Supersmoothness of bivariate splines and geometry of the underlying partition. T. Sorokina ) Abstract. We show that many spaces of bivariate splines possess additional smoothness (supersmoothness) that is not reected in their denitions. Smoothness is treated as vanishing of strongly supported smoothness functionals. We investigate the dependence of the phenomenon of supersmoothness on the geometry of the underlying partition and the degree of splines. In particular, we prove that bivariate splines of suciently low degree cannot have dierent smoothness across collinear edges in the underlying cell. Moreover, such splines do not have non-collinear edges in the cell, that is, such edges can be removed. AMS classication: Primary 4A5, 4A, 65D5, 65D7; Secondary 4A5. Key words and phrases: bivariate splines, smoothness functional, Bernstein-Bezier form, supersmoothness, cell, dimension. x. Introduction Multivariate polynomial splines with additional smoothness have been used in many applications. The most general notion of supersmoothness or superspline spaces was dened by Alfeld and Schumaker in []. Usually additional smoothness conditions are explicitly imposed on spline spaces, thus creating the so-called supersplines with certain desirable properties. However, intrinsic supersmoothness is the phenomenon of additional smoothness that has neither been imposed explicitly, nor has been reected in the standard description of these spaces. The rst observation of this phenomenon for a special case of cubic C splines over the Clough-Tocher split of a triangle goes back to [], where it was noted that the C spline space happens to be C at the split point automatically. The rst recognition of supersmoothness as an intrinsic property of multivariate spline spaces appeared in [5]. In this paper we concentrate on bivariate splines. It is common knowledge that dimensions of many bivariate splines depend on the exact geometry of the underlying partition. The details of this dependence, in general, remain unknown. One of the goals of this paper is to show that the converse is true as well: properties of spline spaces do aect geometry and topology of the underlying partition. For example, in Section 3 we show that splines of suciently low degree lose their ) Department of Mathematics, Towson University, Towson, MD 5, USA, tsorokina@towson.edu. Supported by the Faculty Development and Research Committee grant from Towson University.
spline structure and become polynomials on cells without collinear edges. Results of this kind follow from the intrinsic supersmoothness of bivariate splines { the main subject of our attention in this paper. The rst essential tool for our analysis is the dimension formulae (.3) for bivariate splines on cells. It is a direct generalization of the rst formula of such kind obtained by Schumaker in [4]. The second tool is the software designed by P. Alfeld for computing dimensions and nding minimal determining sets. It is available on www.math.utah.edun pa. We used this software extensively to conjecture our results, which was in some cases more dicult than to prove them. The paper is organized as follows. In Section, we review the notion of strongly supported smoothness functionals and their inuence on the dimension of bivariate polynomial splines on cells from [4] and [3]. Section 3 contains our main result on supersmoothness of bivariate splines on cells in terms of saturated strongly supported smoothness functionals. In Section 4, we consider applications of this result to more conventional spline spaces. Section 5 contains two results on yet entirely dierent types of supersmoothness that do not follow from those of Sections 3 and 4, and thus indicate that the subject is far from being closed. x. Preliminaries The space of bivariate C r -continuous polynomial splines over a triangulation 4 of a polygonal domain in IR will be denoted by S r d(4) := fs C r () : sj T P d for all triangles T 4g; where Pd is the space of polynomials of degree d in two variables. This paper is concerned with bivariate splines over special partitions called cells. Moreover, we study properties of splines on cells that are not aected by the boundary edges. Thus, in contrast to [3] (cf p. 34), we always assume that cells are interior, i.e., the common vertex in a cell is interior. Denition.. Suppose 4 is a partition consisting of a set of triangles which all share one common interior vertex. Suppose every triangle has exactly two neighbors with each of which it shares a common edge. Then we call 4 a cell. We rst describe two common types of supersmoothness { at the interior vertex and across an interior edge. If 4 is a cell with the interior vertex v then we say that every spline in the space S r; d (4) := Sr d(4) \ C (v); > r; has (super)smoothness at the interior vertex v. As always, C (v) is understood as equality of all derivatives of order at the point v. If 4 is a cell, e is one of the interior edges shared by two triangles T and T in 4, and > r, then we say that s S r d (4) has (super)smoothness across the edge e if s C (T [ T ).
V V V V V 4 V 4 Fig.. Tips of smoothness functionals are depicted as lled circles. We next describe a more general notion of supersmoothness assuming that the reader is familiar with the Bernstein-Bezier techniques for analyzing bivariate polynomial splines. An extensive treatment on the subject can be found in [3]. It is well known that spline spaces can be dened using smoothness functionals, see Chapter 5 of [3] for a detailed treatment. In this paper we dene smoothness functionals with an additional specication called level. Suppose T := hv ; v ; v 3 i and T e := hv4 ; v 3 ; v i are two triangles sharing an interior edge e := hv ; v 3 i of 4. Fix n j d. Then for any spline s Sd (4), we call j;e n a smoothness functional of order n on level j from v : n j;e s := c n;d j;j n X ++=n ~c ;j n+;d j+ e B (v ); where fc ijk g i+j+k=d and f~c ijk g i+j+k=d are the B-coecients of sj T and sj ~T, respectively, and f e B n g are the Bernstein basis polynomials of degree n associated with the triangle e T. Note that each smoothness functional n j;e is uniquely associated with its tip: the domain point n;d j;j n in T. Given a set T of smoothness functionals associated with oriented edges of 4, the space of smooth splines S T d (4) is dened as follows S T d := fs S d(4) : s = for all T g: We also need the notion of strongly supported smoothness functionals, see Chapter 9 of [3]. The set T of smoothness functionals dened on a set E of oriented edges of a triangulation is called strongly supported provided that for every e E; n j;e T ) m j i;e T ; i n m and m n: Figure illustrates this concept for d = 7, e = hv ; v 3 i. The gure on the left shows the supports of four smoothness functionals on level six from v, three smoothness functionals on level ve from v, two smoothness functionals on level four from v, and one smoothness functional on level three from v, namely, f n 6;eg 3 n= ; f n 5;eg n= ; f n 4;eg n= ; 3;e : 3
The gure on the right shows the supports of four smoothness functionals on level six from v, two smoothness functionals on level ve from v, two smoothness functionals on level four from v, and one on level three from v, namely, f n 6;eg 3 n= ; f n 5;eg n= ; f n 4;eg n= ; 3;e : The set of smoothness functionals depicted in Figure (left) is strongly supported while the one in Figure (right) is not. Lemma 9.4 in [3] assures that if T is strongly supported then the smoothness functionals are symmetric about the edge. That is, in Figure (left) the tips can be located in T ~ as well as in T. Given a set T of strongly supported smoothness functionals associated with a cell 4 with edges fe i g n i=, we assume that the levels of all of the smoothness conditions are counted from the common point of the cell. Given i n and j d, let r i;j := maxfk : k j;e i T g be the smoothness value in T associated with e i on level j. We note that each smoothness value r i;j is uniquely associated with a domain point on the edge e i at distance j from the common vertex. If T is strongly supported, then it can be described in terms of the smoothness values rather than smoothness functionals: T := fr i;j ; i = ; : : : ; n; j = ; : : : ; dg: The most general formula for the dimension of splines on bivariate cells is given by Theorem 9.5 in [3]. Let a cell 4 have n edges, fe i g n i=, whose slopes are fa ig n i=, respectively. We note that any cell can be rotated so that the slopes are dened. Given a set T of strongly supported smoothness functionals associated with 4 where m i;j := 8 < : dim S T d (4) = nx dx i= j= " j := (j r i;j ) + nx i= dx j= (j + " j ) + ; (:) m i;j ; (:) ; if there exists l with a i = a l and r l;j < r i;j, ; if there exists l > i with a i = a l and r l;j = r i;j, j r i;j ; otherwise. (:3) x3. Saturated smoothness values In this section we investigate when smoothness values in strongly supported conditions can be set higher than declared in the denition of the space. We consider strongly supported functionals only, and use the notation of (.), (.) and (.3). 4
Theorem 3.. Let Sd T (4) with strongly supported T be dened on a cell 4 with n edges. Given f; : : : ; ng and f; : : : ; dg, let r ; < be the smoothness value in T associated with the edge e on level. If T := T [ r ;+ ;e remains strongly supported, then Sd T (4) = ST d (4) if and only if " + ; and either (i) e has no collinear counterpart or (ii) e has a collinear counterpart with strictly higher smoothness value on level. Proof: Since S T (4) S T (4), we analyze when dim S T d (4) = dim ST d T remains strongly supported, we can rewrite (4). If T := fr i;j = r i;j ; if (i; j) 6= (; ) and r ; = r ; + g; and use (.) to compute m i;j := 8 < : dim S T d (4) = nx dx i= j= " j := (j r i;j) + nx i= m i;j ; dx j= (j + " j) + ; (3:) ; if there exists l with a i = a l and rl;j < r i;j, ; if there exists l > i with a i = a l and rl;j = r i;j, j ri;j ; otherwise. Thus from (.) and (3.) it follows that dim S T (4) dim S T (4) = + ( + " ) + ( + " ) + : (3:) We rst consider the case when e has no collinear counterpart. We note that m i;j = m i;j if (i; j) 6= (; ); and m ; = m ; ; Then from (3.) it follows that " j = " j ; if j 6= ; and " = " : dim S T (4) = dim S T (4), ( + " ) + ( + " ) + =, " = +: We next consider the case when e has a collinear counterpart e. Without loss of generality assume >. We note that in this case m i;j = m i;j if (i; j) 6= (; ) and (i; j) 6= ( ; ); while " j = " j ; for all j 6= : 5
We now analyze when (3.) vanishes by cases. ). If r ; r ;, then r ; = r ; + > r ; = r ;, and m ; = m ; = ; m ; = m ; = r ;: Thus " = " and (3.) does not vanish. ). If r ; r ;, then r ; = r ; + < r ; = r ;, and m ; = m ; = r ; ; m ; = m ; = : Thus " = " and (3.) yields dim S T (4) = dim S T (4), (+ " ) + (+ " ) + =, " = +: 3). If r ; = r ;, then r ; = r ; + = r ; = r ;, and m ; = r ; ; m ; = ; m ; = ; m ; = r ; : Thus " = " and (3.) yields again that dim S T (4) = dim S T (4), (+ " ) + (+ " ) + =, " = +: Combining ){3) shows that (3.) vanishes if and only if r ; " = + : This completes the proof. r ; and Examples below illustrate Theorem 3.. The rst example, see Fig., deals with a cell 4 consisting of three triangles. It shows that if s S (4) satises a C smoothness condition across one of the interior edges e i := hv ; v i i; i = ; ; 3, then s automatically satises C smoothness conditions across the other two edges. The corresponding sets of smoothness values are given by T = fr i; = ; i = ; ; 3; r ; = r ; = ; r 3; = g; T = fr i; = ; i = ; ; 3; r ; = ; r ; = r 3; = g; see Fig. (left and middle). Both T and T are strongly supported, and " = 3X i= ( r i; ) = + + : Thus, from Theorem 3. we obtain that S T (4) = S T (4). Next consider another strongly supported set given by smoothness values T = fr i; = ; i = ; ; 3; r ; = r ; = r 3; = g; 6
V V V V V V V V V Fig.. Smoothness values associated with the domain points on the interior edges for s S T (4), s S T (4) and s S T (4), respectively; " = ; " = ; and S T (4) = S T (4) = S T (4). V V V V V V Fig. 3. Smoothness values for the same space of cubic splines; "3 = 4. depicted in Fig. (right). In this case " = 3X i= ( r i; ) = + + ; and Theorem 3. yields S T (4) = S T (4). This space appears as Example 5.8 in [3], where a direct proof based on linear equations for the coecients is given. Another example for the same triangulation is depicted in Fig. 3. Both sets of smoothness functionals are strongly supported, and for = 3 we have " 3 = 3X i= (3 r i;3 ) = + + 4; thus both sets of smoothness values dene the same space of cubic splines. Our next example deals with collinear edges, see Fig. 4, where e := hv ; v i is parallel to e 3 := hv ; v 3 i and e := hv ; v i is parallel to e 4 := hv ; v 4 i. In this case, we have two sets of strongly supported functionals associated with the domain 7
points on two pairs of collinear edges: e := hv ; v i is parallel to e 3 := hv ; v 3 i and e := hv ; v i is parallel to e 4 := hv ; v 4 i. T = fr i; = ; i = ; ; 3; 4; r ; = ; r ; = r 3; = r 4; = g; T = fr i; = ; i = ; ; 3; 4; r ; = r 3; = ; r ; = r 4; = g: Next we compute m ; = ; m ; = ; m 3; = = ; m 4; = = ; " = ; and since " Theorem 3. yields S T (4) = S T (4). V 4 V 4 V V V V V V Fig. 4. Smoothness values associated with the domain points on the interior edges for s S T (4) on the left and s S T (4) on the right; " = ; and S T (4) = S T (4). We end this section by showing that Theorem 3. can be simultaneously applied to smoothness conditions associated with dierent edges. Corollary 3.. Let Sd T (4) with strongly supported T be dened on a cell 4 with n edges. Given f ; : : : ; k g f; : : : ; ng and f; : : : ; dg, let r i ; <, i = ; : : : ; k, be the smoothness value in T associated with the edge e i on level. S Let T := k r ; i + ;e i [T. If for each i = ; : : : ; k, the set of smoothness conditions i= T i := r ; i + ;e i [ T remains strongly supported, then Sd T if " + ; and for each i = ; : : : ; k, either (i) e i has no collinear counterpart or (4) = ST d (4) if and only (ii) e i has a collinear counterpart with strictly higher smoothness value on level. Proof: Applying Theorem 3. to a xed i, we obtain that for each i = ; : : : ; k, S T d (4) = S T i d (4); if and only if " + and either (i) or (ii) holds; and thus S T d (4) ST d (4). 8
x4. Supersmoothness across interior edges In this section we discuss spline spaces of degree d for which it is possible to saturate smoothness across an interior edge. We show that bivariate splines of suciently low degree cannot have dierent smoothness across collinear edges. Additionally, we show that bivariate splines of suciently low degree do not have non-collinear edges in the underlying cell, that is, such edges can be removed. The rst such example is given in Fig., where all three edges are extraneous, and the split can be considered as one triangle. The main result of this section is a generalization of Theorems 3. and 3. in [5]. Although it might be possible to improve some multivariate results from [5], we chose not to deal with spacial dimensions higher than two here since we have not been able to obtain sharp estimates in those cases. In Theorem 4. below, we use the following convention: if an edge e in the cell does not have a collinear counterpart, we add it with associated supermoothness d across e. Thus we shall be taking about a cell with m slopes, not edges. Each slope corresponds to a pair of collinear edges. Moreover, conditions (i) and (ii) in both Theorem 3. and Corollary 3. can be reduced to (ii) alone. Theorem 4.. Let 4 be a cell with m slopes and m pairs of collinear edges. Suppose T is dened by the following smoothness conditions: for each pair of collinear edges (e i ; ~e i ), let (r i ; i ) be the smoothness across e i and ~e i, respectively, with the convention r i i d. Suppose T is dened by the following smoothness conditions: for each pair of collinear edges (e i ; ~e i ), let i be the smoothness across both of them. Then S T d (4) = S T d (4); whenever d d := $P m % r i= i + m Proof: We number the edges so that r = minfr i g m i=. Then (.) yields : " j := ( ; j = ; : : : ; r ; mp j minfj; ri g ; j = r + ; : : : ; d: (4:) i= Next we describe an algorithm that applies Corollary 3. as many times as possible, i.e. M := maxf i r i g m i= times. The algorithm raises smoothness values by one on all m edges e i starting at level d and down to level r, since Sd T (4) has overall smoothness r. Then it goes back to level d and repeats until possible. The algorithm is set up to terminate prematurely (in step 3) when the set of smoothness functionals is about to cease to be strongly supported. Algorithm.. Set k =.. Set j = d. 9
3. If (j; k) := j + m P i= j minfj; i ; r i + kg, go to step 4. If not, terminate. 4. If j > r + k, set j = j and go to step 3. If not, go to step 5. 5. If k < M, set k = k + and go to step. If not, terminate. In order to prove the theorem we need to show that if d d the algorithm does not terminate prematurely, that is the condition in step 3 is always true. For a xed j, the function (j; k) is immediately seen to be a non-decreasing function of k. Thus we know that (j; k) (j; ) = j + " j ; for all j = r + ; : : : ; d: We now use (4.) to obtain an estimate on (j; ). If j = r + ; : : : ; d, then (j; ) = j + = + r = d + mx j minfj; ri g = j + (j r ) mx i= mx i= i= This completes the proof. maxf; j r i g + r mx i= (d r i ) (d r i ) = (d; ) ; for d d : mx j minfj; ri g Example 4.3. For a cell with ve slopes, let two non-collinear edges have smoothness 7 and 6 across them. Let three pairs of collinear edges have the following pairs of smoothness across them (7; 7); (5; 7) and (6; 7). Then for d = 8, we have the following pairs of smoothness values: (5; 7); (6; 8); (6; 7); (7; 8); (7; 7): The overall smoothness is r = 5, and d = 8. Thus, the two non-collinear edges can be removed. Moreover, the smoothness across collinear edges rises. The new cell 4 with three slopes and three pair of collinear edges hosts the space S 7 8 (4). This result can be easily checked with P. Alfeld's software. We conclude this section with two results that follow from Theorem 4.. Corollary 4.4. Let 4 be a cell with n non-collinear edges e i ; i = ; : : : ; n, and let T be dened by P smoothness r i across each e i. Then each spline in Sd T (4) has n supersmoothness ( r i= i + )=(n ) at the interior vertex of 4. Remark 4.5. Let 4 be an arbitrary triangulation 4 in IR used as an underlying partition for Sr+(4). r From Theorem 4. it follows that 4 can be reduced by removing all interior non-collinear edges for cells with m r +. We note that the new reduced partition remains a triangulation if r =. i=
x5. Other types of supersmoothness The goal of this section is to demonstrate that despite the fact that Theorem 3. is the most general result on increased smoothness values, it does not describe all possible types of intrinsic supersmoothness of bivariate splines. Our rst result shows that even in the least favorable situation for supersmoothness, i.e., in the most symmetric one, there is still extra smoothness present. Theorem 5.. Let 4 be a cell with no non-collinear and l collinear edges meeting at v. Then for any s S l @ d (4) any l-th order mixed derivative l s @u i @u (v), where i l u i ; : : : ; u il are pairwise distinct directions of non-collinear edges, exists. Proof: Since any l-th order derivative at v is fully dened by the B-coecients associated with the disc of radius l around v, it suces to prove the theorem for s S l l (4), in which case any l-th order derivative is a constant. We use induction on l. Let l =, and without loss of generality let u and u be the directions of x and y-axes. Then for any quadratic spline s S l l (4) there exist a quadratic polynomial p and constants A and B such that s(x; y)j H4 = p(x; y); s(x; y)j H = p(x; y) + Ay ; s(x; y)j H = p(x; y) + Ay + Bx ; s(x; y)j H3 = p(x; y) + Bx ; where H i is the i-th quadrant, i = ; : : : ; 4. By direct dierentiation it is clear that @ s s(x; y) = @ @x@y s s(x; y) = @ @y@x s p(x; y): @x@y Assume now that the statement is true for l, and consider s S l l (4). The derivative @s @u is in S l (4). Moreover, this derivative is l Cl across both edges with slope u. Thus we can remove those edges from 4, and obtain a new cell 4. The derivative @s @u S l (4 l ) satises the induction hypothesis. Thus, any l -st @ order mixed derivative l s @s @u i @u i l @u ; where all pairwise distinct u ij 6= u. We observe that we can repeat this argument for @s @u i for any i = ; : : : ; l. Our next result describes a type of supersmoothness that has not been encountered anywhere and does follow from any of the results proved in this paper. For an illustration see Fig. 5, and its description following the lemma. Lemma 5.. Let 4 be a cell with four non-collinear edges meeting at the point v. Then there exists a unique straight line passing through v with the property that for any smooth quadratic spline s on 4, the restriction of s on this line is a univariate quadratic polynomial. Proof: Assume v is the origin in IR, and the edge e is aligned with the positive direction of the x-axis. Since 4 comes from a triangulation, every angle in the cell
is less than. Thus, perhaps after reecting 4 about the x-axis, we can align the cell in the counterclockwise order with the following rays of pairwise distinct slopes: e : y = ; x : e : x = a y; y ; e : x = a y; y ; e 3 : x = a 3 y; y : Moreover, in this case a > a, and there are two cases for a 3 : (i) a 3 > a > a or (ii) a > a 3 > a : (5:) It is useful to visualize a ; a ; a 3 as points on the cotangent line of the unit circle. With v being the origin, we next introduce the following notation for the triangles: T := he 3 ; v; e i; T := he ; v; e i; T := he ; v; e i; T 3 := he ; v; e 3 i: Using simple linear algebra, is is easy to see that any smooth quadratic spline s on 4 can be written as follows: s(x; y)j T = p(x; y); s(x; y)j T = p(x; y) + Ay ; s(x; y)j T = p(x; y) + Ay s(x; y)j T3 = p(x; y) + = p(x; y) + Ay A(x a y) (a a )(a a 3 ) ; A(x a 3 y) (a 3 a )(a 3 a ) A(x a y) (a a )(a a 3 ) A(x a y) (a a )(a 3 a ) ; where p is a quadratic polynomial, and A is an arbitrary constant. For all j 6= i, we consider equations of the type s("y; y)j Ti s("y; y)j Tj. Clearly, the only solution on neighboring triangles is A =. So, we are left with two possibilities: s("y; y)j T s("y; y)j T ; or s("y; y)j T s("y; y)j T3 : Taking into account (5.), the rst equation has two solutions the second equation has two solutions " = a p (a a )(a a 3 ); only if (ii); " = a 3 p (a 3 a )(a 3 a ); only if (i): Moreover, only x = a p (a a )(a a 3 ) y lies in both T and T. Indeed, a < a p (a a )(a a 3 ) < a 3 :
Similarly, only the straight line x = a 3 + p (a 3 a )(a 3 a ) y lies in T and T 3, since p (a 3 a )(a 3 a ) + a 3 > a 3 : While it is straightforward to use P. Alfeld's applet to nd examples for the results of Sections 3 and 4, there are no direct provisions for those of Section 5. We describe how to create one such example. The le to be imported contains the following: number of vertices 7, number of triangles 6; coordinates of the vertices: (; ); (; ); (; ); ( 6; 8); ( ; 5); (; ); (; ); and the connectivities are: ( ); ( 3); ( 3 4); ( 4 5); ( 5 6); ( 6 ): Set r =, d =, and supersmoothness two across the edges [v ; v 3 ], and [v ; v 6 ]. This makes the partition into a cell with four interior non-collinear edges, see Fig. 5 (left). The line [v 3 ; v 6 ] is the unique line described by Lemma 5.. It can be checked by setting the three coecients on the edge [v ; v 3 ], and observing that the three coecients on the edge [v ; v 6 ] get determined at the same time, see Fig. 5 (right). This happens because a univariate quadratic polynomial is dened by three coecients. 5 5 6 6 4 3 4 3 Fig. 5. Example created in P. Alfeld's applet to illustrate Lemma 5.. References. Alfeld, P. and L. L. Schumaker, Upper and lower bounds on the dimension of superspline spaces, Constr. Approx. 9 (3), 45{6.. Farin, G., Bezier polynomials over triangles, Report TR/9, Dept. of Mathematics, Brunel University, Uxbridge, UK, 98. 3. Lai, M. J. and L. L. Schumaker, Spline Functions on Triangulations, Cambridge University Press, Cambridge, 7. 4. Schumaker, L. L., On the dimension of spaces of piecewise polynomials in two variables, in Multivariate Approximation Theory, W. Schempp and K. Zeller, eds, Basel, Birkhauser, 979, 396{4. 5. Sorokina, T., Intrinsic supersmoothness of multivariate splines, Numer. Math. 6 (), 4{434. 3