Dimension and Local Bases of Homogeneous Spline Spaces

Dimension and Local Bases of Homogeneous Spline Spaces by

Peter Alfeld 1), Marian Neamtu 2), and Larry L. Schumaker 3) Abstract. Recently, we have introduced spaces of splines de ned on trian-

gulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein-Bezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangulations. Rather than working with splines on sphere-like surfaces directly, we instead investigate more general spaces of homogeneous splines in IR3 . In particular, we present formulae for the dimensions of such spline spaces, and construct locally supported bases for them.

1. Introduction Let
:= fT [i] gN1 be a planar triangulation of a set , and let 0  r  d be

integers. The classical space of splines of degree d and smoothness r is de ned by Sdr (
) := fs 2 C r ( ) : sjT [i] 2 Pd; i = 1; : : : ; N g; (1:1) where Pd is the space of bivariate polynomials of degree at most d. These spaces of spline functions have found numerous applications in interpolation, data tting, nite element solutions of boundary value problems, CAGD, image processing, and elsewhere. There is a well-developed (albeit incomplete) constructive theory for the polynomial spline spaces Sdr (
) which includes 1)

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, [email protected]. Supported by the National Science Foundation under grant DMS-9203859. 2) Department of Mathematics, Vanderbilt University, Nashville, TN 37240, [email protected] 3) Department of Mathematics, Vanderbilt University, Nashville, TN 37240, [email protected]. Supported by the National Science Foundation under grant DMS-9208413.

2 1) 2) 3) 4) 5)

dimension formulae construction of local bases estimates on the approximation power algorithms for manipulating the splines algorithms for interpolation, data tting, etc. Recently [4], we introduced analogous spaces of splines de ned on a triangulation on the sphere or on a sphere-like surface. As suggested by our companion paper [5], we believe that such spaces have important applications, and hence it is important to develop the analogous constructive theory. Following [4], we will analyze spherical splines by investigating a more general class of splines associated with a trihedral decomposition T := fT [i] gN1 of a set  IR3 (see Sect. 2 below). Given such a decomposition, the associated spaces of homogeneous splines are de ned by

Hrd(T ) := fs 2 C r ( ) : sjT [i] 2 Hd; i = 1; : : : ; N g;

(1:2)

where Hd denotes the space of trivariate polynomials of degree d which are homogeneous of degree d (recall that a function f de ned on IR3 is homogeneous of degree d provided f (v) = d f (v) for all real numbers  and all v 2 IR3). Splines on the sphere or on a sphere-like surface S are then obtained by restricting Hrd(T ) to S . The main purpose of this paper is to establish dimension formulae for spaces of homogeneous splines, and to show how to construct bases of locally supported splines. Homogeneous splines can be stored and evaluated using the algorithms presented in [4] for homogeneous polynomials. The question of the approximation power of homogeneous and spherical splines will be dealt with elsewhere. Applications to the interpolation and tting of scattered data on the sphere or on a sphere-like surface are discussed in [5]. Even though we are working in IR3, because of the nature of homogeneous polynomials|which are essentially bivariate functions|the entire development is closely modelled after the analysis of the bivariate spaces of splines Sdr (
) carried out in [7, 15, 16].

2. Homogeneous Spline Spaces

We begin by introducing some notation, closely following [4]. De nition 2.1. Let fv1 ; v2; v3 g be a set of linearly independent unit vectors in IR3. We call

T = fv 2 IR3 : v = b1 v1 + b2 v2 + b3v3 with bi  0g

(2:1)

the trihedron generated by fv1; v2 ; v3g. As in [4], we call the real numbers b1 , b2 , b3 the trihedral coordinates of v with respect to T . They are homogeneous linear functions in the coordinates of v.

3 We call the set fv 2 T : bi = 0g the (i-th) face of T , and the set fvi :   0g the (i-th) ray of T (or the ray generated by vi ). To avoid awkward repetitions, we abuse our notation slightly: in addition to writing v for a unit vector, we also use v to denote the associated point in IR3 and the associated ray generated by v. De nition 2.2. Let T = fT [i] gNi=1 be a non-empty set of trihedra, and let

:= [T [i] . Then we call T a trihedral decomposition of provided 1) the interiors of the trihedra in T are pairwise disjoint, 2) the set \ S is homeomorphic to a two-dimensional disk or equals S , where S is the unit sphere, 3) each face of a trihedron in T is either on the boundary of or it is a common face of precisely two trihedra in T . Each of the T [i] \ S is a spherical triangle, and
= fT [i] \ S gNi=1 is a spherical triangulation, cf. [18]. We say a trihedral decomposition T is total if

= IR3. Otherwise, we say that it is partial. It will be convenient to denote the set of unit vectors de ning the rays of the trihedra in T by V . If T is a partial trihedral decomposition, it is natural to de ne rays to be boundary rays of T provided they are associated with vectors v 2 V which lie on the boundary of . All other rays will be called interior rays. We denote the sets of boundary and interior rays in T by VB and VI , respectively. Clearly, all rays of a total trihedral decomposition are interior rays. Following the notation used for planar triangulations, we denote the number of boundary and interior rays of T by VB and VI , respectively. Similarly, we denote the number of boundary and interior faces of T by EB and EI . For a partial decomposition, the number of rays is given by V := VB + VI , and the number of faces is given by E := EB + EI . For a total decomposition, V = VI and E = EI . Let T be a trihedron generated by fv1; v2 ; v3 g, and let b1 , b2 , b3 denote the corresponding trihedral coordinates as functions of v 2 IR3. The homogeneous Bernstein basis polynomials of degree d associated with T are the polynomials d (v ) := d! bi bj bk ; Bijk i!j !k! 1 2 3

i + j + k = d:

(2:2) ?




The space Hd of trivariate homogeneous polynomials is a d+2 dimensional ?2d+2
linear space, and as observed in [4], it is spanned by the set of 2 Bernstein basis polynomials de ned in (2.2). Thus each p 2 Hd can be written uniquely in the form X d (v ): p(v) = cijk Bijk (2:3) i+j +k=d

In [4], p is referred to as a homogeneous Bernstein-Bezier (HBB-) polynomial of degree d.

4 It will be convenient to de ne the domain points associated with T to be the points Pijk := iv1 + jvd2 + kv3 ; i + j + k = d: (2:4)

In contrast to the case of polynomial splines on planar triangles, this de nition of Pijk is not the only natural one (see Remark 9.3). If we look at all of the domain points for all of the trihedra in a trihedral decomposition, it is clear that the domain points associated with a common face of two trihedra fall on top of each other. If we eliminate such repetitions, we see that for a given trihedral decomposition T , there is ?one
point associated with each ray, d ? 1 points associated with each face, and d?2 1 associated with the interior of each trihedron. Thus the set G of distinct domain points has cardinality   d ? 1 (2:5) #(G ) = V + (d ? 1)E + 2 N: The importance of the HBB-form of homogeneous polynomials is that it provides a simple way to describe when two such polynomials de ned on adjoining trihedra join together smoothly. Indeed, suppose T [1] and T [2] are two trihedra generated by the sets fv1 ; v2; v3 g and fv1 ; v3; v4 g, respectively. Then as shown in [4], the two associated homogeneous polynomials p[1] and p[2] of degree d agree on the face shared by T [1] and T [2] in value and all derivatives up to order r if and only if

c[2] ijk =

X

+ +=k

k c[1] i+;;j + B (v4 ) for all k  r; i + j + k = d;

(2:6)

k are the Bernstein basis polynomials of degree k associated with T [1] . where B By (2.6), p[1] and p[2] join continuously across their common face, if and only if [1] c[2] i + j = d: (2:7) ij 0 = ci0j ; We conclude that a spline s 2 H0d(T ) is uniquely de ned by a set of #(G ) coecients, one associated with each point P 2 G . This implies that the space H0d(T ) is of dimension #(G ). For later use, for each P 2 G , it will be convenient to de ne a linear functional P de ned on H0d(T ) with the property that for any s 2 H0d(T ),

P s = cP ;

(2:8)

where cP is the coecient associated with the point P . We denote the set of all such linear functionals by . Clearly #() = #(G ). For each  2 , there is a unique spline s 2 H0d(T ) such that

s =  ;; all 2 :

(2:9)

5 The spline s has all coecients equal to 0 except for the coecient s which has value 1. By construction, s has one of the following supports: 1) a single trihedron T if the coecient s is associated with a domain point in the interior of T , 2) a pair of adjoining trihedra if the coecient s is associated with a domain point in the interior of a face separating two trihedra, 3) the union of all trihedra which share the ray v if the coecient s is associated with the domain point v. In view of these properties, we say that such splines have local support. The duality property (2.9) assures that the splines s for  2  are linearly independent, and since there are precisely #(G ) of them, they form a basis for H0d(T ). To obtain analogous results for Hrd(T ), we follow [7, 15, 16]. To get an upper bound on dimension, we construct a determining set ?   such that if s 2 Hrd (T ),

s = 0 for all 2 ? implies s  0: (2:10) Then as shown in [7], dim Hrd (T ) is bounded above by the cardinality of ?. We can get a lower bound for the dimension (and construct a basis at the same time) if ? is chosen so that for each  2 ?, there exists a spline s 2 Hrd(T ) satisfying

s =  ;; all 2 ?:

(2:11)

This duality implies that the splines fsg are linearly independent, and it follows that the dimension of Hrd(T ) is equal to the cardinality of ? and that these splines form a basis. Such a set ? is called a minimal determining set. We close this section by presenting the main result of the paper. Its proof will be developed in the following sections. Theorem 2.3. Let r  0 and d  3r+2. Suppose T is a trihedral decomposition of a set  IR3. Let

 :=

X

v2VI

v ; where v :=

d?r X m=1

(r + m + 1 ? mev )+

(2:12)

where ev is the number of distinct planes containing the faces that meet at the ray v. Then dim Hrd (T ) = (d ? r)(d ? 2r)V ? 2d2 + 6dr ? 3r2 + 3r + 2 + ; if T is a total decomposition, and dim Hrd(T ) = (d ? r +21)(d ? r) VB + (d ? r)(d ? 2r)VI 2 2 ? 2d ? 6dr +23r ? 3r ? 2 + ;

(2:13)

(2:14)

6 if T is a partial decomposition. In either case there exists a basis for Hrd(T ) consisting of splines such that the support of each spline is either one trihedron, an adjoining pair, or an orange.

3. Minimal Determining Sets for Splines on Oranges

In [7, 15] the key to analyzing the dimension of bivariate spline spaces was rst to examine the special case of a cell consisting of a set of triangles sharing one vertex. In this section we construct minimal determining sets for spline spaces on the trihedral analog of cells. In the context of tetrahedral decompositions these were called oranges in [9, 19]. Throughout this section we assume only that 0  r < d. De nition 3.1. A trihedral decomposition T consisting of a set of trihedra sharing one ray v is called an orange. We call v the axis of the orange. Suppose the trihedra in T are labeled in counterclockwise order as T [1] , T [2] , : : :, T [N ] as we move around the axis v, where the rays of T [`] are v, v` , and v`+1. If v is an interior ray, we have vN +1 = v1. We can label the domain points in these trihedra as [`] := iv + jv`d+ kv`+1 ; Pijk i + j + k = d: (3:1) Theorem 3.2. If T is an orange associated with a boundary ray v, then     d + 1 ? r d + 2 r : (3:2) dim Hd (T ) = 2 + (N ? 1) 2 If T is an orange associated with an interior ray v, then  X    d?r d ? r + 1 r + 2 r + (r + m + 1 ? me)+; (3:3) dim Hd (T ) = 2 + N 2 m=1 where e denotes the number of distinct planes shared by trihedra in T . Proof: Let  be a plane which intersects the axis of T at a point w which is not the origin, and so that  is perpendicular to the axis. The intersections with  of those faces of T which contain the axis are rays in  emanating from w. If we replace them with unit line segments with one end at w and connect their other endpoints, we get a planar triangulation
consisting of a set of triangles sharing the vertex w. Clearly, the restriction of a spline in Hrd(T ) to
is a spline in Sdr (
). Conversely, by the homogeneity of the splines in Hrd (T ), a spline in Sdr (
) extends uniquely to a spline in Hrd(T ). The two spaces Sdr (
) and Hrd(T ) are therefore isomorphic, and the dimension assertion follows from Theorem 2.2 in [16]. Following the proofs of Lemma 3.1 in [15] and Lemma 3.1 in [7], we now construct minimal determining sets for Hrd(T ) when T is an orange. We need the concept of a ring of domain points around a ray v.

7

De nition 3.3. Let T be an orange as above. Then given an integer d, the m-th ring of T is the set of domain points n

[`]

Pd?m;j;k : j + k = m; ` = 1; 2; : : : ; N

o

(3:4)

The m-disk in T is the union of the 0-th through m-th rings. Theorem 3.4. Suppose T is an orange associated with a boundary ray v. Then the set N n n o [ o [1] [`] Pijk : i + j + k = d [ Pijk : k  d ? r (3:5) `=2

is a minimal determining set for Hrd (T ). Suppose T is an orange surrounding an interior ray v, and let e be as in Theorem 3.2. Let

N ?e+1 < N ?e+2 <


< N ?1 = N; N = N + 1; and let

1 < 2 <


< N ?e be the complementary set so that

(3:6) (3:7)

f1; 2 ; : : : ; N g = f2; 3; : : : ; N + 1g:

(3:8)

Let ?0   be the set of functionals corresponding to domain points in the trihedron T [1] . In addition, for each m = 1; : : : ; d ? r, let ?m be the set of functionals corresponding to the rst Nm ? (r + m + 1) + (r + m + 1 ? me)+ points in the ordered set

fPd[?1m] ?r;m?1;r+1 ; : : : ; Pd[?1m] ?r;0;m+r ; : : : ; Pd[?Nm]?r;m?1;r+1 ; : : : ; Pd[?Nm]?r;0;m+r g:

(3:9)

Then ?=

d[ ?r m=0

(3:10)

?m

is a minimal determining set for Hdr (T ). Proof: We prove the result only for the case where the axis of the orange is an interior ray; the other case is similar. It is easy to check that the cardinality of ? is given by the formula (3.3), and so we only need to show that ? is a minimal determining set. To that end, consider the plane  that is perpendicular to the vector v and passes through the point v. Explicitly, 



 = u 2 IR3 : (u ? v)
v = 0 ;

(3:11)

8 where
denotes the ordinary dot product. Let w` denote the orthogonal projection of v` onto , i.e.,   w` = v` + 1 ? vv`

vv v; ` = 1; 2; : : : ; N: (3:12) The intersection of T with  forms a two-dimensional cell
in the sense of [16]. Let ?
denote the functionals de ned on Sdr (
) corresponding to the projections of the points de ning ?. In view of the correspondence between bivariate polynomials and trivariate homogeneous polynomials, by Theorem 3.3 of [16], ?
is a (minimal) determining set of Sdr (
). The fact that ? is a determining set for Hrd(T ) now follows from a careful comparison of the smoothness conditions for Sdr (
) and Hrd(T ). Any spline s 2 H0d(T ) can be expressed on the trihedron `] . To obtain the smoothness conditions for T [`] in the form (2.3) with cijk = c[ijk Hrd(T ) we write v`+1 = r`v + s` v`?1 + t`v`; (3:13) where for convenience we treat all rays, domain points, and coecients cyclically as we move around v (so that v0 = vN for example). By (2.6), a spline in Hd0 (T ) belongs to Hdr (T ) if and only if for all ` = 1; 2; : : : ; N , k!    `+1] = X c[`] (3:14) c[ijk i+;;j + ! !! r` s` t` ; + +=k for k  r and i + j + k = d. Consider now the corresponding smoothness conditions for Sdr (
). It can be checked that the projections of the v` satisfy w` = r~` w + s` w`?1 + t`w`+1; (3:15) where w is the projection of v and r~` := 1 ? s` ? t`: (3:16) Using a tilde to denote the coecients of a spline in Sdr (
) we obtain the conditions k!    `+1] = X c~[`] c~[ijk ` = 1; : : : ; N: (3:17) i+;;j + ! !! r~` s` t` ; + +=k

We now show that ? is a determining set for Hrd(T ). Consider a spline s 2 Hrd(T ). We work our way through the rings of the orange. The 0-th ring is v itself. It is in ? and therefore the coecient corresponding to it must be zero. Suppose now that the coecients corresponding to the rst m rings are all zero and consider the (m + 1)-th ring and the smoothness conditions (3.14) and (3.17) for k = m. In spite of r` and r~` being dierent, these equations are equivalent since the terms where r` 6= 0 and r~` 6= 0 contain coecients which are zero by the induction hypothesis. Thus, the coecients of s must vanish on the (m +1)-th ring and it follows that ? is a determining set. Since it has cardinality equal to the dimension of Hdr (T ), it follows that ? is minimal.

9

Remark 3.5. The argument used in the proof of Theorem 3.4 applies to all

minimal determining sets which have #?m points on the (r + m)-th ring for m = 1; : : : ; d ? r. However, it is not true in general that the analog of a minimal determining set for a two dimensional cell is also a minimal determining set for a corresponding orange as is shown in the following example.

Fig. 1. A non-determining set.

Example 3.6. Let T be an orange with N = 4 and v3 = ?v1 and v4 = ?v2 . Discussion: Figure 1 shows a minimal determining set for S21(
) that is not determining for H21 (T ). Points corresponding to functionals not in the set are

marked with a dot. Functionals corresponding to all other points are in the set. Note that in particular the functional corresponding to the center point (which is at v) is not in the set. Clearly, in the two-dimensional cell the coecients at the points marked with a crosshair () or a triangle (4) determine the coecient at v. In fact we have (3:18) cv = c +2 c4 : On the other hand, the relevant smoothness condition for H21 (T ) is

c = ?c4;

(3:19)

and so the two points marked with  or 4 cannot both be in the minimal determining set. It is of course easy to construct sets that are minimal determining for both S21(
) and H21(T ). An example (conforming to Theorem 3.4) can be obtained from Figure 1 by replacing the point marked with  with v.

10

4. A Minimal Determining Set for Hrd(T ) when d  3r + 2 In this section we construct a minimal determining set ? for Hrd(T ) in the case where d  3r + 2. As in the bivariate case [7, 15], the key to the construction is

to partition the Bezier coecients into suitable subsets. Suppose the trihedron T is generated by the vectors v1, v2 , and v3, and let P := fPijk gi+j+k=d be the associated set of Bezier coecients. To make the description of ? easier, we recall the correspondence between coecients, domain points, and the associated linear functionals cijk  Pijk  Pijk ; (4:1) and work only with domain points Pijk here. We de ne the distance of Pijk from the ray v to be ( d ? i if v = v1 dist (Pijk ; v) := d ? j if v = v2 (4:2) d ? k if v = v3 . For i = 1; 2; 3, let

D (vi ) := P 2 P : dist(P; vi )   A(vi ) := P 2 P : dist(P; vi )  ; dist( P; vi+1 )  d ? r; dist(P; vi+2 )  d ? r  F (vi ) := P 2 P : dist(P; vi )  d ? r E (vi ) := P 2 F (vi ) : jdist(P; vi+1 ) ? dist(P; vi+2 )j   ? r BL (vi ) := [F (vi ) \ D2r (vi+1 )] n [D (vi+1 ) [ A(vi+1 ) [ E (vi )] BR (vi ) := [F (vi ) \ D2r (vi+2 )] n [D (vi+2 ) [ A(vi+2 ) [ E (v i )] C := P 2 P : dist(P; vj )  d ? r; j = i; i + 1; i + 2 ; where

(4:3)

 := r + b r +2 1 c; (4:4) and we identify v4 = v1 and v5 = v2 . The set D (vi ) contains the points in a disk around vi of radius . A(vi ) (called a cap in [15]) is the set of points not in D (vi ) but whose corresponding coecients are involved in smoothness conditions of order up to r across the two faces sharing vi . The sets E (vi ), BL (vi ) and BR (vi ) include only domain points whose corresponding coecients are involved in smoothness conditions across the face opposite the ray vi . Finally, C corresponds to coecients which do not enter any smoothness conditions. In Figure 2 we have marked the domain points associated with one trihedron for the case d = 23 and r = 6 to show which of the above sets they belong to. Dots correspond to points in the sets D (vi ), circles to points in the sets E (vi ), asterisks to points in the caps A(vi ), plus signs to points in the sets BL(vi ) and BR(vi ), and 's to points in the set C .

11

Fig. 2. Division of domain points, d = 23, r = 6,  = 9. As in the bivariate case, in order to describe a minimal determining set for

Hrd(T ), we have to take account of certain degenerate faces. In [15] an edge F of

a planar triangulation is de ned to be degenerate at one of its endpoints v if the edges preceding and succeeding F and connected to v are collinear. We require a similar concept for trihedral decompositions: De nition 4.1. Let F be an interior face of a trihedral decomposition T , and let v be one of the two rays generating it. We say that F is degenerate at v if the faces other than F of the two trihedra sharing F and meeting in v are coplanar. We also need to adapt the familiar concept of a singular vertex. De nition 4.2. An interior ray v of a trihedral decomposition T is said to be singular if it has precisely four faces meeting at v which lie in two distinct planes. In contrast to the planar case where an edge can be degenerate at only one endpoint, for trihedral decompositions, it is possible for a face to be degenerate at both of the rays de ning it, see Example 7.1 below. We are now ready to describe a minimal determining set ? for Hrd(T ) in the case d  3r + 2. Algorithm 4.3. If d  3r + 2, choose the set ? as follows: 1) For each interior ray v of T , choose a minimal determining set for Hr (Tv ) as described in Theorem 3.4, where Tv is the orange surrounding v. 2) For each boundary ray v of T , choose a minimal determining set for Hr (Tv ) as described in Theorem 3.4. where Tv is the orange containing v. 3) For each trihedron T in T , choose the functionals corresponding to C and all three of the sets A(vi ) associated with T .

12 4) For each face F in T , include the functionals corresponding to the set E (v) associated with a ray v in an adjoining trihedron and opposite to F . If F is a boundary face, there is only one such trihedron, while if it is an interior face, we can work with either of the two trihedra sharing it. If F is a boundary face, also include the functionals associated with the two sets BL (v) and BR(v). 5) Suppose T [1] and T [2] are two trihedra generated by the vectors v1 , v2, v3 and v1 , v3 , v4, respectively. Suppose the common face F is degenerate at v1. Without loss of generality, suppose that in the previous step ? includes the set E (v2 ) in T [1] . Then in de ning ?, remove the cap A(v1 ) in T [2] and replace it with the set BL (v1 ) in T [1] . If the cap A(v1 ) in T [2] has already been moved because the other face of T [2] passing through the ray v1 is degenerate at v1 , take no action. If F is degenerate at both v1 and v3 , carry out this step for v3 as well. (The eect of this step is illustrated in Figures 1 and 2 in [15]). 6) If v is singular, add the functionals corresponding to one cap A(v) in one of the trihedra containing v.

Theorem 4.4. Let T be a trihedral decomposition and let d  3r +2 and r  0.

Then the set ? constructed in Algorithm 4.3 is a minimal determining set for Hrd(T ), and its cardinality is given by (2.13) if T is total, and by (2.14) if T is partial. For each  2 ?, let s 2 Hrd(T ) be the unique spline such that (2.11) holds. Then fsg2? forms a basis for Hrd (T ) such that the support of each spline is either one trihedron, an adjoining pair, or an orange. Proof: We give the proof only in the case where T is total as the case where it is partial is very similar. First we observe that the cardinalities of the sets de ned in (4.3) are as follows:   + 2 #D(vi ) = 2   2 r ?  + 1 #A(vi ) = #BL(vi ) = #BR(vi ) = 2 (4:5) 9 15 2 2 #E (vi ) = dr + d + 6r ? 2 r ? 1 ? 2 r ? 2   d ? 3 r ? 1 : #C = 2 Moreover, the sets are pairwise disjoint and their union is the set of all domain points in the trihedron T . Next, we show that the cardinality of the set ? is given by (2.13) when T is total. It can be shown that in this case 

N = 2(V ? 2) and E = 3(V ? 2);

(4:6)

13 where E is the number of faces of T . Note that step 5 of Algorithm 4.3 does not change the cardinality of ?, and that step 2 does not contribute since there are no boundary rays. With these observations it follows from Algorithm 4.3 and Theorem 3.4 that 









r + 2 + E  ? r + 1 + ~ #? = v v 2 v2VI 2    2 r ?  + 1 d ? 3 r ? 1 +3 + N 2 2   9 15 2 2 + E dr + d + 6r ? 2 r ? 1 ? 2 r ? 2   2 r ?  + 1 ; + K 2 X

(step 1) (step 3) (step 4)

(4:7)

(step 6)

where Ev is the number of interior faces meeting at the ray v, K is the number of singular rays, and

~v := Using

X ?r m=1

(r + m + 1 ? mev )+ :

X

v2T

Ev = 2E

(4:8)

(4:9)

and (4.6), the equality of the right hand sides of (4.7) and (2.13) follows after a straightforward manipulation. (Note that for singular rays, the ~v and the factor multiplying K combine to produce v ). We now show that ? is a determining set for Hrd(T ). In the absence of degenerate faces this follows as in [15]. For a degenerate face, note that the coecients corresponding to points in the cap about to be moved in step 5 of Algorithm 4.3 are implied to be zero by the smoothness conditions (2.6) across the degenerate face, independent of the possible relocation of other caps. To complete the proof, we now construct a basis for Hrd(T ) satisfying (2.11). Clearly, for a given  2 ?, we can set the coecient s = 1 and all other coecients corresponding to 2 ? with 6=  to zero, we can solve for the remaining coecients using the smoothness conditions. If the domain point P corresponding to  is contained in a set C , then the resulting spline s has support on the trihedron T containing P . If P is in a set of the form E (vi ), then s has support on the union of the two trihedra containing the face opposite vi. In all other cases, s has support on an orange.

14

Remark 4.5. Rather than constructing an explicit basis, it is also possible to

prove the dimension statement in Theorem 4.4 by showing that the expressions in (2.13) provides a lower bound on dim Hrd(T ) as was done in [1] in the planar case. This is done by thinking of Hrd (T ) as a subspace of H0d (T ), enforcing the smoothness conditions in the -disks via Theorem 3.4, and then subtracting the number of appropriate smoothness conditions (2.6) needed to enforce smoothness across the interior faces of T .

5. A Minimal Determining Set for Hrd(T ) when d  4r + 1

As in the case of splines de ned on a planar triangulation [7], the construction of a minimal determining set can be greatly simpli ed if d  4r + 1. In this case the disks of radius 2r around rays of T do not overlap, and the remaining smoothness conditions across faces of T decouple. In that case the following much simpler algorithm can be used: Algorithm 5.1. If d  4r + 1, choose the set ? as follows: 1) For each interior ray v of T , choose a minimal determining set for Hr2r (Tv ) as described in Theorem 3.4, where Tv is the orange surrounding v. 2) For each boundary ray v of T , choose a minimal determining set for Hr2r (Tv ) as described in Theorem 3.4. where Tv is the orange containing v. 3) For each trihedron T in T , choose C . 4) For each face in T , choose E~(v1 ) = F (v1 ) n [D2r (v2 ) [ D2r (v3 )] ; (5:1) where v1, v2, v3 de ne a trihedron such that v2 and v3 span the face. For the case d = 23 and r = 5, Figure 3 shows the choice of the domain points for a single trihedron T using Algorithm 5.1. As in Figure 2, dots correspond to points in sets of the form D2r (vi ) and circles correspond to points in E~(ui), while 's mark the points in C .

6. The case d  3r + 1

As in the planar case, it is also possible to treat spline spaces for d  3r + 1 provided we restrict the class of trihedral decompositions somewhat. Theorem 6.1. Let d = 3r +1, and suppose that the trihedral decomposition T does not possess any degenerate faces. Then the dimension of H3rr+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. De nition 6.2. A trihedral decomposition T is said to be generic with respect

to r and d provided that for all suciently small perturbations of the directions of the rays of T , the resulting trihedral decomposition T~ satis es

dim Hdr (T~ ) = dim Hdr (T ):

(6:1)

Theorem 6.3. Fix d 2 f2; 3; 4g, and suppose T is a generic trihedral decomposition with respect to r = 1 and d. Then the dimension of Hd1 (T ) is given by (2.13) or (2.14), depending on whether T is total or not. Proof: The space Hd1 (T ) is isomorphic to the space Sd1(
), where
is the generalized triangulation (see [9]) obtained by projecting the points in V through the origin onto a plane that does not contain the origin and is not parallel to any of the rays in T . The result then follows from Theorems 27 and 33 in [9].

The proof of Theorem 6.3 does not involve nding a minimal determining set. For d = 4, it may be possible to construct one using the techniques in [6]. However, in the case d 2 f2; 3g, no general procedure for nding a minimal determining set is known even in the (generic) planar case.

7. Doubly degenerate faces

While the structure of bivariate splines on planar triangulations and homogeneous splines on trihedral decompositions in IR3 are very similar, there is a

16 situation which can occur in the homogeneous case but cannot occur in the planar case: it is possible for a face to be degenerate at both rays. We illustrate this in the following example.

Example 7.1. Let vi = ?vi+3 = ei ; i = 1; 2; 3;

(7:1)

where ei denote the standard unit vectors, and let T  be the set of trihedra generated by the sets

fv1; v2 ; v3g ; fv1 ; v2; v6 g ; fv1; v3 ; v5g ; fv1 ; v5; v6 g ; fv2; v3 ; v4g ; fv2 ; v4; v6 g ; fv3; v4 ; v5g ; fv4 ; v5; v6 g :

(7:2)

Discussion: The convex hull of these points forms a regular octahedron. In the

resulting trihedral decomposition each face is degenerate at each of its two rays, and at each ray each face sharing the ray is contained in one of only two planes. Thus, all rays of T  are singular. As a check on our formulae and to provide actual numbers for comparison purposes, we have computed the dimensions of Hdr (T ) for 1  r  5 and 1  d  15 by setting up the smoothness conditions and numerically computing the rank of the matrix describing the smoothness conditions using the Goliath package [3,2] and other special purpose software. For the trihedral decomposition T  there are 6 singular rays. Thus for d  2r the expression (2.13) becomes d?r X r 2 2 d := 4d ? 12dr + 9r + 3r + 2 + 6 (r + 1 ? m)+ m=1 ?


= 2 2d2 ? 6dr + 6r2 + 3r + 1 :

(7:3)

This gives

? 12d + 20; ? 24d + 62; rd = > 4d2 ? 36d + 128; 2 > > : 4d ? 48d + 218; 4d2 ? 60d + 252; 8 2 4d > > > < 4d2

if d  2 and r = 1 if d  4 and r = 2 if d  6 and r = 3 if d  8 and r = 4 if d  10 and r = 5.

(7:4)

17 In Table 1 we have used an asterisk to mark those cases where the computed dimensions of Hdr (T  ) dier from the values of rd. As a curiosity, we note that for the trihedral decomposition T  , the formulae are in fact correct for d = 3r +1 (and of course all larger values) but not for d  3r, even though T  is not generic and all faces are degenerate.

d: r = 1: r = 2: r = 3: r = 4: r = 5:

1 2 3 4 5 6 7 3 9 19 36 60 92 132 3 6 13 24 39 61 90 3 6 10 18 30 46 66 3 6 10 15 24 37 54 3 6 10 15 21 31 45

8 9 10 11 12 13 14 15 180 236 300 372 452 540 636 740 126 170 222 282 350 426 510 602 93 127 168 216 272 336 408 488 75 100 132 171 217 270 330 398 63 85 111 141 178 222 273 331

Table 1. Dimensions of Hdr (T  ) on the regular octahedron. 8. Super Splines As in the planar case [15], the above methods can also be used to compute the dimension and to construct locally supported bases for spaces of homogeneous super splines: r v Hr; d (T ) := fs 2 Hd (T ) : s 2 C (v ); v 2 Vg;

(8:1)

where  := fv gv2V and r  v < d for all v. Here s 2 C v (v) means that all of the derivatives up to order v of the pieces of s which join at v have a common value at v. We assume throughout that the  and v disks around neighboring rays do not overlap, i.e., max f; ug + max f; v g < d

(8:2)

for all pairs of vertices u and v which generate a face of T , where  is de ned in (4.4).

18

Theorem 8.1. Let T be a partial trihedral decomposition and suppose that

(8.2) holds. Then

r + 2 V 2       X X X v + 2  ? r + 1  ? r + 1 v v ? (Ev ? 1) ? Ev + 2 2 2 v2VB v2VI v2V 

dim Hr; d (T ) = (d ? r)(d ? 2r) ?



? 2d2 + 6dr ? 3r2 + 3r + 2 +

X

d?r X

v2V m=v ?r+1

(r + m + 1 ? mev )+ (8:3)

if T is a total trihedral partition, and 

(d ? r + 1)(d ? r) V + (d ? r)(d ? 2r) ? dim Hr; B d (T ) = 2 2 2 ? 2d ? 6dr +23r ? 3r ? 2   X   + 1 ? r v + (1 ? E )

i2VB

+ +

v

X v

v2VI X



2



+ 2 ? E v + 1 ? r v 2 2

d?r X

v2VI m=v ?r+1





r+2 V I 2

(8:4)



(r + m + 1 ? mev )+

if T is a partial trihedral decomposition. Here Ev is the number of interior faces attached to the vertex v for each v 2 V . Moreover, there exists a basis of splines for Hr; d (T ) such that the support of each spline is either one trihedron, an adjoining pair, or an orange.

Proof: We give the proof for the case of a partial trihedral decomposition.

The proof when the decomposition is total is similar (and simpler). The key observation is that the set of points chosen by Algorithm 4.3 and lying inside the disk Dv (v) is a minimal determining set for Hrv (Tv ), where Tv is the orange with axis v. Thus, if we now?impose v continuity at v, then we can replace
 +2 r v those dim Hv (Tv ) points by 2 points lying in one trihedron in Tv . This shows that for each ray v, the change in the number of points in the minimal determining set ? constructed by Algorithm 4.3 is given by dim Hr (Tv ) ? dim H (Tv ) = dim Hr (Tv ) ?





v + 2 : 2

(8:5)

19 Thus, 

r (T ) ? X dim H r (Tv ) ? dim Hr; ( T ) = dim H v d d v2V

= (d ? r +21)(d ? r) VB + (d ? r)(d ? 2r)VI



v + 2 2

d?r 2 2 XX ? 2d ? 6dr +23r ? 3r ? 2 + (r + m + 1 ? mev )+ v2V m=1

"

 X   v ?r  ? r + 1 r + 2 v + (r + m + 1 ? mev )+ + Ev ? 2 2 m=1 v2VI   X   X v + 2  + 2  + 1 ? r v v + ? 2 : 2 2 + (Ev ? 1) v2V i2VB X

#

(8:6)

Now combining terms, we get (8.4). Our new minimal determining set can now be used to construct a basis of locally supported splines as was done in the proof of Theorem 2.3. The case where all v are equal is of particular interest. Corollary 8.2. Suppose

v =   r; v 2 V ;

(8:7)

and that 2 < d. Then (2d2 ? 6dr ? 3r2 + 12r + 3r ? 52 ? 3) V dim Hr; ( T ) = d 2 2 2 + (?2d + 6rd + 3r ? 3r + 62 ? 12r + 6 + 2) +

X

d?r X

v2V m=?r+1

(8:8)

(r + m + 1 ? mev )+

if T is a total trihedral partition, and (d2 ? 2rd ? r2 + d + r ? 22 + 4r ? 2) V dim Hr; ( T ) = B d 2 2 2 2 + (2d ? 6rd ? 3r + 32r ? 5 + 12r ? 3) VI 2 2 2 + (?2d + 6rd + 3r ? 3r2+ 6 ? 12r + 6 + 2) +

X

d?r X

v2VI m=?r+1

(r + m + 1 ? mev )+

(8:9)

20 if T is a partial trihedral decomposition. Moreover, there exists a basis of splines for Hr; d (T ) such that the support of each spline is either one trihedron, an adjoining pair, or an orange. Proof: Substituting (8.7) in (8.3) and using (4.6) and (4.9) leads to (8.8). For a partial decomposition, the classical Euler relations for a triangulation imply X Ev = 2E = 2(2VB + 3VI ? 3): (8:10) v2V

Now substituting (8.7) in (8.4) and using (8.10) leads to (8.9). In both Theorem 8.1 and Corollary 8.2, the formula for a total trihedral decomposition can be obtained from the formula for a partial one by dropping the term with VB and doubling the constant term. Moreover, if we set  = r in the corollary, of course we recover the formulae in Theorem 2.3.

9. Remarks Remark 9.1. The proof of Theorem 3.4 is based on the proof of Theorem 3.3

of [16] for polynomial splines on planar triangulations. The description of the minimal determining set for a cell given there is not quite correct in that it allows N ?1 < N which could lead to the same point being included in ? twice. This is easily xed by requiring that N ?1 = N as we have done here.

Remark 9.2. As in our paper [4] it is possible to develop a theory of homo-

geneous splines de ned on a (total or partial) decomposition of IR2 by wedges (the two-dimensional analogs of trihedra). Such splines can be restricted to a circle or a similar curve to obtain univariate functions along the curve. The corresponding dimensions and minimal determining sets can be obtained in a straightforward manner by considering a single ring in Theorem 3.4. We refrain from giving a detailed description. Remark 9.3. In the bivariate polynomial spline case, there is no question that the right way to de ne domain points Pijk associated with the Bernstein-Bezier coecients of a polynomial is by the formula in (2.4). In that case the set of pairs fP; cP gP 2G is called the Bezier net of s, and has an important geometric interpretation. However, in the trihedral setting, it is not so clear what is the best way to de ne the analogous points. As discussed in [ANS1], there are reasonable alternatives, although it appears that there is no de nition which carries the full geometric signi cance of the domain points in the planar case. Our choice here is a useful way to label control coecients. Remark 9.4. For polynomial spline spaces on planar triangulations, there are well-known lower and upper bounds on the dimension of Sdr (
) which are of interest for d < 3r + 2, see e.g. [17] and references therein. Similar bounds can be derived for our homogenous spline spaces, and will be treated elsewhere.

21

Remark 9.5. The formula (8.4) given in Theorem 8.1 for a partial trihedral decomposition is much simpler than the corresponding formula in Theorem 2.4 of [15]. Since our proof of Theorem 8.1 can also be used in the bivariate case, the simpler formula (8.4) is also valid there.

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