on polynomial interpolation. First, let us consider the case 1) (Fig. 4). Note that 1 = s, 1 = s i=1 Ω i and T 0 is a common vertex to all Ω i. The edges e l := T l T 0, l = 1, 2,..., s 1 Figure 4: Part of the triangulation at the boundary vertex T 0. are chosen to be the inner edges of 1 and the edges e l, l = s, s + 1,..., 2s 1, belong to B, the common border of 1 and \ 1. The choice of vertices t l is given as t := (t l ) = (T 1, T 2,..., T s 1, T 1, T 1, T 2,..., T s 1,,,..., ), }{{} E I 2s+1 and the vertices z k that simplify M 12 are selected as z := (z k ) = (T 1, T 1, T 2,..., T s 1,,,..., ). }{{} N s Let us show that M s ( 1, ), s 4, is of the full rank. If s = 1, the matrix M 1 ( 1, ) = [ Q11,11 Q22,11] 9 4 7

10 Q 11,63 Q 11,74 Q 22,21 Q 22,32 Q 22,63 Q 22,74 is of the full rank, since rank Q kk,lj = 2. Let us recall Fig. 2. Then [ ] Q22,li det = (v Q l v m ) 4 0, (9) 22,mi and det Q 11,li Q 11,mi Q 11,lj Q 11,nj But (9) and (10) imply that the matrix M 2 ( 1, ), 2 = (v l v m )(v m v n )(v n v l ) 0. (10) Q 11,11 Q 11,12 Q 22,11 Q 22,12 3 fm 2 ( 1, ) = 64 Q 11,21 Q 11,32 75, is of the full rank. If s = 3, M 3 ( 1, ) is given as 2 fm 3 ( 1, ) = Q 11,31 Q 11,11 Q 11,12 Q 11,22 Q 11,23 Q 12,22 Q 22,11 Q 22,12 Q 22,22 Q 22,23 Q 11, Q 11,53 Q 22,31 75 Q 22,42 Q 22, By (9) one can omit block rows 3,4,8,10, and block columns 4 and 6, and consider the rest of the matrix. If the last column of the matrix Q 11,23 Q 12,22 Q 11,53 75 Q 22,42 is omitted, one gets 6 6 minor 6 7 α 3 4 e 4 (v 4 v 2 )(v 2 v 5 )(v 5 v 4 ) 0. Together with (10) this establishes that the matrix M 3 ( 1, ) is of the full rank. In the last case, s = 4, the matrix M 4 ( 1, ) is a square matrix, fm 4 ( 1, ) = 2 Q 11,41 Q 11,11 Q 11,12 Q 11,22 Q 11,23 Q 12,22 Q 11,33 Q 11,34 Q 12,33 Q 22,11 Q 22,12 Q 22,22 Q 22,23 Q 22,33 Q 22,34 Q 11,52 3, 64 Q 22,41 Q 22, and has determinant det M 4 ( 1, ) = e 5 e 6 ( (v 1 v 4 ) 5 (v 1 v 5 )(v 2 v 5 ) 3 (v 2 v 6 ) 3 (v 3 v 6 )(v 3 v 7 ) 5 (v 4 v 5 )(v 5 v 6 ) 2 (v 6 v 7 ) ) 0. 10

11 Since rank M s ( 1, ) = 4 (2s 1), 1 = s, V B ( \ 1 ) = V B ( ) + s 2, V I ( \ 1 ) = V I ( ) s + 1, and σ( \ 1 ) = σ( ) by assumption, (7) is satisfied. This concludes the first part of the proof of Theorem 2.2. Now let us consider the second part of the proof. Let s = 2. Then 1 = 4 and 1 is a cell of degree 4 (Fig. 5). Here the points on the edges and the additional Figure 5: Cell of degree 4 at the boundary (s = 2). points for the faces of the triangulation are chosen in a slightly different way, such that the smoothness conditions for the inner vertex of degree 4 are included in M( 1 ) : e 1 : T 1, e 2 : T 1, e 3 : T 1, e 4 : T 1, e 5 : T 2, e 6 : T 2, and for the faces: Ω 1 : T 1, Ω 2 : T 1, Ω 3 : T 1, Ω 4 : T 1 (Fig. 5). With this choice one is left only with two nonzero blocks Q 12,53, Q 12,64 in M 12. The matrix M 2 ( 1, ) reads: 2 Q 11,11 Q 11, Q 11,21 0 Q 11, Q 11,32 0 Q 11, Q 11,43 Q 11, Q 11, Q 12, Q 11, Q 12, Q 22,11 Q 22, Q 22,21 0 Q 22, Q 22,32 0 Q 22, Q 22,43 Q 22, Q 22, Q 22,64 If one omits the columns 6, 9, 19, 20, one gets a minor of size 24: α 4 3α 5 α 6 (v 2 v 1 ) 5 (v 3 v 4 )(α 2 α 3 (v 4 v 1 ) + α 1 α 4 (v 3 v 2 ))(v 5 v 4 ) 2 (v 6 v 4 ) 2 (v 6 v 5 ) 2 (β 5 (c 1 c 2 ) + α 5 ( d 1 + d 2 ))(β 6 (c 1 c 2 ) + α 6 ( d 1 + d 2 )). (11) Since a vector w l := (β l, α l ) is orthogonal to v l = (α l, β l ), the geometry of the triangulation implies β l (c 1 c 2 ) + α l ( d 1 + d 2 ) = w l, T 1 T 2 = 0, l = 5,

12 Therefore the minor (11) is nonzero, except in the case, when α 2 α 3 (v 4 v 1 ) + α 1 α 4 (v 3 v 2 ) = 0. This happens iff T 1 is a singular vertex. So, in the nonsingular case, the matrix is of the full rank, otherwise the rank changes for at least 1. But, if one takes submatrix (by omitting the row 1, and the columns 3,6,9,19,20) one gets the minor 2α 1 α 2 2α 5 3α 4 α 5 α 6 (v 1 v 2 ) 4 (v 3 v 4 )(v 5 v 4 ) 2 (v 6 v 4 ) 2 (v 6 v 5 ) 2 (β 5 (c 1 c 2 ) α 5 (d 1 d 2 ))(β 6 (c 1 c 2 ) α 6 (d 1 d 2 )) 0. By using Gaussian elimination on the first 8 12 block M( 1 ) it is shown, that M( 1 ) is of the full rank 8, except in the singular case, when the rank equals 7. The independence of the rest of the matrix follows from the structure of the matrix and the results on the rank. Now let s = 3. The assumptions imply that 1 is a cell of degree 4 with an additional triangle (Fig. 6). Similarly to the case s = 2, the points on the Figure 6: Cell of degree 4 with an additional triangle at the boundary (s = 3). edges are chosen as e 1 : T 1, e 2 : T 1, e 3 : T 1, e 4 : T 1, e 5 : T 2, e 6 : T 3, e 7 : T 3, e 8 : T 2 and for the faces as Ω 1 : T 1, Ω 2 : T 1, Ω 3 : T 1, Ω 4 : T 1, Ω 5 : T 2 (Fig. 6). This particular choice leaves only 3 nonzero blocks in M 12, namely Q 12,52, Q 12,63, Q 12,74. Let us shorten the notation: Q kl ij := Q kl,ij. Now the matrix M 3 ( 1, ) reads 12

13 2 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q The change of the rank in M( 1 ) in the singular case can be proved as in the previous case. The special structure of the matrix allows a simplification of the computation of the rank. By (9) the submatrix [ Q T 22,55 Q T 22,85] T is of the full rank, therefore the rows 25,26,31,32 and columns 32,33,34,35 can be omitted. Similarly, the submatrix [ Q T 22,11 Q T 22,21] T is of the full rank and the rows 17,18,19,20 and columns 16,17,18,19 of the reduced matrix can be omitted. One is left with the matrix of size By omitting the columns 1,10 and 17 one obtains the minor 2 e 3 e 4 2 α 3 β 2 3β 6 β 7 ( α 4 β 1 β 2 β 3 +α 3 β 1 β 2 β 4 α 2 β 1 β 3 β 4 +α 1 β 2 β 3 β 4 )(v 3 v 1 ) (v 4 v 2 )(v 3 v 5 )(v 6 v 4 ) 3 (v 7 v 4 ) 3 (v 7 v 6 ) 2 (v 3 v 8 )(v 5 v 8 ). (12) Assume v 3 v 8 0. Thus the minor (12) is nonzero, except in the case, when α 4 β 1 β 2 β 3 + α 3 β 1 β 2 β 4 α 2 β 1 β 3 β 4 + α 1 β 2 β 3 β 4 = 0. This happens iff T 1 is a singular vertex. But, if one takes a subdeterminant, by omitting the row 1 and the columns 1,10,17,24, one gets e 3 e 4 α 2 3β 2 2β 2 3β 4 β 4 7(v 1 v 3 )(v 3 v 5 )(v 6 v 4 ) 5 (v 7 v 4 )(v 6 v 7 )(v 3 v 8 )(v 5 v 8 ) 0. If the edges e 3 and e 8 are collinear one must consider two cases. If the edges e 2 and e 3 lie in a general position (v 2 v 3 0), then the omission of the columns 16,19, and 24 yields the minor 3 e 4 α 2 3β 2 3β 3 7(v 2 v 1 )(v 3 v 1 )(v 2 v 3 )(v 3 v 4 )(v 3 v 5 ) (v 4 v 6 ) 5 (v 4 v 7 )(v 5 v 7 )(v 6 v 7 ) 0. If the edges e 2, e 3 and e 8 are collinear, the omission of the columns 13,16,17 yields the minor e 4 2 β 6 2(v 1 v 2 )(v 4 v 1 )(v 4 v 2 ) 2 (v 5 v 2 )(v 4 v 6 ) 3 (v 7 v 4 ) 2 (v 5 v 7 )(v 6 v 7 ) 3. (13) 3 13

14 If T 1 is a singular vertex, the minor (13) vanishes, but its subminor, obtained by deletion of the row 3 and the columns 1,13,16,17, reads e 4 α 4 β 2 1β 6 2(v 4 v 2 ) 2 (v 5 v 2 )(v 6 v 4 ) 3 (v 7 v 4 ) 2 (v 5 v 7 )(v 6 v 7 ) 3 0, thus the rank of the matrix M 3 ( 1, ) is reduced by 1. So the matrix M 3 ( 1, ) is of the full rank in general, and the rank is reduced by 1 iff T 1 is a singular vertex. Since rank M s ( 1, ) = 4 (2s + 2) σ( 1 ), 1 = s + 2, V B ( \ 1 ) = V B ( ) + s 2, V I ( \ 1 ) = V I ( ) s, and σ( \ 1 ) = σ( ) σ( 1 ) by assumption, (7) is satisfied. This concludes the proof of Theorem Remarks 1.) The algorithm given in the section 3 determines dim S 1 n( ), n 3 for a large class of triangulations. Since it proves that dim S 1 3( ) is equal to the Schumaker s lower bound, a slightly modified Thm. 3.3 in [6] can be applied. Thus for n 4 our results coincide with the general results on the dimension, given in [11] and [12]. 2.) The methods for the calculation of the rank of large matrices, described in the proof of Theorem 2.2 in section 4, could be used for analyzing larger subtriangulations 1 that satisfy the assumptions of Theorem 2.1. The main problem is the growing size of the matrices, and consequently the calculation of the rank. 3.) The described approach of reducing the triangulation and thus determining the dimension of the corresponding spline space S3( ) 1 could be reversed. This allows the construction of triangulations with the dimension of S3( ) 1 equal to the Schumaker s lower bound (1), therefore useful for the interpolation with cubic C 1 splines at the vertices of the triangulation. Even more, with careful positioning of vertices one can construct highly applicable Delaunay triangulations. The idea of the construction is as follows: One starts with one triangle, and in every step adds a new vertex, and attaches it to the already constructed triangulation (this step is a reduction of Theorem 2.2, applied in reverse). In every step, the new vertex should be positioned in such a way, that the conditions of Theorem 2.2 are satisfied. Thus the sequence of added vertices, read in reverse, determines the reduction path for the triangulation, and the dimension dim S3( ) 1 is equal to the Schumaker s lower bound. 4.) In [3] the dimension of the spline space S3( ) 1 over the nested polygon triangulation is given. The algorithm in [3] depends on the construction of the triangulation and certain conditions should be satisfied. It is interesting to compare our results. Algorithm in [3] can be used only for nested polygon 14

15 triangulations, ours can be used on general triangulations. In the case of nested polygon triangulations one can construct examples of triangulations where only one of the approaches can be applied. In Fig. 7 there is an example, where an approach from [3] fails, but our method can be applied. In Fig. 8 there is an example in the opposite direction. Our method fails, but the method of [3] produces the result Figure 7: The approach from [3] fails, but our approach could be applied. Figure 8: An example, where our method fails, but the method of [3] can be used. 5.) At last, consider an example. Let be a triangulation, shown in Fig. 7. The algorithm applies a sequence of reductions, starting with the vertex 2 and continuing in 1, 0, 6, 5, 8, 12, 10, 3, 11. The resulting triangulation is a triangle. Thus the dimension of the spline space S 1 3( ) is equal to the Schumaker s lower bound, i.e., 34. References [1] L. L. Schumaker, On the dimension of the space of piecewise polynomials in two variables, in Multivariate Approximation Theory, W. Schempp and K. Zeller (eds.), Birkhauser, Basel, 1979, [2] L. L. Schumaker, Bounds on the dimension of spaces of multivariate piecewise polynomials, Rocky Mountain J. of Math., 14 (1984), [3] O. Davydov, G. Nürnberger, and F. Zeilfelder, Cubic spline interpolation on nested polygon triangulations, in Curve and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L. L. Schumaker, (eds.), , Vanderbilt University Press, [4] C. Manni, On the Dimension of Bivariate Spline Spaces on Generalized Quasi-cross-cut Partitions, J. of Approximation Theory 69, 1992,

16 [5] D. J. Riepmeester, Upper bounds on the dimension of bivariate spline spaces and duality in the plane, in M. Daehlen, T. Lyche, L. L. Schumaker (eds.), Mathematical methods for curves and surfaces, Vanderbilt University Press, Nashville, 1995, [6] Z. B. Chen, Y. Y. Feng, J. Kozak, The Blossom Approach to the Dimension of the Bivariate Spline Space, J. Comput. Math., vol. 18, No. 2 (2000), [7] K. Mehlhorn, S. Näher, M. Seel, C. Uhrig, The LEDA user manual, version 4.1, manuscript, [8] S. Karlin, Total Positivity, Stanford University press, Stanford, [9] E. Weisstein, World of Mathematics, [10] C. Krattenthaler, Advanced determinant calculus, Séminaire Lotharingien Combin. 42 (The Andrews Festschrift), [11] P. Alfeld, B. Piper, L. L. Schumaker, An explicit basis for C 1 quartic bivariate splines, SIAM J. Numer. Anal., 24 (1987), no. 4, [12] A. K. Ibrahim, L. L. Schumaker, Super Spline Spaces of Smoothness r and Degree d 3r + 2, Constr. Approx. 7 (1991),