Available online at www.sciencedirect.com Progress in Natural Science 19 (2009) 25–31 www.elsevier.com/locate/pnsc Comparison of CSC method and the B-net method for deducing smoothness condition Renhong Wang, Kai Qu * Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China Received 31 March 2008; received in revised form May 2008; accepted May 2008 Abstract The first author of this paper established an approach to study the multivariate spline over arbitrary partition, and presented the so-called conformality method of smoothing cofactor (the CSC method) Farin introduced the B-net method which is suitable for studying the multivariate spline over simplex partitions This paper indicates that the smoothness conditions obtained in terms of the B-net method can be derived by the CSC method for the spline spaces over simplex partitions, and the CSC method is more capable in some sense than the B-net method in studying the multivariate spline Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences Published by Elsevier Limited and Science in China Press All rights reserved Keywords: Multivariate spline; Smoothing cofactor; Global conformality condition; B-net method; Smoothness condition Introduction Splines are piecewise polynomials with certain smoothness The first author of this paper established the basic theory on multivariate spline over arbitrary partition, and presented the so-called conformality method of smoothing cofactor (the CSC method) which is suitable for studying the multivariate spline over arbitrary partition [1] In this paper we take the bivariate spline as an example to prove that the CSC method and the B-net method are equivalent over simplex partitions The CSC method and the B-net method on bivariate spline spaces are presented in Section In Section 3, we derive the smoothness conditions over triangulation with the CSC method, which are the same as the smoothness conditions presented by Farin [2,3] Finally, we indicate that the CSC method and the * Corresponding author Tel.: +86 411 81892893 E-mail address: qukai8@yahoo.cn (K Qu) B-net method are equivalent for multivariate spline spaces over simplex partitions Bivariate spline spaces Let D be a domain in R2 , P k the collection of all these bivariate polynomials with real coefficients and total degree no more than k, i.e., ( ) k X kÀi X i j cij x y jcij R P k :¼ p ¼ i¼0 j¼0 Using a finite number of irreducible algebraic curves to carry out the partition D of the domain D, then the domain D is divided into N sub-domains d1 ; ; dN , each of such subdomains is called a cell of D These line segments that form the boundary of each cell are called the edges, intersection points of the edges are called the vertices If two vertices are two end points of a single edge, then these two vertices are called the adjacent vertices The vertices which are not lying on the boundary of domain D are called interior 1002-0071/$ - see front matter Ó 2008 National Natural Science Foundation of China and Chinese Academy of Sciences Published by Elsevier Limited and Science in China Press All rights reserved doi:10.1016/j.pnsc.2008.05.030 26 R Wang, K Qu / Progress in Natural Science 19 (2009) 25–31 vertices The space of bivariate spline with degree k and smoothness l over D is defined by S lk Dị :ẳ fs C l Dịjsjdi P k ; i ¼ 1; ; N g Lemma [4] d lk ðN Þ ẳ ! lỵ1 kl N ỵ ! lỵ1 N 1ịk N ỵ 1ịl ỵ N 3ị ỵ N 1ị N À1 ð3Þ 2.1 The conformality method of smoothing cofactor Theorem [1] Let the representation of z ẳ sx; yị on the two arbitrary adjacent cells Di , and Dj be z ẳ pi x; yị; and z ẳ pj x; yị where z ẳ pi x; yị, and z ¼Spj ðx; yÞ P k , respectively In order to let sðx; yÞ C l ðDi Dj Þ, if and only if there is a polynomial qij ðx; yị P klỵ1ịd , such that lỵ1 pi x; yị pj x; yị ẳ ẵlij x; yị qij ðx; yÞ Theorem [4] Let Dqc be a quasi-cross-cut partition of a simply connected region, Dqc have L1 cross-cuts, L2 rays, and V interior vertices A1 ; ; AV Denote by N i ; i ¼ 1; ; V the number of cross-cuts, and rays passing through Ai We have dimS lk Dqc ị ẳ kỵ2 þ L1 kÀlþ1 þ V X d lk N i ị iẳ1 4ị 1ị where Di , and Dj have the common interior edge Cij : lij ðx; yị ẳ where d lk N ị is given in Eq (3) 2.2 The B-net method and the irreducible algebraic polynomial lij ðx; yÞ P d The polynomial qij ðx; yÞ defined by Eq (1) in Theorem is called the smoothing cofactor of sðx; yÞ across Cij from Dj to Di Let A be a given interior vertex over partition D, the conformality condition at A is dened by X lỵ1 ẵlij x; yị qij ðx; yÞ A P where A presents the summation of all the interior edges around A, and qij ðx; yÞ is the smoothing cofactor across Cij Let A1 ; ; AM be all the interior vertices over partition D The global conformality condition is dened by X ẵlij x; yịlỵ1 qij x; yị 0; v ẳ 1; ; M ð2Þ Av Theorem [1] Let D be any partition of D The bivariate spline function sðx; yÞ S uk ðMÞ exists, if and only if for every interior edge, there exists a smoothing cofactor of sðx; yÞ, and the global conformality condition Eq (2) is satisfied Definition [1] The partition D is called a cross-cut partition, if all the edges are lying on some straight lines crosscutting domain D We call a partition to be quasi-cross-cut denoted by Dqc , if each edge in this partition is either a part of cross-cut or a part of rays in D The B-net method is suitable for studying the spline functions over arbitrary simplex partition Now we introduce the main idea of the B-net method of bivariate spline spaces over simplices [3] It is well known that any point x in the plane can be uniquely expressed in terms of barycentric coordinates with respect to any nondegenerate triangle M with vertices v1 ; v2 ; v3 (see Fig 1, left): x ẳ s1 v1 ỵ s2 v2 ỵ s3 v3 where s :¼ ðs1 ; s2 ; s3 ị is usually normalized by the requirement s1 ỵ s ỵ s ẳ and the coecients s :ẳ s1 ; s2 ; s3 ị are called the barycentric coordinates of x over the triangle M We have detðv2 À x; v3 À xÞ ; detðv2 À v1 ; v3 À v1 Þ detðv1 À x; v2 À xị s3 ẳ detv1 v3 ; v2 v3 Þ s1 ¼ detðv1 À x; v3 À xÞ ; detðv1 À v2 ; v3 À v2 Þ An important property of barycentric coordinates is affine invariance v1 Definition [1] The union of all the cells sharing the same interior vertex V is called the relative region (or star-region) of the interior vertex V Let V N be the solution space corresponding to the conformality condition at an interior vertex, where N is the number of lines passing though this interior vertex, and having different slopes The dimension of V N is presented as follows s2 ¼ v2 v1 T Tˆ x v2 vˆ1 v3 v3 Fig Triangle M (left) and two adjacent triangles, T and Tb (right) R Wang, K Qu / Progress in Natural Science 19 (2009) 25–31 Let k :¼ ðk1 ; k2 ; k3 ị; jkj ẳ k1 ỵ k2 ỵ k3 ẳ n; k! ¼ k1 !k2 !k3 ! Bernstein polynomials of degree n over a triangle are defined by n! k n! s ¼ sk1 sk2 sk3 ; k! k1 !k2 !k3 ! ki Z ỵ ; i ẳ 1; 2; Bnk sị ẳ k1 ỵ k2 þ k3 ¼ n; There are many properties of Bernstein polynomials [5], such as (1) (2) (3) (4) Bnk ðsÞ P 0, if s M ẳ ẵv1 ; v2 ; v3 P n jkjẳn Bk sị n fBk sị; jkj ẳ ng is a basis of the polynomial space P n Bnk ðsÞ has a unique maximum value at point s ¼ kn From property (3), we have Lemma [5] Any polynomial P P n can be uniquely expressed as X bk Bnk sị 5ị P sị ẳ jkjẳn where fbk ; jkj ¼ ng are called the Be´zier coordinates of P ðsÞ over M, the piecewise linear function interpolating to fðkn ; bk ị : jkj ẳ ng is called the Bezier net of P ðsÞ over M, B-net for shot Let v1 ; v2 ; v3 be the vertices of triangle T, and bv ; v2 ; v3 be the vertices of triangle Tb T and Tb have the common boundary v2 v3 (see Fig 1, right) The smoothness conditions of polynomials of degree n over two adjacent triangles are presented as follows Theorem [3] Let P ðsÞ and Pb ðsÞ denote polynomials of degree n defined on T ẳ ẵv1 ; v2 ; v3 , and Tb ¼ ½bv ; v2 ; v3 , respectively Let fbk ; jkj ¼ ng and fb b k ; jkj ẳ ng be the Bezier coordinates of P sị over T and Pb ðsÞ over Tb , respectively A necessary and sufficient condition for P ðsÞ and Pb ðsÞ to be C r across the common boundary is ^ bkt ẳ btk0 rị; t ẳ 0; 1; Á ; r ð6Þ Deriving the B-net method with the conformality method of smoothing cofactor By the definition of the barycentric coordinates, we have Lemma Let bk ð^sÞ, and ck ðsÞ denote polynomials of degree k defining over two adjacent triangles Tb ẳ ẵ^v1 ; v2 ; v3 and T ẳ ẵv1 ; v2 ; v3 , respectively Denote by rðr1 ; r2 ; r3 Þ the barycentric coordinates of v1 over Tb The relations between the barycentric coordinates over the adjacent triangles are as follows bs ẳ r2 s1 ỵ s2 ; bs ¼ r1 Á s1 ; bs ¼ r3 s1 ỵ s3 8ị 3.1 S 13 Dị over two adjacent triangles Denote by b3 ðbs Þ and c3 ðsÞ the bivariate polynomials of degree defining over two adjacent triangles Tb ẳ ẵbv ; v2 ; v3 and T ẳ ẵv1 ; v2 ; v3 , respectively (see Fig 2) Let fbg : jgj ¼ 3g and fck : jkj ¼ 3g be the Be´zier coordinates of b3 ð^sÞ over Tb and c3 ðsÞ over T, respectively Denote by rðr1 ; r2 ; r3 Þ the barycentric coordinates of v1 over Tb The expression of b3 bs ị is X b3 ^sị ẳ bg B3g ^sị ẳ b3;0;0^s31 ỵ 3b2;1;0^s21^s2 ỵ 3b1;2;0^s1^s22 jgjẳ3 ỵ b0;3;0^s32 þ 3b0;2;1^s22^s3 þ 3b0;1;2^s2^s23 þ b0;0;3^s33 þ 3b1;0;2^s1^s23 þ 3b2;0;1^s21^s3 þ 6b1;1;1^s1^s2^s3 By Lemma 3, we have ^s1 ¼ r1 s1 ; ^s2 ẳ r2 s1 ỵ s2 ; ^s3 ẳ r3 s1 ỵ s3 Denote m1 ẳ r31 b3;0;0 ỵ 3r21 r2 b2;1;0 ỵ 3r1 r22 b1;2;0 ỵ r32 b0;3;0 ỵ 3r22 r3 b0;2;1 ỵ 3r2 r23 b0;1;2 ỵ r33 b0;0;3 ỵ 3r1 r23 b1;0;2 ỵ 3r21 r3 b2;0;1 ỵ 6r1 r2 r3 b1;1;1 m2 ẳ r21 b2;1;0 ỵ 2r1 r2 b1;2;0 ỵ r22 b0;3;0 ỵ 2r2 r3 b0;2;1 ỵ r23 b0;1;2 ỵ 2r1 r3 b1;1;1 m3 ẳ r21 b2;0;1 ỵ 2r1 r2 b1;1;1 ỵ r22 b0;2;1 ỵ 2r2 r3 b0;1;2 ỵ r23 b0;0;3 ỵ 2r1 r3 b1;0;2 where brk rị ẳ 27 X bkỵl Brl rị; jkj ẳ n r 7ị jljẳr r is the barycentric coordinate of bv kt ¼ ðt; k2 ; k3 ị; k0 ẳ 0; k2 ; k3 ị; k2 ỵ k3 ẳ n t over v2 T, sxị ẳ 0; 8x K ) sxị ẳ 0; 8x C K is a minimal determining set if there is no smaller determining set 2,1,0 3,0,0 v1 1,2,0 Definition [6] Let D denote the simplex partition on domain D, and let C denote the set of control points of a spline in S lk ðMÞ A subset K # C is a determining set for S lk ðMÞ if 1,2,0 0,3,0 0,2,1 1,1,1 2,1,0 2,0,1 1,1,1 3,0,0 vˆ1 0,1,2 2,0,1 1,0,2 0,0,3 v3 Fig S 13 ðMÞ 1,0,2 28 R Wang, K Qu / Progress in Natural Science 19 (2009) 2531 where lx; yị ẳ is the equation of v2 v3 (ii) From Eq (8), b ^sị ẳ m1 s31 ỵ ỵ 3m2 s21 s2 r3 b0;2;1 ịs1 s22 ỵ 3m3 s21 s3 þ 3ðr1 b1;2;0 þ r2 b0;3;0 þ 3ðr1 b1;0;2 þ r2 b0;1;2 ỵ r3 b0;0;3 ịs1 s23 ỵ 6r1 b1;1;1 þ r2 b0;2;1 þ r3 b0;1;2 Þs1 s2 s3 þ b0;3;0 s32 ckt ẳ btk0 rị; t ẳ 0; 1; ; l 10ị where kt ẳ t; k2 ; k3 ị; k0 ẳ 0; k2 ; k3 ị; k2 ỵ k3 ẳ k t ỵ 3b0;2;1 s22 s3 ỵ 3b0;1;2 s2 s23 ỵ b0;0;3 s33 Proof By Lemma 2, bk ð^sÞ and ck ðsÞ can be expressed as X X k! bk ^sị ẳ bg Bkg ^sị ẳ bg ^sg 11ị g! jgjẳk jgjẳk The expression of c3 sị is X c3 sị ẳ ck B3k sị ẳ c3;0;0 s31 ỵ 3c2;1;0 s21 s2 ỵ 3c1;2;0 s1 s22 jkjẳ3 ỵ c0;3;0 s32 ỵ 3c0;2;1 s22 s3 þ 3c0;1;2 s2 s23 þ c0;0;3 s33 þ 3c1;0;2 s1 s23 ỵ 3c2;0;1 s21 s3 ỵ 6c1;1;1 s1 s2 s3 and ck sị ẳ Notice that the expression of the common boundary v2 v3 is s1 ¼ Let b3 ðbs Þ, and c3 ðsÞ be C across the common boundary By Theorem 1, there is a polynomial qðsÞ of degree 1, such that ck Bkk sị ẳ jkjẳk X jkjẳk ck k! k s k! 12ị From Eq (8) X k! g g g ðr1 Á s1 Þ r2 s1 ỵ s2 ị r3 s1 ỵ s3 ị !g g !g3 ! jgj¼k g2 g3 X X X k! g2 g3 g Àj rg11 sg11 ¼ bg ri2 si1 s2g2 Ài rj3 sj1 s33 i j !g !g ! g i¼0 j¼0 jgj¼k g3 g2 X X X k! g2 g3 g j ẳ bg rg11 ri2 rj3 s1g1 ỵiỵj s2g2 Ài s33 j g1 !g2 !g3 ! i¼0 j¼0 i bk ^sị ẳ c3 sị b3 bs ị ẳ qsịs21 So c0;3;0 ẳ b0;3;0 ; c0;2;1 ẳ b0;2;1 ; c0;1;2 ¼ b0;1;2 ; c0;0;3 ¼ b0;0;3 c1;2;0 ¼ r1 b1;2;0 ỵ r2 b0;3;0 ỵ r3 b0;2;1 bg jgjẳk c1;0;2 ẳ r1 b1;0;2 ỵ r2 b0;1;2 ỵ r3 b0;0;3 ẳ c1;1;1 ẳ r1 b1;1;1 ỵ r2 b0;2;1 ỵ r3 b0;1;2 X jgj¼k It indicates that the necessary and sufficient conditions for polynomials of degree defining over two adjacent triangles to be C across the common boundary are that the Be´zier coordinates of the two polynomials satisfy the relations above This is the same as Theorem Moreover, we obtain the expression of the smoothing cofactor across the common boundary v2 v3 qsị ẳ c3;0;0 m1 ịs31 ỵ 3c2;1;0 m2 ịs21 s2 ỵ 3c2;0;1 m3 ịs21 s3 Next, we derive the smoothness conditions obtained from the B-net method with the conformality method of smoothing cofactor bg g3 g2 X k! X g g ỵiỵj g2 Ài g3 Àj r ri rj s s2 s3 g1 ! iẳ0 jẳ0 g2 iị!g3 jị!i!j! Denote r :¼ ðr1 ; r2 ; r3 ị :ẳ g1 ; i; jị; j r j¼ k1 ; g2 À i ¼ k2 ; g3 À j ¼ k3 It is clear that g2 ¼ k2 ỵ i; g3 ẳ k3 ỵ j; g :ẳ g1 ; g2 ; g3 ị :ẳ 0; k2 ; k3 ị ỵ r1 ; r2 ; r3 ị So bk ð^sÞ can be simplified as X X k! r r sk bg bk ^sị ẳ !k ! r!k jkjẳk jrjẳk ẳ 3.2 S lk Dị over two adjacent triangles X X b0;k2 ;k3 ịỵr1 ;r2 ;r3 Þ jkj¼k jrj¼k1 Theorem Let bk ðbs Þ and ck ðsÞ denote polynomials of degree k defining over two adjacent triangles Tb ẳ ẵbv ; v2 ; v3 and T ẳ ẵv1 ; v2 ; v3 , respectively Let fbg : jgj ¼ kg and fck : jkj ¼ kg be the Be´zier coordinates of bk ðbs Þ over Tb and ck ðsÞ over T, respectively Denote by rðr1 ; r2 ; r3 ÞSthe barycentric coordinates of v1 over Tb Let M = Tb T , sðx; yÞ S lk ðMÞ, p1 ðx; yÞ, and p2 ðx; yÞ be the expressions of sðx; yÞ over Tb and T, respectively, where p1 ðx; yÞ and p2 ðx; yÞ P k Then the following conditions are equivalent to each other (i) There is a smoothing cofactor qðx; yÞ P kÀlÀ1 across the common boundary v2 v3 , such that p2 ðx; yÞ À p1 x; yị ẳ qx; yị lx; yị X lỵ1 ð9Þ k! rr s k r!k2 !k3 ! Comparing Eq (12) with Eq (13), we have k k c ðsÞ b ^sị ẳ X jkjẳk X ck b0;k2 ;k3 ịỵr1 ;r2 ;r3 ị rr r! k1 ! jrjẳk 13ị ! k! k s k2 !k3 ! Let k1 ẳ t, then ck sị bk ^sị ¼ k X X À Á ckt À btk0 ðrÞ t¼0 jkj¼k l X X À Á ¼ ckt À btk0 rị tẳ0 jkjẳk ỵ k! st sk2 sk3 t!k2 !k3 ! k! st sk2 sk3 t!k2 !k3 ! k XÀ X Á ckt btk0 rị tẳlỵ1 jkjẳk k! st sk2 sk3 14ị t!k2 !k3 ! R Wang, K Qu / Progress in Natural Science 19 (2009) 25–31 29 Deriving (ii) with (i) There is a smoothing cofactor qðx; yÞ P kÀlÀ1 across the common boundary v2 v3 , such that p2 x; yị p1 x; yị ẳ qx; yị lx; yị V2 lỵ1 where lx; yị ¼ is the equation of v2 v3 , and its barycentric coordinate over T is s1 ¼ So the first part of Eq (14) should be zero, that is ckt ẳ btk0 rị; T1 T2 V0 t ẳ 0; 1; Á Á Á ; l T3 where jkj ¼ k; kt ¼ ðt; k2 ; k3 Þ; k0 ẳ 0; k2 ; k3 ị; k2 ỵ k3 ẳ k À t Deriving (i) with (ii) It is known that V3 V1 Fig Triangle Mà ckt ¼ btk0 rị; t ẳ 0; 1; ; l; jkj ¼ k; kt ¼ ðt; k2 ; k3 ị; k0 ẳ 0; k2 ; k3 ị; k2 ỵ k3 ¼ k À t r1 ðr11 ; r12 ; r13 Þ; r2 ðr21 ; r22 ; r23 Þ; and r3 ðr31 ; r32 ; r33 Þ the barycentric coordinates of three vertexes V ; V ; and V over T ; T ; and T , respectively We have So the first part of Eq (14) is zero Moreover, there is a polynomial qsị ẳ k X X ckt btk0 rịị tẳlỵ1 jkj¼k Lemma k! stÀlÀ1 sk22 sk33 t!k2 !k3 ! r11 r21 ¼ 1; r12 r31 ¼ 1; r23 ¼ À1; r11 ¼ r13 r32 ¼ r33 ; r12 þ r11 r22 ¼ 0; r13 þ r12 r32 ¼ ð15Þ such that Proof Let bðiÞ ðsi Þ ðsi :ẳ si1 ; si2 ; si3 ị; i ẳ 1; 2; 3Þ be the polynomials of degree k defining over T i By Lemma 3, we have ck ðsÞ bk ^sị ẳ qsịslỵ1 Obviously, qsị is the smoothing cofactor across the common boundary v2 v3 Theorem indicates that both the existence of the smoothing cofactor and the smoothness conditions obtained from the B-net method are equivalent over two adjacent triangles h s11 ¼ r11 s21 ; s31 ẳ r31 s12 ; s21 ẳ s31 ỵ r21 s32 ; s32 ẳ s11 ỵ r32 s12 So s11 ẳ r11 s31 ỵ r21 s32 ị ẳ r11 s31 ỵ r11 r21 s32 3.3 S lk D ị ẳ r11 r31 s12 ỵ r11 r21 s11 ỵ r32 s12 ị on the star-region over triangulation ẳ r11 r21 s11 ỵ r11 r31 ỵ r11 r21 r32 ịs12 Let Dà be a triangulation shown in Fig 3, and V be the common vertex of triangles T ; T , and T Denote by b0(1,3) , Obviously, r11 r21 ¼ Others can be proved similarly V2 *b ( 2) , 3, b0(1, 2) ,1 b ( ) , ,1 * b1(,22), (1) 1, , b * b1(,11),1 * (1) ,1, b b1(,10), b2(1,0) ,1 b3(1, 0) , V1 ( 3) b0,3, * b0(1,1), b ( ) ,1, * b1(,21,)1 b0(1, 0) ,3 b ( ) 0, 0,3 * b0( 3, 0),3 V0 * b2( 2,1), b1(,20), b1(,30), b0( 3,1), * b2( ,20),1 * b0( 3, 2),1 b2( 3,0),1 * b2( 3,1), b1(,32), Fig S 13 ðMÃ Þ b3(,20), *V b3(,30), h 30 R Wang, K Qu / Progress in Natural Science 19 (2009) 25–31 We can get some conditions of the Be´zier coordinates between two adjacent simplexes which satisfy certain smoothness Then we find all the conditions of the Be´zier coordinates over the whole partition Taking S 13 ðDÃ Þ for example (see Fig 4), K is one of minimal determining sets [6] for S 13 D ị, where jKj ẳ 12, we mark all the control points belonging to K with à We also have dimS 13 D ị ẳ 12 by Theorem Using the barycentric coordinates, of li x; yị ẳ 0; i ẳ 1; 2; are l1 x; yị ẳ : s21 ẳ 0; l3 x; yị ẳ : s12 ẳ q2 x; yịl2 x; yị qi x; yịli x; yị lỵ1 ẳ0 where li x; yị ¼ 0; i ¼ 1; 2; are the equations of V V i ; i ¼ 1; 2; (III) 2ị 1ịt bgt ẳ bg1 ;0;g3 ị r2 Þ; ð1Þ ð3Þt t ¼ 0; 1; Á Á Á ; l bnt ẳ b0;n1 ;n3 ị r3 ị; 3ị k X X 3ị b gt tẳlỵ1 jgjẳk k! g t r31 s12 ị s11 ỵ r32 s12 ị g1 !t!g3 ! k X X g s13 ỵ r33 s12 Þ À ð2Þt bðg1 ;0;g3 Þ ðr2 ị tẳlỵ1 jgjẳk g3 k t X XX X k! 3ị g sg311 st32 s333 ẳ bg1 ;t;g3 ị g1 !t!g3 ! tẳlỵ1 jgjẳk iẳ0 jẳ0 k! g g þiþj g3 Àj r ri rj stÀi s s13 g1 !i!j!ðt À iÞ!ðg3 À jÞ! 31 32 33 11 12 k X ð2Þt X k! g sg311 st32 s333 ð23Þ À bðg1 ;0;g3 Þ ðr2 Þ g !t!g ! tẳlỵ1 jgjẳk Denote iẳ1 bkt ẳ b0;k2 ;k3 ị r1 ị; l2 x; yị ẳ : s32 ẳ 0; lỵ1 ẳ q2 s3 ịs32 ẳ (I) s S lk ðDÃ Þ (II) There are smoothing cofactors qi ðx; yÞ P kÀlÀ1 ; i ¼ 1; 2; such that X expressions Substituting Eq (22) into Eq (18), we have lỵ1 Theorem Suppose that V is the common interior vertex of triangles T ; T , and T in the partition Dà Let bðiÞ ðsi Þ denote polynomials of degree k defining over T i , and sjT i ẳ biị si ị; si :ẳ si1 ; si2 ; si3 ị; i ẳ 1; 2; Denote by r1 ; r2 ; and r3 the barycentric coordinates of three vertices V ; V ; and V over T ; T ; and T , respectively Then the following propositions are equivalent: the ð2Þt It is obvious that we have ð16Þ Proof The equivalence of (I) and (II) can be obtained by Theorem directly Moreover, we can derive (III) with (II) by Theorem Now we will derive (II) with (III) By the proof of Theorem 5, we know that the expressions of qi ; i ¼ 1; 2; are k X ð2Þ X ð1Þt bkt À bð0;k2 ;k3 ị r1 ị q1 s2 ị ẳ tẳlỵ1 jkjẳk tẳlỵ1 jgjẳk g1 ẳ r ; t ẳ n1 þ r ; g ¼ n3 þ r From Eqs (15) and (16), the representation Eq (23) can be simplied as q2 x; yịl2 x; yịlỵ1 ẳ k X XX t0 ẳlỵ1 jnjẳk jrjẳt0 3ị br1 ;n1 þr2 ;n3 þr3 Þ k! sn1 st sn3 n1 !t!n3 ! 11 12 13 k X X ð2Þt À bðg1 ;0;g3 Þ ðr2 Þ t0 ! r r r!  k! stÀlÀ1 sk222 sk233 t!k2 !k3 ! 21 ð17Þ k X ð3Þ X ð2Þt bgt À bg1 ;0;g3 ị r2 ị q2 s3 ị ẳ r :ẳ r1 ; r2 ; r3 ị ẳ g1 ; i; jị; n1 ẳ t i; n3 ẳ g3 j; n :ẳ n1 ; t; n3 ị; t0 ẳ g1 ỵ i ỵ j tẳlỵ1 jgjẳk ẳ k! g g s stÀlÀ1 s333 g1 !t!g3 ! 31 32 k X X t0 ẳlỵ1 jnjẳk k X ð3Þt0 bð0;n1 ;n3 Þ ðr3 Þ X ð2Þt k! sn1 st sn3 n1 !t!n3 ! 11 12 13 bðg1 ;0;g3 ị r2 ị tẳlỵ1 jgjẳk k! g sg1 st s g1 !t!g3 ! 31 32 33 k! g sg1 st s g1 !t!g3 ! 31 32 33 18ị q3 s1 ị ẳ k X 1ị X 3ịt bnt b0;n1 ;n3 ị r3 ị tẳlỵ1 jnj¼k k! sn1 stÀlÀ1 sn133 n1 !t!n3 ! 11 12 19ị By Lemma 3, we have s21 ẳ s31 ỵ r21 s32 ; In a similar way, we have lỵ1 q3 x; yịl3 x; yịlỵ1 ẳ q3 s1 ịs12 ẳ s22 ẳ r22 s32 ; s23 ẳ s33 ỵ r23 s32 20ị s21 ẳ r21 s11 ; s22 ẳ s12 þ r22 s11 ; s23 ¼ s13 þ r23 s11 21ị s31 ẳ r31 s12 ; s32 ẳ s11 ỵ r32 s12 ; s33 ẳ s13 ỵ r33 s12 22ị k X X 1ịt b0;k2 ;k3 ị r1 ị tẳlỵ1 jkjẳk k X X tẳlỵ1 jnjẳk 3ịt k! st sk2 sk3 t!k2 !k3 ! 21 22 23 bð0;n1 ;n3 Þ ðr3 Þ k! sn1 st sn3 n1 !t!n3 ! 11 12 13 R Wang, K Qu / Progress in Natural Science 19 (2009) 25–31 and q1 ðx; yÞl1 ðx; yị Acknowledgements lỵ1 ẳ ẳ q1 s2 ịslỵ1 21 k X X 2ịt bg1 ;0;g3 ị r2 ị tẳlỵ1 jgjẳk À k X X ð1Þt k! g g s st s g1 !t!g3 ! 31 32 33 bð0;k2 ;k3 ị r1 ị tẳlỵ1 jkjẳk k! st sk2 sk3 t!k2 !k3 ! 21 22 23 Therefore q1 ðx; yÞl1 x; yị 31 lỵ1 ỵ q2 x; yịl2 x; yị lỵ1 ỵ q3 x; yịl3 x; yị lỵ1 lỵ1 ẳ q1 s2 ịslỵ1 21 ỵ q2 s3 ịs32 ỵ q3 s1 ịs12 ẳ lỵ1 By Theorem and Theorem 6, we have Theorem For any given simplex partition, the smoothness conditions obtained, respectively, by the conformality method of smoothing cofactor and the B-net method are equivalent This work was supported by National Natural Science Foundation of China (Grant Nos 60533060, 60373093, 10726067, 10726068 and 10801024) References [1] Wang RH The structural characterization and interpolation for multivariate splines Acta Math Sin 1975;18(2):91–106 (in Chinese) [2] Farin G Subsplines ueber Dreiecken Ph.D thesis, Technical University Braunschweig, Germany, 1979 (in German) [3] Farin G Triangular Bernstein–Be´zier patches CAGD 1986;3(2): 83–127 [4] Wang RH Multivariate spline function and their applications Beijing/ New York/London: Science Press/Kluwer Acad Pub; 2001, pp 37–42 [5] Lorentz GG Bernstein polynomials 2nd ed New York: Chelsea Publishing Company; 1986 [6] Alfeld P Bivariate spline spaces and minimal determining sets J Comput Appl Math 2000;119(1–2):13–27