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mathematics of computation volume 57,number 195 july 1991,pages 299-308 THE STRUCTUREOF MULTIVARIATESUPERSPLINE SPACES OF HIGH DEGREE PETER ALFELDAND MARITZASIRVENT Abstract. We consider splines (of global smoothness r, polynomial degree d, in a general number k of independent variables, defined on a k-dimensional triangulation ¡T of a suitable domain ii ) which are r2 ~m~ -times differentiate across every m-face { m = 0, ■■■ , k - 1) of a simplex in !T . For the case d > rl we identify a structure that allows the construction of a minimally supported basis. 1. Introduction A k-simplex K (0 < k < k) is the convex hull of k + X points in R called the vertices of K. K is nondegenerate if its k-dimensional volume is nonzero, and degenerate otherwise. The dimension of a nondegenerate k-simplex is k . The convex hull of a subset of p + 1 vertices of K is a //-face of K . Let FcR be a given set of A^ distinct points. A triangulation J~ of the set W is a set of nondegenerate /c-simplices satisfying the following requirements: 1. All vertices of each simplex in ¡T are elements of W. 2. The interiors of the simplices in y are pairwise disjoint. 3. The set (1) £1- \J KcRk is homeomorphic to [0, 1] . 4. Each (k - l)-face of a simplex in ST is either on the boundary of Q, or else is a common face of exactly two simplices in ÏT. 5. No simplex in ¿7" contains any points of "V other than its vertices. Note that a /¿-face of a simplex in y is itself a /¿-dimensional simplex. On the triangulation !T we define a spline space Srd(J~) as usual by (2) Srd(F) = {s £ Cr(Q) : s\r £ &Ï Vt e ^}, where ¿Pd is the (kdd)-dimensional linear space of all /c-variate polynomials of total degree less than or equal to d. Received February 27, 1990; revised September 21, 1990. 1980MathematicsSubjectClassification(1985 Revision).Primary 65D07, 41A63,41A15. ©1991 American Mathematical Society 0025-5718/91 $1.00+ $.25 per page 299 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PETER ALFELD AND MARITZA SIRVENT 300 In this paper, we consider subspaces of Srd(!T) obtained by increasing the smoothness requirements across faces of the underlying simplices. More precisely, denoting by 5? the set of all /¿-faces of the simplices in ¿7" ( p = 0, ... , k - 1) and letting 5? = LLIo^u' we define the (superspline) space §d(&~) as a subspace of Srd(&~)as follows: §d(^) = {5 £ Sd(¿7~): s is /Mimes differentiable across a Va £ 5*}, ^ ' , ~,k—dimrj—1 where p = r2 The concept of supersplines was introduced in Chui and Lai [8], [9]. The area in between finite elements and full spline spaces was further explored by Schumaker [16] and Ibrahim and Schumaker [11]. 2. The generalized Bézier-Bernstein form Crucial to analyzing the dimension of spline spaces is the Bézier-Bernstein form of a multivariate polynomial. In the case k < 2 this form is used widely and is well known. A review of the Bézier-Bernstein form for a general number of variables is in de Boor [6]. In this paper, we use a notation that is particularly suitable for our purposes. However, generalized barycentric coordinates and global control nets have also been proposed in Alfeld [2] and de Boor [6]. We use T" as an index set and denote by N the set of nonnegative integers. For vectors I = [iv]veT £ N (4) and a = [av]veT el we define III= £ ', > vëv (5) ■l=iiiin«vní.,> vçT ' VÇ.T where (6) 0°:=1. We also use the notation (7) a(\) = conv{u : iv > 0}, cr(a) = conv{v : av ^ 0} where conv X denotes the convex hull of a point set X . We now define generalized barycentric coordinates as cardinal piecewise linear functions bv £ Sx (Q) by the requirement ( X if v = w (8) W4.-{0 else *.«'• Clearly, in each /c-simplex K £ ¡T the functions bv , where v is a vertex of K , reduce to the ordinary barycentric coordinates. Globally, i.e., for all x £ Q, they satisfy (9) ¡>>„ = 1, bv>0 \lv£T, and x = ^ ver License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use veîr bv(x)v. THE STRUCTURE OF MULTIVARÍATESUPERSPLINESPACESOF HIGH DEGREE 301 For a given polynomial degree d, we use the domain index set ;i0) \dd = {l£HN:\l\ = dando(l)£Si'\j3r} Letting (11) b = b(x) = [bv(x))ver, it is clear that every function s £ Sd(&~) can be written as (12) 5 = EcibIlei, The coefficients cx are the Bézier ordinales of 5. 3. The de Casteljau algorithm Let 5 e Sd(^), K £ ZT, and s\K=p £&>%. Without loss of generality we may relabel the vertices and assume that K is the k-simplex (13) K = {a = (ax,--- , ak+x): ax + ••• + %+, =d, a} >0, a.eR}. Then, with V] denoting the vertices of K, we have k+l (14) k+l p= J2 c.b1 where x = £>,*}' ie/cni^ ;=1 T,bj = L ;=1 The Bernstein polynomials b (x) satisfy a recurrence relation, see Farin [10] fc+i (15) b\x) = J2b/~eJ(x), ;=i \l\=d, and eJ = (0,--- , X,-- ,0). This relation allows one to expand p in terms of Bernstein polynomials of lower degree with (polynomial) coefficients p[(b). Theorem 1. We have (16) p= £ p,(b)b\ 0<r<¿, |I|=rf-r where (17) rf(b)= c„ fc+i (18) prl(b)= ^2b/l^l(b), |I| = rf-r, 0<r<d. The intermediate coefficients p[(b) can also be written explicitly as (19) P,»=Eci/' |I| = d-r. |M|=r License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PETER ALFELD AND MARITZA SIRVENT 302 The formulas in Theorem 1 may be used to evaluate p at a given point, and are referred to as the de Casteljau Algorithm. 4. Smoothness across an interface Given a vector e £ R , the directional derivative of p in the direction of a = db.k+l i?'---' denoted Da , is given by (Alfeld [2]) ' de i ' (20) Dap = d £ p\{0L)b\ HM-i where (21) p\(a)= £ cl+MaM, \I\=d-X. |M| = 1 In general, the rth directional derivative of p in the direction of a is given by (Farin [10]) (22) D>= jrhv S *W, 1 '■ \l\=d-r where (23) p[(a) = £ cl+MaM, \I\ = d - r, and p°(a) = cv |M|=r The following theorem was proved by Farin [ 10] in the bivariate case; here we state without proof the result in any number of variables. Let x be an m-face of K ; without loss of generality, we will assume that (24) r = {*£K: am+2 = ■• • = ak+x = 0} . Theorem 2. Let p, q £ ¿?d be such that p\T = q\x. Then Dsap = Dsaq on t for all directions a, 0 < s < r, if and only if Pj = qr3for all J = (/',,... 0,... ,0), |J| =d-r. 5. Subsimplices and subpolynomials Let K £ y (25) be a fc-simplex and x be a face of K with dim x = n , K=U£RN: { (26) Y,av=d,av>o\ vex x=U£K: Y,av= A{ V€T J For J e K n ld , so that Y,veT jv = d- p, define (2V) , J Tj = {a = [av]v£K: jv>av,v£x}. Clearly, Tj is a subsimplex of K similar to K . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use , jm+x, THE STRUCTUREOF MULTIVARIATESUPERSPLINESPACESOF HIGH DEGREE 303 Let p £ ¿Pd , and (I, cjjK be the control points of p with respect to K. The control points (I, c,)l€T define a polynomial p} of degree p. We call Tj the subsimplex of K associated with J, and p} the subpolynomial of p associated with J. De Boor [6] introduces the concepts of subsimplices and subpolynomials and proves most of the results of this section. Since our definitions are slightly different, we restate two key facts involving subpolynomials. Theorem 2 can be restated as follows: Theorem 3. Let p, q £ 3?d , and x be a face of K with dim x = m such that P\T = a\T. Then (28) Dsap = Dsaq onx, for all directions a and 0 < s < p, if and onlyif (29) Pi = ai Vie A"with dist(I, x) = p. The above theorem immediately yields the following: Theorem 4. Let K, K1 eJ, x C K ni', p, q £&>d , and let s £ Sd(Q) with s\K = p, s\Ki = q. Then s £ Cp(x) if and only if px = qx VI e K1 with dist(I, x) = p. 6. Determining sets We now generalize the concept of a determining set known from the bivariate case (see Alfeld and Schumaker [4]). Definition 5. A set D c Id is a determining set of §rd(&~)if, for all s £ Sd(^), (30) c, = 0 VI G D => s = 0. D is a minimal determining set if there is no determining set which has fewer elements than D. It is clear from elementary linear algebra that the number of elements in a determining set of Sd(Sf) provides an upper bound on the dimension of S^(y), and that the number of elements in a minimal determining set is unique and equals the dimension of S^(^). 7. A minimal determining set of Sd(^) For each simplex ueyuJ let p = r2 ~dim(T~1as before; we define two sets of domain indices recursively by (31) 2(o) = \l£\d: { ^iv>d-p\ vea and (32) 2(o) = 3(o)\{j2(x), T.<a where x is a proper face of a . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ) 304 PETER ALFELD AND MARITZASIRVENT Definition 6. Let it 6 yu^. 2(a) if, for all s £ STd{&~), (33) A set s/(a) c D(o) is a determining set of cx = 0 V\£S¿(o)U(2(o)\2(o))^cl = 0 VleW{o). The set s/ is a minimal determining set of ^"(a) if there is no determining set of 3¡(a) with fewer elements. Lemma 7. Let d > r2 . Then, for all I £ Id there exists a unique ireyuJ such that I £ 3(a). Proof. To prove the lemma, we have to show first that for all a, x £ S* U!T : (34) o¿x^2(o)n2(x) = 0. To establish this, suppose there is a domain index I £ 2(a) (~)2(x), for two simplices a ,x £S^ \J¡T, such that dim a > dim x, and x is not a face of a . Thus, (35) J2iv>d-r2h-dima-1 w6cr and £/„ > rf - r2^dimT-1, «6r which implies /t^, (36) V~* • E i„ +, V^ £i„ • ^.> -> 2dj - r2»Ar-dimrJ-l -rl -fc-dimr-l vEo vÇx Moreover, (37) £'« +£', = E '«+ E '»■ Now, rearranging (37), substituting (36) into (37), and using J2veaUTiv < d, we obtain (38) £ iv>d-r2k-dimr. v€of~ir So there exists a face f of t such that I £ 2(x), which is a contradiction. Finally, we need to prove that ld = \Jae^>IJ^-2(a). To see this, we only need to show that for each domain index le I¿ there exists a simplex a G S? U !T such that I 6 2(a). However, this is trivial, since for each domain index I there exists at least one Ä>simplex K such that J2veKiv = d. Thus, I G 2(K), and there must be a simplex a -< K such that I £ 2(a). D Lemma 8. Let d > r2k, p = r2k~d>ma-[, and sé'(a) := 2(a) n_K, where K £ y and a <K. Then sé (a) is a minimal determining set of 2(a). Proof. First we establish that sf(o) is determining. Let s G Sd(^), and let s\K = p , with p g &>d. If c, = 0 VI G s/(a) U (2'(a) \2(a)), then p, = 0 Vl£s/(o)U(2(a)\2(o)), License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE STRUCTUREOF MULTIVARIATESUPERSPLINESPACESOF HIGH DEGREE 305 where px is the subpolynomial of p of degree < p associated with I (notice that dist (I, a) < p VI £sé(a) ). Let K1 be any other /c-simplex of 3~ such that a <K' and K ¿ K', and let q = s\K,, q £ 9°k . Now, s £ C(a) ; then VI G (2(a) n K') U (2(a)\2(a))pl = ql, where ql is the subpolynomial of q of degree < p associated with I. But, (39) p, = 0 Vl£(2(o)nK)u(2(o)\2(a)) (40) =► q1= 0 VL£(2(a)nK')[J(2(a)\2(o)) (41) =* Cj= 0 VI G (2(a) r\K')ö (2(a) \2(a)) ( (42) =* c, = 0 VIG \ (J 2(a) HK U(2(a)\2(a)) 1 K£3- \^K (43) =» J C, = 0 VlG^'((T)U(^((T)\Sr((7)). Therefore, (44) cx= 0 Yl£2(o). To see that sé (a) is indeed minimal, take I G sé (a) and consider the set sé (a) = sé (a) \ {1}, define a polynomial on ATwhose Bézier coefficients are equal to zero except at the domain point I, where c, = 1, and extend this polynomial globally on the rest of the triangulation. The coefficients cs are equal to zero on 2(a) \2(a), since the smoothness conditions there only involve domain indices in 2(x) for x <a . Hence, c} = 0 V/ GsT(a) U (2(a) \2(a)), Cj = 0 V7 G2(a) since in particular / G 2(a) but this does not imply that and c¡ = X. Therefore, sé (a) cannot be determining and sé (a) is a minimal determining set of 2(a). D The following theorem is the central result of this paper. Theorem 9. Let r > 0, d > r2k and let sé (a) = 2(a) n K, where K £ 9~ is a k-simplex so that a is a face of K. Then (45) sé := (J sé (a) is a minimal determining set of §d(¿7~)■ Proof. Let 5 e §¿(7) and assume that c, = 0 V/ g sé . Using induction on dim a , we first establish that sé is a determining set. (1) If dim a = 0 then c, = 0 V/ Gsé (a) implies c, = 0 VI G2(a), since 2(a) = 2(a). (2) We assume now that c, = 0 V/ G 2(a) and Va with dimrj < n . License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 306 PETER ALFELD AND MARITZA SIRVENT (3) Let dim ff = n, c, = 0 V/ G sé (a). Let 3ê(a) = sé(a)U(2(a)\2(a)) ; notice that B(a) csé(a)o (\J^a2(x)). By the induction hypothesis, c, = 0 VI G \JT^,a2(x) implies that cx = 0 VI G ¿®(o). But se (a) is a determining set of 2(a) ; thus, c¡ = 0 V/ G S^ff), hence (46) cI = 0 V/G [J 2(a), rre^ur and by Lemma 7 we get that (47) cx= 0 Vlelrf. To show that j/ is indeed minimal, we give arbitrary Bézier ordinates cx V\£sé , and we construct the rest of the Bézier ordinates in such a way that the piecewise polynomial they determine is a superspline. So, we assume Cj VI G sé are given, and let J G \d . Then by Lemma 7 there is a unique a GS? U T such that J G 2(a). We define Cj inductively over dim a . (1) If ff is a vertex, then there exists K G y so that ct -<ATand sé(a) c K. Let ffj be the subsimplex of K associated with J, a3 c sé(a). Then the polynomial ps is defined; so, for J g 2(a) n AT'with K' g y, A"' ?¿ A, let tj be the subsimplex of K' associated with J. Then there is a unique way to give Bézier ordinates to the domain points in Tj in such a way that they define a polynomial q} which equals p3. Furthermore, in this manner, c3 can be defined for all J G2(a)\sé(a). (2) Suppose Cj has been defined for all J G \J¿[ma<n^(^) ■ (3) Let J g 2(a) with a an «-simplex and K g y such that se (a) c K. And let again a-, be the subsimplex of K associated with J. Note that (48) oJc2(a)r)Kcsé(o)\j{\J2(x)nK.\ . Then by the induction hypothesis and by the fact that the Bézier ordinates have been defined on sé , we have that all of the Bézier ordinates on rjj are defined; therefore p3 is defined. So as before, for J G 2(a) n K', K' ^ K, we can define q} in the same way we did when a was a vertex. Thus, cx has been defined for all I G Id . Next, we need to show that the piecewise polynomial function defined by the Cj's is well defined. Suppose pL = qh for L G K and £u6i lv = d - p . Let J G 2(a) n K' and suppose that J G £L with J 7¿L, where (49) {l = {l6Jf':/,>/,OE{1 íeyur, and (50) rJ = {leK':iv>jv v£o}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use and ff ^¿} THE STRUCTURE OF MULTIVARIATESUPERSPLINESPACESOF HIGH DEGREE 307 Then J G £L implies jv > lv , which in turn implies Tj c ¿;l . Therefore, qs is a subpolynomial of qh . Similarly, p3 is a subpolynomial of pL . Hence, p3 = q-, since qLUph£ C°°(Ç). Therefore, the Bézier ordinates are well defined and by construction, the piecewise polynomial defined is a superspline. D Corollary 10. We have dimS^(y) = E^es^r W(<?)\■ In view of the above corollary, to compute dimS^(y) we need only to know the cardinality of sé (a) for every a G y. Clearly, |j/(ff)| depends only on m = dim a . The following theorem is proved in Alfeld and Sirvent [5]. r Theorem 11. We have dimS¿(y) k = \^m=0(t>(m)fm, where fm is the number of m-simplices of S? Uy, and for m = 0, ... ,k, (¡)(m) = (f>m(0),where, with k-m-l , the quantities <p™(p)are defined recursively by Pm = rl pq-p j + m-q j *» = E ;=0 (51) P-J + Q Q 9-1 <7+ l i+X tíü + j) if 0 < q < m: and Po-P (52) ¿o» = E (53) •rí(O) j +m j Po - p + m m k-l -E d+k k m=0 8. Minimally k+ X m + X ¿(0). supported bases Definition 12. The star of a simplex a £ S? U T, denoted star(a), is the set of all fc-simplices A"G y such that a is a face of K. Definition 13. A basis {I : p = X, 2, ... , dimS^(y)} is said to be minimally supported if for each basis function / there exists a simplex a £ S? U T such that the support of / is contained in star (a). The basis functions constructed in the proof of Theorem 9 are minimally supported: using the same construction, we can define cardinal supersplines (54) so that, if \£Sé(a), /jeS^y): /I(J) = áIJ V\,i£sé then cA= 0 VJ G ld \ star(ff) and c, = 1. Acknowledgments This research was supported by the National Science Foundation with grant DMS-8701121 to the University of Utah. The authors also appreciate the careful work of the editor and the thoughtful comments on an earlier version of this paper by an anonymous referee. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use PETER ALFELDAND MARITZASIRVENT 308 Bibliography 1. P. Alfeld, Scattered data interpolation in three or more variables, Mathematical Methods in Computer Aided Geometric Design (Tom Lyche and Larry L. Schumaker eds.), Academic Press, New York, 1989, pp. 1-34. 2. _, A bivariate C Clough-Tocherscheme, Comput. Aided Geom. Design 1 (1984), 257267. 3. Peter Alfeld, 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. Design4 (1987), 105-123. 4. Peter Alfeld and L. L. 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Department of Mathematics, University of Utah, Salt Lake City, Utah 84112 E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use