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Dimension of trivariate C1 splines on bipyramid cells
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Computer Aided Geometric Design www.elsevier.com/locate/cagd
Dimension of trivariate C 1 splines on bipyramid cells Julien Colvin 1 , Devan DiMatteo 1 , Tatyana Sorokina ∗,2 Towson University, 7800 York Road, Towson, MD 21252, United States
a r t i c l e
i n f o
a b s t r a c t We study the dimension of trivariate C 1 splines on bipyramid cells, that is, cells with n + 2 boundary vertices, n of which are coplanar with the interior vertex. We improve the earlier lower bound on the dimension given by J. Shan. Moreover, we derive a new upper bound that is equal to the known lower bound in most cases. In the remaining cases, our upper bound is close to the known lower bound, and we conjecture that the dimension coincides with the upper bound. We use tools from both algebraic geometry and Bernstein–Bézier analysis. © 2015 Elsevier B.V. All rights reserved.
Article history: Available online xxxx Keywords: Bipyramid Dimension Trivariate spline Cell Supersmoothness
1. Introduction Smooth piecewise polynomial functions, or splines, are used widely in approximation theory, computer aided geometric design, image analysis, and numerical analysis. Key tools for analyzing multivariate spline spaces come from two fields: classical approximation theory and applied algebraic geometry. The two fields and the two corresponding research communities use entirely different approaches: Bernstein–Bézier analysis (see the survey of Alfeld, 2015, in this issue) and homological techniques (see the survey of Schenck, 2015, in this issue). For a full treatise on Bernstein–Bézier analysis, we refer the reader to Lai and Schumaker (2007). The homological approach to multivariate splines is explained in several papers, see Mourrain and Villamizar (2014), Schenck and Stilman (1997) and Billera (1988). In this paper we are concerned with trivariate splines. Let Pd denote the set of polynomials in three variables of degree ≤ d. A spline is a piecewise-polynomial function defined on a domain ⊂ R3 that belongs to a certain smoothness class. More precisely, for a fixed tetrahedral partition of the underlying domain , our spline space of interest is defined as
S dr () = {s ∈ C r () : s| T ∈ Pd
for each tetrahedron
T ∈ }.
(1)
Finding the exact dimensions of spaces S dr () of trivariate splines remains an open problem. While the dimension of any particular spline can be computed using software such as Macaulay2, or the java applet (Alfeld), general formulae are still unknown. This problem is inherited from bivariate splines, where the dimension remains unknown for d ≤ 3r + 1. As it is shown in section 17.8 in Lai and Schumaker (2007), there is a tight connection between trivariate and bivariate cases, and there is no hope to resolve the trivariate case before the dimension of bivariate splines is known for all values of d and r. We note that despite this difficulty, the dimensions of trivariate splines are known for few very special partitions, including the so-called macro-element or finite-element spaces. For example, trivariate splines on the Alfeld, the Clough–Tocher and
* 1 2
Corresponding author. E-mail addresses: [email protected] (J. Colvin), [email protected] (D. DiMatteo), [email protected] (T. Sorokina). Partially supported by a grant from the Towson University Fisher College of Science and Mathematics. Partially supported by a grant from the Simons Foundation (#235411 to Tatyana Sorokina).
http://dx.doi.org/10.1016/j.cagd.2015.12.001 0167-8396/© 2015 Elsevier B.V. All rights reserved.
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Fig. 1. Orange in R3 .
Worsey–Piper splits, see Lai and Schumaker (2007) and references therein. Other examples do not qualify as finite-element spaces, but are built on very special grids, see Schumaker and Sorokina (2004) for an example. The first and only formula for a dimension of a trivariate spline space on a tetrahedral partition of a somewhat general nature is for the so-called oranges, see Theorem 17.30 in Lai and Schumaker (2007). An orange, see Fig. 1, is a union of tetrahedra all sharing the same edge, [u , v ] in Fig. 1, and pairwise sharing common triangular faces. The goal of this paper is to provide a second example of such a general formula for a partition that we call a bipyramid cell, see Fig. 2. A bipyramid cell is a special case of the so-called interior cell. An interior cell is a collection of tetrahedra all sharing an interior vertex (cf. Section 17.8.1 in Lai and Schumaker, 2007). Note that a bipyramid cell is very different from an orange. An orange is not an interior cell, and all smoothness conditions associated with it are essentially bivariate. The smoothness conditions associated with the bipyramid cell include purely trivariate ones. Because of this difficulty, we have been able to analyze the case for r = 1 in (1) only. A large part of research in the area of trivariate splines has focused on obtaining upper and lower bounds on the dimension of trivariate spline spaces. Several results have been obtained for bounding the dimension of general splines either below or above, see Mourrain and Villamizar (2014), Lau (2005), Alfeld and Schumaker (2008), Alfeld (1996), and Shan (2014). None of the bounds cited above are tight enough to obtain exact dimensions. In this paper, we prove new upper bounds for the bipyramid cells that allow us to obtain exact dimensions in three out of four subcases; in the last subcase our new upper bound is very close to the known lower bound, and we conjecture that the dimension is equal to the upper bound. Finally, we prove that the lower bound in Shan (2014) has a typo; fixing this leads to an improved lower bound. We compare the improved lower bound to our new upper bound. The lower bound is obtained using homological techniques, while the upper bound is proved using Bernstein–Bézier analysis. The paper is organized as follows. In Section 2 we introduce bipyramid cells, and study their geometry. In Section 3 we fix the bound in Shan (2014), and derive the lower bound for the dimension of S d1 () on bipyramid cells. In Section 4 we prove a result on supersmoothness needed to obtain our new upper bound. Section 5 contains several results on the dimension of bivariate splines that are also needed to derive the new upper bound. The main results of this paper concerning the new upper bound and the exact dimension of S d1 () on bipyramid cells are in Section 6. We conclude with remarks and conjectures in Section 7. 2. Preliminaries To begin, we define a class of partitions in R3 called bipyramid cells, see e.g. Fig. 2. Definition 2.1. A bipyramid cell is a tetrahedral partition such that:
• there is exactly one interior vertex v 0 ; • n boundary vertices v 1 through v n are coplanar along with v 0 , and form a polygon surrounding v 0 in the base plane B := [ v 1 , . . . , v n ]; • each vertex v i , i = 1, . . . , n, is connected to v 0 by the interior edges [ v 0 , v i ]; • two boundary vertices v n+1 and v n+2 lie outside the base plane B, are on opposite sides of B, and are connected to v 0 by the interior edges [ v 0 , v n+1 ] and [ v 0 , v n+2 ]; and • vertices v n+1 and v n+2 connect to the boundary vertices v i , i = 1, . . . , n. It is easy to check that these conditions force to be shellable with no holes or cavities, see Lai and Schumaker (2007) for a discussion of the importance of these facts. For simplicity, and without loss of generality, we can assume that the interior vertex v 0 is located at the origin, and the n coplanar boundary vertices lie in the base xy-plane. For the remainder
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Fig. 2. Collinear case.
Fig. 3. Coplanar II case.
Fig. 4. Supersmoothness.
Fig. 5. No supersmoothness.
3
of this paper, we will assume that is a bipyramid cell. Our goal is to find the dimension of the spline space S d1 (). The next lemma summarizes several immediately obvious facts about . Lemma 2.2. In a bipyramid cell the following holds: (1) there are 2n tetrahedra, 3n interior triangular faces, and n + 2 interior edges; (2) the number of different slopes m formed by the coplanar interior edges [ v 0 , v i ], i = 1, . . . , n, in the base plane satisfies 2 ≤ n2 ≤ m ≤ n; and (3) the number of distinct planes h containing the interior triangular faces is as follows: Case 1 (Generic): v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 does not contain any other v i . Then h = 2m + 1; Case 2 (Coplanar): v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 contains at least one other v i , i = 1, . . . , n. Then h = 2m; Case 3 (Collinear): v n+1 and v n+2 are collinear with v 0 . Then h = m + 1. For further investigation, we split the coplanar case into two subcases as follows:
• Coplanar Case I: v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 contains exactly one other v i , i = 1, . . . , n. • Coplanar Case II: v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 contains exactly two other v i , i = 1, . . . , n. Fig. 2 shows an example of the collinear case; Fig. 3 shows an example of the coplanar case II, where vertices v 0 , v 2 , v 4 , v 7 , v 8 lie in the same plane (shaded). Fig. 4 shows an example of the generic case, and finally Fig. 5 shows an example of the coplanar case I. Next, we need to classify small values of h for the results in the next section. If h ≥ 6, the results in the next sections do not require knowledge of the particular geometry of . Since in the collinear case we will compute the dimension of S d1 () without using any lower bounds, the following lemma is stated for the non-collinear cases only.
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Lemma 2.3. In a bipyramid cell with non-collinear v 0 , v n+1 and v n+2 , the following holds: If h = 4, then it is the coplanar case II with m = 2, and n = 4. If h = 5, then it is the generic case with m = 2, and n = 4. Proof. If h = 3, then we must be in the collinear case where h = m + 1, m = 2, and n = 4. Then v 5 and v 6 are collinear with v 0 , and there are only two different slopes. Thus, the partition must form an octahedron whose main diagonals intersect at the interior vertex v 0 . If h = 4, then we have two possibilities. One possibility is the coplanar case II with h = 2m, m = 2, and n = 4. The base vertices form a quadrilateral where opposite vertices are collinear with v 0 . The vertices v 5 , v 6 and v 0 are not collinear, and the plane containing v 5 , v 6 , and v 0 also contains two opposite v i . The other possibility is the collinear case with h = m + 1, m = 3, and 3 ≤ n ≤ 6. Then v n+1 and v n+2 are collinear with v 0 , and there are three different slopes in the base plane. If h = 5, then we have two possibilities. One possibility is the collinear case with h = m + 1, with m = 4 and 4 ≤ n ≤ 8. In this case, v n+1 , v n+2 and v 0 are collinear. The other possibility is the generic case with h = 2m + 1 with m = 2, and n = 4. Then v 5 and v 6 are not collinear with v 0 , and the plane containing v 5 , v 6 , and v 0 does not contain any other vertices. This is another octahedral case. 2 3. Known lower bounds on the dimension Results in this section follow from both Mourrain and Villamizar (2014) and Shan (2014). We use the terminology from the latter since it is better suited for our partition. Remark 7.1 in Section 7 shows how to use Mourrain and Villamizar (2014) to obtain the lower bound of this section. To simplify the notation in Shan (2014), we introduce the quantity J (h, k), which depends on the number of distinct planes h, containing the interior triangular faces, and a parameter, k, that runs from 0 to the degree of the spline, d. For a ring R of trivariate polynomials, we define J (h, k) to be the dimension of the graded module ( R / J )k , where J is the ideal generated by the squares of the linear forms which define the h distinct planes passing through the interior vertex v 0 , and k is the degree of the polynomials, see the survey of Schenck (2015) in this issue for additional information on the module. Lemma 3.1. In a bipyramid cell with non-collinear v 0 , v n+1 and v n+2 , the following holds:
J (h, 0) = 1 for any h; J (4, 2) = 2;
J (h, 1) = 3 for any h;
J (5, 2) = 1;
J (h, k) = 0 otherwise.
Proof. We first reproduce the more general result for cells that appears in Shan (2014), changing to our notation:
J (h, 0) = 1, for any h,
J (3, k) =
J (5, k) =
3, if k = 2, 1, if k = 3, 0, if k ≥ 4; 1 or 2, 0,
if k = 2, if k ≥ 3.
J (h, 1) = 3, for any h,
J (4, k) =
2, if k = 2, 0, if k ≥ 3;
J (h, k) = 0, for any k ≥ 2, h ≥ 6.
Note that, in the more general case covered in Shan (2014), J (5, 2) is multi-valued. For bipyramid cells, it follows from Lemma 2.3 that the only case with h = 5 is the generic case. In this case, after a change of variables, we have J = x2 , y 2 , z2 , ( z − α x)2 , ( z − β y )2 , for some non-zero α , β ∈ R. Clearly, dim( R / J )2 = 1 = J (5, 2). We next note that, for all bipyramid cases except the collinear one, the minimal number of generators of J is equal to either the number of distinct planes containing the interior triangular faces (i.e. h), or 6 if h > 6. Since we do not discuss the collinear case in this lemma, this completes the proof. 2 Remark 3.2. It is easy to see that in the collinear case, the line formed by v n+1 , v 0 , and v n+2 might be shared by an arbitrarily large number of planes. However, in this case J cannot be six-generated, as it is claimed in Shan (2014). In Section 6, we obtain the exact dimension of trivariate splines on bipyramids in the collinear case without referring to any lower bounds. Therefore, we omit the discussion of a lower bound for this case. We next discuss the lower bound for splines on any cell proved in Shan (2014). Let f i denote the number of i-dimensional interior faces of the partition: f 3 is the number of tetrahedra, f 2 is the number of interior triangular faces, and f 1 is the number of interior edges. Additionally, let he be the number of planes containing all triangular faces attached to the edge e, let f 1,2 be the number of interior edges with he = 2, let f 1,3 be the number of interior edges with he = 3, and let f 1,4 be the number of interior edges with he ≥ 4. The following bound is proved in (Shan, 2014):
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Theorem 3.3. If is a cell, dim S d1 () is bounded below as
dim S d1 () ≥ ( f 3 − f 2 + f 1 )
d+3
3
d+1 + ( f 2 − 2 f 1,2 − 3 f 1,3 − 3 f 1,4 ) 3
d d d−1 + 2( f 1,3 + f 1,4 ) + f 1,2 − J (h, k). 3
3
k =0
Next we apply Theorem 3.3 to the case of bipyramid cells. From Lemma 2.2, we know that
f 3 = 2n,
f 2 = 3n,
f 1 = n + 2,
and
f 1,3 + f 1,4 = n + 2 − f 1,2 .
We can thus rewrite the bound in Theorem 3.3 as follows
dim S d1 () ≥ 2
d+3
d+1 d + ( f 1,2 − 6) + (2n + 4 − 2 f 1,2 )
3
3
d−1
+ f 1,2
3
−
d
3
J (h, k).
k =0
Applying a variety of combinatorial identities, Lemma 2.3 and Lemma 3.1, we obtain Corollary 3.4. Let be a bipyramid cell. Then for d ≥ 2 the following holds:
dim S d1 () ≥ 6d − 2 + f 1,2 (d − 1) + 2n
d
3
−
d
J (h, k).
k =2
In the coplanar case I we have f 1,2 = 1, h ≥ 6. Thus
dim S d1 () ≥ 7d − 3 + 2n
d
3
.
In the coplanar case II, if m ≥ 3, then f 1,2 = 2, and h ≥ 6. If m = 2, then f 1,2 = 4 and h = 4. Thus
dim
S d1 ()
≥ 2n
d
3
+
10d − 8, 8d − 4,
if m = 2, if m ≥ 3.
Moreover, in the generic case if m ≥ 3, we have f 1,2 = 0. If m = 2, then f 1,2 = 2. Thus
dim S d1 () ≥ 2n
d
3
+
8d − 5, if m = 2, 6d − 2, if m ≥ 3.
Corollary 3.4 can be obtained from Mourrain and Villamizar (2014) as well, see Remark 7.1. 4. Supersmoothness The result in this section is essential for obtaining our upper bound in Sections 5 and 6. We use the notation of Definition 2.1, and square brackets for the convex hull. As in Lai and Schumaker (2007), a bivariate (interior) cell is a collection of triangles all sharing an interior vertex. Theorem 4.1. Any spline s in the space S d1 () of trivariate C 1 splines on a bipyramid with the base B := [ v 1 , . . . , v n ] and interior vertex v 0 ∈ B has the following supersmoothness: if a vertex v i , 1 ≤ i ≤ n, is not coplanar with [ v 0 , v n+1 , v n+2 ], then all derivatives of s of order two across [ v 0 , v i ] in any direction coplanar with B are continuous. Proof. We first note that if v 0 is collinear with v n+1 and v n+2 , then there is a plane containing v 0 , v n+1 , v n+2 and v i , for any of the remaining v i . Thus, we let v 0 be not collinear with v n+1 and v n+2 . Without loss of generality, let v 5 be not coplanar with [ v 0 , v n+1 , v n+2 ], see Fig. 4. Fix an arbitrary point u 5 in the interior of the edge [ v 0 , v 5 ], and consider any cross-section of the double pyramid by a plane passing through u 5 , parallel to [ v 0 , v n+2 ], and not containing [ v 0 , v 5 ]. In Fig. 4, it is the quadrilateral [u 1 , u 6 , u 4 , u 7 ]. Such a cross-section contains a bivariate cell Q with the interior vertex u 5 and four interior edges. Two of them lie in B and are collinear, and the other two are not collinear. In Fig. 4, the collinear ones are [u 5 , u 1 ] and [u 5 , u 4 ], while [u 5 , u 6 ] is parallel to [ v 0 , v 6 ] by construction, and thus [u 5 , u 7 ] is not collinear with [u 5 , u 6 ] .
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It clearly suffices to prove that the bivariate spline s˜ = s| Q has supersmoothness two at u 5 in the direction of the two collinear edges. In Fig. 4, we need to prove that s˜ has a second order continuous derivative at u 5 in the direction [u 5 , u 1 ]. The required supersmoothness immediately follows from Theorem 3.2 in Sorokina (2010). Since u 5 is an arbitrary point on the edge [ v 0 , v 5 ] such bivariate supersmoothness holds across the whole edge [ v 0 , v 5 ]. To complete the proof, we note that if v i is coplanar with [ v 0 , v n+1 , v n+2 ], as in Fig. 5 v 1 is coplanar with [ v 0 , v 6 , v 7 ], then Q is formed by two pairs of collinear edges. In Fig. 5, u 6 , u 1 , u 7 are collinear, and so are u 2 , u 1 and u 5 . The bivariate spline s˜ = s| Q does not have continuous second order derivatives in either direction at u 1 , see e.g. section 4 of Sorokina (2010). Thus, the non-coplanarity condition on the statement of the theorem is essential. 2 5. Dimension of bivariate splines over the base cell To prove our upper bound in the next section, we need to compute the dimension of bivariate C 1 splines S k1 ( B ) over the triangulation B of the base B of the bipyramid, 0 ≤ k ≤ d. We note that B is an interior bivariate cell, i.e., a collection of triangles sharing the interior vertex v 0 . Lemma 5.1. Let B be a bivariate cell with n edges and m slopes. Then
dim S 01 ( B ) = 1,
dim
S k1 ( B )
=
dim S 11 ( B ) = 3, and for k ≥ 2,
k , if m = 2, 2 k 3 + n 2 , if m ≥ 3. 4+4
(2)
Proof. Clearly, splines in S 01 ( B ) and S 11 ( B ) are bivariate polynomials whose dimensions are 1 and 3, respectively. Using Theorem 9.3 from Lai and Schumaker (2007) (originally proved in Schumaker, 1979), for k ≥ 2, we obtain
dim S k1 ( B ) = 3 + n
k
+
2
k −1
(2 − j (m − 1))+ .
(3)
j =1
We next simplify (3) for various values of m. If m = 2, then n = 4 and k −1 k −1 (2 − j (m − 1))+ = (2 − j )+ = 1. j =1
j =1
If m ≥ 3, then 2 − j (m − 1) ≤ 0 ⇐⇒ 2 ≤ j (m − 1). Since j starts at 1 and m ≥ 3, the minimum value of j (m − 1) is 2, so the inequality is always true. Therefore (2 − j (m − 1))+ = 0, so the sum evaluates to 0, and the result follows. 2 For k = d, due to additional supersmoothness described in Theorem 4.1, we can no longer use Theorem 9.3 in Lai and 1,∗ Schumaker (2007). Let S d ( B ) be the restriction of S d1 () on B, i.e., the subspace of S d1 ( B ) with the supersmoothness described in Theorem 4.1. It then follows from Theorem 9.25 in Lai and Schumaker (2007) that 1,∗
dim S d ( B ) =
n d d ( j − ri ) + 3 + ( j + 1 − j )+ , i =1 j =2
j =2
where r i is the smoothness across the edge [ v 0 , v i ],
mi , j :=
0, 0, j − ri ,
j :=
(4)
n
i =1 mi , j ,
and
if there exists l with αi = αl and rl < r i , if there exists l > i with αi = αl and rl = r i , otherwise.
We next simplify (4) further by considering each case of Section 2. 1,∗
Lemma 5.2. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of S d1 () on B. In the generic case,
1,∗ dim S 2 ( B )
= 6, dim
1,∗ S 3 ( B )
⎧ d−1 ⎪ ⎪ ⎨ 9 + 4 2 , if m = 2 ∧ d ≥ 4, 1,∗ 1 dim S d ( B ) = 7 + n d− , if m = 3 ∧ d ≥ 3, 2 ⎪ ⎪ ⎩ 6 + n d−1, if m ≥ 4. 2
= 12, if m = 2 and for all remaining cases
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Proof. In this case, Theorem 4.1 yields r i = 2, for all i = 1, . . . , n, and j and ri in (4), we obtain 1,∗
dim S d ( B ) =
7
j = m( j − 2) for all j ≥ 2. Substituting the values for
d n d ( j − 2) + 3 + ( j + 1 − m( j − 2))+ i =1 j =3
=n
d−1 2
j =2
+6+
d
(2m + 1 − j (m − 1))+ .
j =3
Since m ≥ 2, we have
2m + 1 − j (m − 1) ≤ 0 ⇐⇒ 2m + 1 ≤ j (m − 1) ⇐⇒ j ≥ 2 +
3 m−1
.
Since j starts at 3, once m ≥ 4, the inequality is always true. Therefore, (2m + 1 − j (m − 1))+ = 0 if m ≥ 4, and the d cases m = 2 and m = 3 must be computed separately. If m = 3, then j = 3 is the only positive term and j =3 (2m + 1 − d j (m − 1))+ = 1. If m = 2, we have two non-negative terms j = 3, and j = 4, and j =3 (2m + 1 − j (m − 1))+ = 3.
n
2
2
For d = 2, the dimension formula yields j =2 ( j − r i ) + 3 + j =2 ( j + 1 − m( j − 2))+ . Since r i = 2 for all i, and i =1 j = 2, the first sum is a summation of zeros, and the second summation has only one term, namely (2 + 1 − m(2 − 2))+ = 3. Therefore, the dimension is 6. 3 n 3 For d = 3, the dimension formula gives j =2 ( j − r i ) + 3 + j =2 ( j + 1 − m( j − 2))+ . Again, since r i = 2 for all i, i =1
n
the double summation becomes i =1 (3 − 2) = n. The second sum is equal to 3 + (4 − m)+ , so the value depends on m. Considering the cases m = 2, m = 3, and m ≥ 4 completes the proof. 2 1,∗
Lemma 5.3. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of
S d1 ()
on B. In the first coplanar case I, we have
dim
1,∗ S d ( B )
=
d
2
d−1 + (n − 1) + 5. 2
Proof. In this case, the plane containing v n+1 , v n+2 , and v 0 contains exactly one boundary vertex of B. Without loss of generality, assume that it is v 1 . Then Theorem 4.1 yields
r i := Moreover,
1, if i = 1, 2, if 2 ≤ i ≤ n.
j = ( j − 1) + (m − 1)( j − 2), for all j ≥ 2. We substitute the values for j and ri in (4) 1,∗
dim S d ( B ) =
d d n d ( j − 1) + ( j − 2) + 3 + (2 − (m − 1)( j − 2))+ j =2
i =2 j =3
j =2
d d d−1 + (n − 1) +5+ = (2m − j (m − 1))+ . 2
2
j =3
Since m ≥ 3, we have
2m − j (m − 1) ≤ 0 ⇐⇒ 2m ≤ j (m − 1) ⇐⇒ j ≥ 2 +
2 m−1
.
Since j starts at 3 and m ≥ 3, the inequality is always true. Therefore, (2m − j (m − 1))+ = 0, and the result follows.
2
1,∗
Lemma 5.4. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of
S d1 ()
on B. In the second coplanar case II,
1,∗ dim S 2 ( B )
= 7, and for d ≥ 3, we have
d d−1 6, if m = 2, 1,∗ + (n − 2) + dim S d ( B ) = 2 5, if m ≥ 3. 2 2
Proof. In this case, the plane containing v n+1 , v n+2 , and v 0 contains exactly two boundary vertices of B. Without loss of generality, assume they are v 1 and v 2 , and they are collinear with v 0 . Then Theorem 4.1 yields
r i :=
1, if i = 1 or i = 2, 2, if 3 ≤ i ≤ n.
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Moreover, j = ( j − 1) + (m − 1)( j − 2), for all j ≥ 2. Note that j does not change between coplanar cases I and II, but the difference in r i must be taken into account for the first part of the formula. Another difference with the previous case is that the value of r2 = 1. Thus 1,∗
dim S d ( B ) = 2
d d n d ( j − 1) + ( j − 2) + 3 + (2 − (m − 1)( j − 2))+ j =2
i =3 j =3
j =2
d d d−1 + (n − 2) +5+ =2 (2m − j (m − 1))+ , 2
2
j =3 1,∗
and the result follows for d ≥ 3. When d = 2, the double summation has only two non-zero terms of 1, so dim S 2 ( B ) = 2 + 3 + 2 = 7. 2 In the collinear case, there is no supersmoothness associated with the base cell and one can use Lemma 5.1 to compute the dimension. 1,∗
Lemma 5.5. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of
S d1 ()
on B. In the collinear case, we have
dim
1,∗ S d ( B )
=n
d
2
+
4, 3,
if m = 2, if m ≥ 3.
6. New upper bounds and the dimension of splines on bipyramids In this section we obtain a new upper bound using the results from Section 5 with the help of Bernstein–Bézier methods (see the survey of Alfeld, 2015 for the terminology). The advantage of this upper bound over the upper bound of Mourrain and Villamizar (2014) is that our upper bound does not require knowing a priori that the module of C r splines over bipyramids is free. We start by defining levels in a bipyramid. Definition 6.1. Let B := [ v 1 , . . . , v n ] be the base, v 0 be the interior vertex, and v n+1 , v n+2 be the two remaining vertices of a bipyramid. The k-th level of (denoted k ) is the cross-section of by the plane parallel to B and passing through the point
(d−k) v
0 +kv n+1
d
, if 0 ≤ k ≤ d,
d
, if − d ≤ k < 0.
(d+k) v 0 −kv n+2
Each k is a bivariate cell of the same geometry as the base cell B =: 0 . To explain the method, we first note that all domain points of s ∈ S dr () can be placed on the 2d + 1 levels parallel to the base B. Clearly, if we simply ignore all smoothness conditions involving domain points from different levels and consider each level separately, the dimension of S d1 () can only decrease. Thus, we obtain the following upper bound on the dimension of splines over bipyramids
dim S d1 () ≤
d k=−d
dim S d1−|k| (k ),
(5)
where S d1−|k| (k ) is the space of bivariate splines of degree d − |k| and smoothness 1 over k . We then note that the bound (5) can be immediately improved. First, if we set all coefficients corresponding to the domain points in the levels 0 , and 1 , then all coefficients associated with the level −1 are defined by C 1 smoothness conditions across the base B. Since k and −k are simply shifts of the same bivariate cell B , we obtain
dim S d1−|k| (k ) = dim S d1−|k| (−k ) = dim S d1−|k| ( B ), ∀k, and dim S 01 ( B ) = 1, dim S 11 ( B ) = 3. Additionally, Theorem 4.1 combined with the results in Section 5 implies that 1,∗
dim S d1 ( B ) = dim S d ( B ), 1,∗
where S d ( B ) is defined in Section 5. Thus, for d ≥ 2, the bound (5) improves to
dim
S d1 ()
≤
1,∗
dim S 2 ( B ) + 5, dim
1,∗ S d ( B ) + dim
S d1−1 ( B ) + 2
d−2 k =2
if d = 2, dim
S k1 ( B ) + 8,
if d ≥ 3.
(6)
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To further simplify (6), we note that d −2
dim
k d −2 4 + 4 2 , if m = 2, = k 3 + n 2 , if m ≥ 3. k =2 d d−1 4(d − 3) + 4 3 − 4 2 , if m = 2, = d d−1 3(d − 3) + n 3 − n 2 , if m ≥ 3.
S k1 ( B )
k =2
Thus, for d ≥ 3, the bound (6) simplifies to 1,∗
dim S d1 () ≤ dim S d ( B ) + dim S d1−1 ( B )
+
− 8 d−2 1 , if m = 2, d d−1 6d − 10 + 2n 3 − 2n 2 , if m ≥ 3. 8d − 16 + 8
d 3
(7)
We are now ready to prove the main results of our paper. Theorem 6.2. Let be a bipyramid cell. In the collinear case, for d ≥ 2, the following holds
d
+ 12(d − 1) + (3 − d)+ , if m = 2, d 2n 3 + n(d − 1) + 6d − 4, if m ≥ 3. 8
dim S d1 () =
3
Proof. In this case, the upper bound (6) is equal to the dimension since the only C 1 smoothness condition involving coefficients associated with different levels has been taken into account. As a result, for d = 2, using Lemma 5.1 in the bound (6), we have
dim S 21 () =
13, 8 + n,
if m = 2, if m ≥ 3.
For d ≥ 3, combining Lemma 5.5 and Lemma 5.1 in the bound (7), we obtain
dim
S d1 ()
=
=
d
+ 4 d−2 1 , if m = 2, 2 d d−1 6 + n 2 + n 2 , if m ≥ 3. d d−1 8d − 16 + 8 3 − 8 2 , if m = 2, + d d−1 6d − 10 + 2n 3 − 2n 2 , if m ≥ 3. d 8 3 + 12(d − 1), if m = 2, d 2n 3 + n(d − 1) + 6d − 4, if m ≥ 3. 8+4
Combining the formulae for d = 2, and d ≥ 3 completes the proof.
2
Remark 7.2 shows how to extend this result to higher values of r. Theorem 6.3. Let be a bipyramid cell. In the coplanar case I, for d ≥ 2, the following holds
dim S d1 () = 2n
d
3
+ 7d − 3.
Proof. For d = 2, using the bound (6) and Lemma 5.3 we obtain the following upper bound
dim S 21 () ≤ 11, which is equal to the lower bound of Lemma 3.4. We now consider d ≥ 3. Recall that m = 2 in this case. Using Lemma 5.3 and Lemma 5.1 the bound (7), we obtain the following upper bound
d−1 d−1 d d−1 + (n − 1) +n + 6d − 2 + 2n − 2n 2 2 2 3 2 d . = 7d − 3 + 2n
dim S d1 () ≤
d
3
Combining the results for d = 2, and d ≥ 3, and comparing them with the corresponding lower bound in Lemma 3.4 completes the proof. 2
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Theorem 6.4. Let be a bipyramid cell. In the coplanar case II, for d ≥ 2, the following holds
d S d1 ()
dim
=
+ 10d − 8, if m = 2, d 2n 3 + 8d − 4, if m ≥ 3. 8
3
Proof. For d = 2, using the bound (6) and Lemma 5.4 we obtain the following upper bound
dim S 21 () ≤ 12, which is equal to the lower bound of Lemma 3.4 if m ≥ 3. We now consider d ≥ 3. Using Lemma 5.3 and Lemma 5.1 the bound (7), we obtain the following upper bound
d−1 10 + 4 2 , if m = 2, d−1 + (n − 2) + ≤2 d−1 2 2 8+n 2 , if m ≥ 3, d d−1 8d − 16 + 8 3 − 8 2 , if m = 2, + d d−1 6d − 10 + 2n 3 − 2n 2 , if m ≥ 3. d 8 3 + 10d − 8, if m = 2, = d 2n 3 + 8d − 4, if m ≥ 3.
S d1 ()
dim
d
Combining the results for d = 2, and d ≥ 3, and comparing them with the corresponding lower bound in Lemma 3.4 completes the proof. 2 Theorem 6.5. Let be a bipyramid cell. In the generic case, for d ≥ 2, and m ≥ 3, the following bounds hold
d
2n
+ 6d − 2 ≤ dim
3
S d1 ()
d
≤ 2n
3
+ 6d − 1 + (4 − m)+ .
Moreover, for m = 2, dim S 21 () = 11, and for d ≥ 3, the following bounds hold
d
8
3
+ 8d − 5 ≤ dim
S d1 ()
≤8
d
+ 8d − 3 − (4 − d)+ .
3
Proof. In the generic case, there are more subcases to consider following the structure in Lemma 5.2. We start with d = 2. Using the bound (6), and Lemma 5.2 for the upper bound and Lemma 3.4 for the lower bound we obtain the following estimate
11 ≥ dim S 21 () ≥
11, if m = 2, 10, if m ≥ 3.
Similarly, in the case d = 3, and m = 2, we have
27 ≤ dim S 31 () ≤ 28. We now consider m = 2, and d ≥ 4. Combining Lemma 3.4 and (6) we obtain
8
d
3
+ 8d − 5 ≤ dim S d1 () ≤ 8
d
3
For m = 3, and d ≥ 3, we obtain
2n
d
3
+ 8d − 3.
+ 6d − 2 ≤ dim S d1 () ≤ 2n
d
3
+ 6d.
We finally consider m ≥ 4, and d ≥ 3. Using (7), Lemma 5.2 and Lemma 3.4 we obtain the following tight bounds
2n
d
3
+ 6d − 2 ≤ dim
S d1 ()
≤ 2n
d
3
+ 6d − 1.
Combining all these subcases completes the proof.
2
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7. Remarks and conclusions Through the combination of homological and Bernstein–Bézier methods, we have been able to provide exact dimensions of trivariate splines on bipyramid cells in most cases, and tight bounds in one subcase. While limited to the case r = 1 and a specific type of partition, our hope is that the methods contained in this paper can be refined and extended to other classes of trivariate splines. We conclude the paper with several remarks. Remark 7.1. Theorem 5.1 in Mourrain and Villamizar (2014) applied directly to the bipyramid case yields a weaker lower bound than the one in Shan (2014). However, working through the proof and making necessary adjustments improves the result and matches the bound in Theorem 3.3. Indeed, in the proof of Theorem 5.1 in Mourrain and Villamizar (2014) we find 0
dim
S dr
f0 2 i J (γi )k , ≥ dim Rk + (−1) dim J (β)k + dim i =1
(8)
i =1
β∈03−i
where the notation matches the one in Section 4 of Schenck (2015). In the case of a cell partition, f 00 = 1, and J (γ ) = J (γ ), where γ is the single interior vertex. Moreover, for r = 1, and particularly in the bipyramid case, the values of J (γ )k can be explicitly computed. This is exactly what is done in Shan (2014), and stated in Lemma 3.1. Using Lemma 3.1 in bound (8) in the proof of Theorem 5.1 in Mourrain and Villamizar (2014), yields the lower bound of Corollary 3.4. Remark 7.2. In the collinear case, one can easily generalize the result to an arbitrary value of r ≤ d. Indeed, using the addition of the dimension associated with the bivariate layers k , k = −d, . . . , 0, . . . , d and smoothness r across the base 0 we obtain
dim S dr () =
r k =0
dim S dr −k (k ) + 2
d k=r +1
dim S dr −k ,
where each dimension can be computed using Theorem 9.3 in Lai and Schumaker (2007). For r = 1, Theorem 6.2 provides a closed-form expression for the sum. For higher values of r, the closed-form expression can be obtained as well; however, it does not look as elegant as the one of Theorem 6.2. Remark 7.3. In all examples of the generic case of Theorem 6.5 that we have checked using software, the dimension of the spline space has matched our upper bound. Per Shan (2014), when the dimension does not match the lower bound used in d this paper, the implication is that k=0 dim H 2 ( R / J )k = 0. Remark 7.4. In comparison to the upper bound of Mourrain and Villamizar (2014), if the module of splines is not known to be free, our upper bound of Section 6 is a significant improvement, though it is limited to the case of bipyramid cells. The upper bound of Mourrain and Villamizar (2014) is a powerful general tool because it can be applied in many more situations than ours. References Alfeld, P., MDS software. www.math.utah.edu/~pa. Alfeld, P., 1996. Upper and lower bounds on the dimension of multivariate spline spaces. SIAM J. Numer. Anal. 33 (2), 571–588. Alfeld, P., 2015. Multivariate splines and the Bernstein–Bézier form of a polynomial. Comput. Aided Geom. Des. http://dx.doi.org/10.1016/j.cagd.2015.11.007. In this issue. Alfeld, P., Schumaker, L.L., 2008. Bounds on the dimension of trivariate spline spaces. Adv. Comput. Math. 29, 315–335. Billera, L., 1988. Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. Am. Math. Soc. 310, 325–340. Lai, M.-J., Schumaker, L.L., 2007. Spline Functions on Triangulations. Cambridge University Press, Cambridge. Lau, W., 2005. A lower bound for the dimension of trivariate spline spaces. Constr. Approx. 23 (1), 23–31. Mourrain, B., Villamizar, N., 2014. Dimension of spline spaces on tetrahedral partitions: a homological approach. Math. Comput. Sci. 8 (2), 157–174. Schenck, H., 2015. Algebraic methods in approximation theory. Comput. Aided Geom. Des. http://dx.doi.org/10.1016/j.cagd.2015.11.001. In this issue. Schenck, H., Stilman, M., 1997. Local cohomology of bivariate splines. J. Pure Appl. Algebra 117–118, 535–548. Shan, J.J., 2014. Lower bound on the dimension of trivariate splines on cells. In: Fasshauer, Gregory E., Schumaker, Larry L. (Eds.), Approximation Theory XIV. San Antonio 2013. In: Springer Proceedings in Mathematics, vol. 83. Springer, pp. 309–333. Schumaker, L.L., 1979. On the dimension of spaces of piecewise polynomials in two variables. In: Schempp, W., Zeller, K. (Eds.), Multivariate Approximation Theory. Birkhäuser, Basel, pp. 396–412. Schumaker, L.L., Sorokina, T., 2004. C 1 quintic splines on type-4 tetrahedral partitions. Adv. Comput. Math. 21, 421–444. Sorokina, T., 2010. Intrinsic supersmoothness of multivariate splines. Numer. Math. 116, 421–434.
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Contents lists available at ScienceDirect
Computer Aided Geometric Design www.elsevier.com/locate/cagd
Dimension of trivariate C 1 splines on bipyramid cells Julien Colvin 1 , Devan DiMatteo 1 , Tatyana Sorokina ∗,2 Towson University, 7800 York Road, Towson, MD 21252, United States
a r t i c l e
i n f o
a b s t r a c t We study the dimension of trivariate C 1 splines on bipyramid cells, that is, cells with n + 2 boundary vertices, n of which are coplanar with the interior vertex. We improve the earlier lower bound on the dimension given by J. Shan. Moreover, we derive a new upper bound that is equal to the known lower bound in most cases. In the remaining cases, our upper bound is close to the known lower bound, and we conjecture that the dimension coincides with the upper bound. We use tools from both algebraic geometry and Bernstein–Bézier analysis. © 2015 Elsevier B.V. All rights reserved.
Article history: Available online xxxx Keywords: Bipyramid Dimension Trivariate spline Cell Supersmoothness
1. Introduction Smooth piecewise polynomial functions, or splines, are used widely in approximation theory, computer aided geometric design, image analysis, and numerical analysis. Key tools for analyzing multivariate spline spaces come from two fields: classical approximation theory and applied algebraic geometry. The two fields and the two corresponding research communities use entirely different approaches: Bernstein–Bézier analysis (see the survey of Alfeld, 2015, in this issue) and homological techniques (see the survey of Schenck, 2015, in this issue). For a full treatise on Bernstein–Bézier analysis, we refer the reader to Lai and Schumaker (2007). The homological approach to multivariate splines is explained in several papers, see Mourrain and Villamizar (2014), Schenck and Stilman (1997) and Billera (1988). In this paper we are concerned with trivariate splines. Let Pd denote the set of polynomials in three variables of degree ≤ d. A spline is a piecewise-polynomial function defined on a domain ⊂ R3 that belongs to a certain smoothness class. More precisely, for a fixed tetrahedral partition of the underlying domain , our spline space of interest is defined as
S dr () = {s ∈ C r () : s| T ∈ Pd
for each tetrahedron
T ∈ }.
(1)
Finding the exact dimensions of spaces S dr () of trivariate splines remains an open problem. While the dimension of any particular spline can be computed using software such as Macaulay2, or the java applet (Alfeld), general formulae are still unknown. This problem is inherited from bivariate splines, where the dimension remains unknown for d ≤ 3r + 1. As it is shown in section 17.8 in Lai and Schumaker (2007), there is a tight connection between trivariate and bivariate cases, and there is no hope to resolve the trivariate case before the dimension of bivariate splines is known for all values of d and r. We note that despite this difficulty, the dimensions of trivariate splines are known for few very special partitions, including the so-called macro-element or finite-element spaces. For example, trivariate splines on the Alfeld, the Clough–Tocher and
* 1 2
Corresponding author. E-mail addresses: [email protected] (J. Colvin), [email protected] (D. DiMatteo), [email protected] (T. Sorokina). Partially supported by a grant from the Towson University Fisher College of Science and Mathematics. Partially supported by a grant from the Simons Foundation (#235411 to Tatyana Sorokina).
http://dx.doi.org/10.1016/j.cagd.2015.12.001 0167-8396/© 2015 Elsevier B.V. All rights reserved.
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Fig. 1. Orange in R3 .
Worsey–Piper splits, see Lai and Schumaker (2007) and references therein. Other examples do not qualify as finite-element spaces, but are built on very special grids, see Schumaker and Sorokina (2004) for an example. The first and only formula for a dimension of a trivariate spline space on a tetrahedral partition of a somewhat general nature is for the so-called oranges, see Theorem 17.30 in Lai and Schumaker (2007). An orange, see Fig. 1, is a union of tetrahedra all sharing the same edge, [u , v ] in Fig. 1, and pairwise sharing common triangular faces. The goal of this paper is to provide a second example of such a general formula for a partition that we call a bipyramid cell, see Fig. 2. A bipyramid cell is a special case of the so-called interior cell. An interior cell is a collection of tetrahedra all sharing an interior vertex (cf. Section 17.8.1 in Lai and Schumaker, 2007). Note that a bipyramid cell is very different from an orange. An orange is not an interior cell, and all smoothness conditions associated with it are essentially bivariate. The smoothness conditions associated with the bipyramid cell include purely trivariate ones. Because of this difficulty, we have been able to analyze the case for r = 1 in (1) only. A large part of research in the area of trivariate splines has focused on obtaining upper and lower bounds on the dimension of trivariate spline spaces. Several results have been obtained for bounding the dimension of general splines either below or above, see Mourrain and Villamizar (2014), Lau (2005), Alfeld and Schumaker (2008), Alfeld (1996), and Shan (2014). None of the bounds cited above are tight enough to obtain exact dimensions. In this paper, we prove new upper bounds for the bipyramid cells that allow us to obtain exact dimensions in three out of four subcases; in the last subcase our new upper bound is very close to the known lower bound, and we conjecture that the dimension is equal to the upper bound. Finally, we prove that the lower bound in Shan (2014) has a typo; fixing this leads to an improved lower bound. We compare the improved lower bound to our new upper bound. The lower bound is obtained using homological techniques, while the upper bound is proved using Bernstein–Bézier analysis. The paper is organized as follows. In Section 2 we introduce bipyramid cells, and study their geometry. In Section 3 we fix the bound in Shan (2014), and derive the lower bound for the dimension of S d1 () on bipyramid cells. In Section 4 we prove a result on supersmoothness needed to obtain our new upper bound. Section 5 contains several results on the dimension of bivariate splines that are also needed to derive the new upper bound. The main results of this paper concerning the new upper bound and the exact dimension of S d1 () on bipyramid cells are in Section 6. We conclude with remarks and conjectures in Section 7. 2. Preliminaries To begin, we define a class of partitions in R3 called bipyramid cells, see e.g. Fig. 2. Definition 2.1. A bipyramid cell is a tetrahedral partition such that:
• there is exactly one interior vertex v 0 ; • n boundary vertices v 1 through v n are coplanar along with v 0 , and form a polygon surrounding v 0 in the base plane B := [ v 1 , . . . , v n ]; • each vertex v i , i = 1, . . . , n, is connected to v 0 by the interior edges [ v 0 , v i ]; • two boundary vertices v n+1 and v n+2 lie outside the base plane B, are on opposite sides of B, and are connected to v 0 by the interior edges [ v 0 , v n+1 ] and [ v 0 , v n+2 ]; and • vertices v n+1 and v n+2 connect to the boundary vertices v i , i = 1, . . . , n. It is easy to check that these conditions force to be shellable with no holes or cavities, see Lai and Schumaker (2007) for a discussion of the importance of these facts. For simplicity, and without loss of generality, we can assume that the interior vertex v 0 is located at the origin, and the n coplanar boundary vertices lie in the base xy-plane. For the remainder
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Fig. 2. Collinear case.
Fig. 3. Coplanar II case.
Fig. 4. Supersmoothness.
Fig. 5. No supersmoothness.
3
of this paper, we will assume that is a bipyramid cell. Our goal is to find the dimension of the spline space S d1 (). The next lemma summarizes several immediately obvious facts about . Lemma 2.2. In a bipyramid cell the following holds: (1) there are 2n tetrahedra, 3n interior triangular faces, and n + 2 interior edges; (2) the number of different slopes m formed by the coplanar interior edges [ v 0 , v i ], i = 1, . . . , n, in the base plane satisfies 2 ≤ n2 ≤ m ≤ n; and (3) the number of distinct planes h containing the interior triangular faces is as follows: Case 1 (Generic): v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 does not contain any other v i . Then h = 2m + 1; Case 2 (Coplanar): v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 contains at least one other v i , i = 1, . . . , n. Then h = 2m; Case 3 (Collinear): v n+1 and v n+2 are collinear with v 0 . Then h = m + 1. For further investigation, we split the coplanar case into two subcases as follows:
• Coplanar Case I: v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 contains exactly one other v i , i = 1, . . . , n. • Coplanar Case II: v n+1 and v n+2 are not collinear with v 0 , and the plane containing v n+1 , v n+2 and v 0 contains exactly two other v i , i = 1, . . . , n. Fig. 2 shows an example of the collinear case; Fig. 3 shows an example of the coplanar case II, where vertices v 0 , v 2 , v 4 , v 7 , v 8 lie in the same plane (shaded). Fig. 4 shows an example of the generic case, and finally Fig. 5 shows an example of the coplanar case I. Next, we need to classify small values of h for the results in the next section. If h ≥ 6, the results in the next sections do not require knowledge of the particular geometry of . Since in the collinear case we will compute the dimension of S d1 () without using any lower bounds, the following lemma is stated for the non-collinear cases only.
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Lemma 2.3. In a bipyramid cell with non-collinear v 0 , v n+1 and v n+2 , the following holds: If h = 4, then it is the coplanar case II with m = 2, and n = 4. If h = 5, then it is the generic case with m = 2, and n = 4. Proof. If h = 3, then we must be in the collinear case where h = m + 1, m = 2, and n = 4. Then v 5 and v 6 are collinear with v 0 , and there are only two different slopes. Thus, the partition must form an octahedron whose main diagonals intersect at the interior vertex v 0 . If h = 4, then we have two possibilities. One possibility is the coplanar case II with h = 2m, m = 2, and n = 4. The base vertices form a quadrilateral where opposite vertices are collinear with v 0 . The vertices v 5 , v 6 and v 0 are not collinear, and the plane containing v 5 , v 6 , and v 0 also contains two opposite v i . The other possibility is the collinear case with h = m + 1, m = 3, and 3 ≤ n ≤ 6. Then v n+1 and v n+2 are collinear with v 0 , and there are three different slopes in the base plane. If h = 5, then we have two possibilities. One possibility is the collinear case with h = m + 1, with m = 4 and 4 ≤ n ≤ 8. In this case, v n+1 , v n+2 and v 0 are collinear. The other possibility is the generic case with h = 2m + 1 with m = 2, and n = 4. Then v 5 and v 6 are not collinear with v 0 , and the plane containing v 5 , v 6 , and v 0 does not contain any other vertices. This is another octahedral case. 2 3. Known lower bounds on the dimension Results in this section follow from both Mourrain and Villamizar (2014) and Shan (2014). We use the terminology from the latter since it is better suited for our partition. Remark 7.1 in Section 7 shows how to use Mourrain and Villamizar (2014) to obtain the lower bound of this section. To simplify the notation in Shan (2014), we introduce the quantity J (h, k), which depends on the number of distinct planes h, containing the interior triangular faces, and a parameter, k, that runs from 0 to the degree of the spline, d. For a ring R of trivariate polynomials, we define J (h, k) to be the dimension of the graded module ( R / J )k , where J is the ideal generated by the squares of the linear forms which define the h distinct planes passing through the interior vertex v 0 , and k is the degree of the polynomials, see the survey of Schenck (2015) in this issue for additional information on the module. Lemma 3.1. In a bipyramid cell with non-collinear v 0 , v n+1 and v n+2 , the following holds:
J (h, 0) = 1 for any h; J (4, 2) = 2;
J (h, 1) = 3 for any h;
J (5, 2) = 1;
J (h, k) = 0 otherwise.
Proof. We first reproduce the more general result for cells that appears in Shan (2014), changing to our notation:
J (h, 0) = 1, for any h,
J (3, k) =
J (5, k) =
3, if k = 2, 1, if k = 3, 0, if k ≥ 4; 1 or 2, 0,
if k = 2, if k ≥ 3.
J (h, 1) = 3, for any h,
J (4, k) =
2, if k = 2, 0, if k ≥ 3;
J (h, k) = 0, for any k ≥ 2, h ≥ 6.
Note that, in the more general case covered in Shan (2014), J (5, 2) is multi-valued. For bipyramid cells, it follows from Lemma 2.3 that the only case with h = 5 is the generic case. In this case, after a change of variables, we have J = x2 , y 2 , z2 , ( z − α x)2 , ( z − β y )2 , for some non-zero α , β ∈ R. Clearly, dim( R / J )2 = 1 = J (5, 2). We next note that, for all bipyramid cases except the collinear one, the minimal number of generators of J is equal to either the number of distinct planes containing the interior triangular faces (i.e. h), or 6 if h > 6. Since we do not discuss the collinear case in this lemma, this completes the proof. 2 Remark 3.2. It is easy to see that in the collinear case, the line formed by v n+1 , v 0 , and v n+2 might be shared by an arbitrarily large number of planes. However, in this case J cannot be six-generated, as it is claimed in Shan (2014). In Section 6, we obtain the exact dimension of trivariate splines on bipyramids in the collinear case without referring to any lower bounds. Therefore, we omit the discussion of a lower bound for this case. We next discuss the lower bound for splines on any cell proved in Shan (2014). Let f i denote the number of i-dimensional interior faces of the partition: f 3 is the number of tetrahedra, f 2 is the number of interior triangular faces, and f 1 is the number of interior edges. Additionally, let he be the number of planes containing all triangular faces attached to the edge e, let f 1,2 be the number of interior edges with he = 2, let f 1,3 be the number of interior edges with he = 3, and let f 1,4 be the number of interior edges with he ≥ 4. The following bound is proved in (Shan, 2014):
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Theorem 3.3. If is a cell, dim S d1 () is bounded below as
dim S d1 () ≥ ( f 3 − f 2 + f 1 )
d+3
3
d+1 + ( f 2 − 2 f 1,2 − 3 f 1,3 − 3 f 1,4 ) 3
d d d−1 + 2( f 1,3 + f 1,4 ) + f 1,2 − J (h, k). 3
3
k =0
Next we apply Theorem 3.3 to the case of bipyramid cells. From Lemma 2.2, we know that
f 3 = 2n,
f 2 = 3n,
f 1 = n + 2,
and
f 1,3 + f 1,4 = n + 2 − f 1,2 .
We can thus rewrite the bound in Theorem 3.3 as follows
dim S d1 () ≥ 2
d+3
d+1 d + ( f 1,2 − 6) + (2n + 4 − 2 f 1,2 )
3
3
d−1
+ f 1,2
3
−
d
3
J (h, k).
k =0
Applying a variety of combinatorial identities, Lemma 2.3 and Lemma 3.1, we obtain Corollary 3.4. Let be a bipyramid cell. Then for d ≥ 2 the following holds:
dim S d1 () ≥ 6d − 2 + f 1,2 (d − 1) + 2n
d
3
−
d
J (h, k).
k =2
In the coplanar case I we have f 1,2 = 1, h ≥ 6. Thus
dim S d1 () ≥ 7d − 3 + 2n
d
3
.
In the coplanar case II, if m ≥ 3, then f 1,2 = 2, and h ≥ 6. If m = 2, then f 1,2 = 4 and h = 4. Thus
dim
S d1 ()
≥ 2n
d
3
+
10d − 8, 8d − 4,
if m = 2, if m ≥ 3.
Moreover, in the generic case if m ≥ 3, we have f 1,2 = 0. If m = 2, then f 1,2 = 2. Thus
dim S d1 () ≥ 2n
d
3
+
8d − 5, if m = 2, 6d − 2, if m ≥ 3.
Corollary 3.4 can be obtained from Mourrain and Villamizar (2014) as well, see Remark 7.1. 4. Supersmoothness The result in this section is essential for obtaining our upper bound in Sections 5 and 6. We use the notation of Definition 2.1, and square brackets for the convex hull. As in Lai and Schumaker (2007), a bivariate (interior) cell is a collection of triangles all sharing an interior vertex. Theorem 4.1. Any spline s in the space S d1 () of trivariate C 1 splines on a bipyramid with the base B := [ v 1 , . . . , v n ] and interior vertex v 0 ∈ B has the following supersmoothness: if a vertex v i , 1 ≤ i ≤ n, is not coplanar with [ v 0 , v n+1 , v n+2 ], then all derivatives of s of order two across [ v 0 , v i ] in any direction coplanar with B are continuous. Proof. We first note that if v 0 is collinear with v n+1 and v n+2 , then there is a plane containing v 0 , v n+1 , v n+2 and v i , for any of the remaining v i . Thus, we let v 0 be not collinear with v n+1 and v n+2 . Without loss of generality, let v 5 be not coplanar with [ v 0 , v n+1 , v n+2 ], see Fig. 4. Fix an arbitrary point u 5 in the interior of the edge [ v 0 , v 5 ], and consider any cross-section of the double pyramid by a plane passing through u 5 , parallel to [ v 0 , v n+2 ], and not containing [ v 0 , v 5 ]. In Fig. 4, it is the quadrilateral [u 1 , u 6 , u 4 , u 7 ]. Such a cross-section contains a bivariate cell Q with the interior vertex u 5 and four interior edges. Two of them lie in B and are collinear, and the other two are not collinear. In Fig. 4, the collinear ones are [u 5 , u 1 ] and [u 5 , u 4 ], while [u 5 , u 6 ] is parallel to [ v 0 , v 6 ] by construction, and thus [u 5 , u 7 ] is not collinear with [u 5 , u 6 ] .
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It clearly suffices to prove that the bivariate spline s˜ = s| Q has supersmoothness two at u 5 in the direction of the two collinear edges. In Fig. 4, we need to prove that s˜ has a second order continuous derivative at u 5 in the direction [u 5 , u 1 ]. The required supersmoothness immediately follows from Theorem 3.2 in Sorokina (2010). Since u 5 is an arbitrary point on the edge [ v 0 , v 5 ] such bivariate supersmoothness holds across the whole edge [ v 0 , v 5 ]. To complete the proof, we note that if v i is coplanar with [ v 0 , v n+1 , v n+2 ], as in Fig. 5 v 1 is coplanar with [ v 0 , v 6 , v 7 ], then Q is formed by two pairs of collinear edges. In Fig. 5, u 6 , u 1 , u 7 are collinear, and so are u 2 , u 1 and u 5 . The bivariate spline s˜ = s| Q does not have continuous second order derivatives in either direction at u 1 , see e.g. section 4 of Sorokina (2010). Thus, the non-coplanarity condition on the statement of the theorem is essential. 2 5. Dimension of bivariate splines over the base cell To prove our upper bound in the next section, we need to compute the dimension of bivariate C 1 splines S k1 ( B ) over the triangulation B of the base B of the bipyramid, 0 ≤ k ≤ d. We note that B is an interior bivariate cell, i.e., a collection of triangles sharing the interior vertex v 0 . Lemma 5.1. Let B be a bivariate cell with n edges and m slopes. Then
dim S 01 ( B ) = 1,
dim
S k1 ( B )
=
dim S 11 ( B ) = 3, and for k ≥ 2,
k , if m = 2, 2 k 3 + n 2 , if m ≥ 3. 4+4
(2)
Proof. Clearly, splines in S 01 ( B ) and S 11 ( B ) are bivariate polynomials whose dimensions are 1 and 3, respectively. Using Theorem 9.3 from Lai and Schumaker (2007) (originally proved in Schumaker, 1979), for k ≥ 2, we obtain
dim S k1 ( B ) = 3 + n
k
+
2
k −1
(2 − j (m − 1))+ .
(3)
j =1
We next simplify (3) for various values of m. If m = 2, then n = 4 and k −1 k −1 (2 − j (m − 1))+ = (2 − j )+ = 1. j =1
j =1
If m ≥ 3, then 2 − j (m − 1) ≤ 0 ⇐⇒ 2 ≤ j (m − 1). Since j starts at 1 and m ≥ 3, the minimum value of j (m − 1) is 2, so the inequality is always true. Therefore (2 − j (m − 1))+ = 0, so the sum evaluates to 0, and the result follows. 2 For k = d, due to additional supersmoothness described in Theorem 4.1, we can no longer use Theorem 9.3 in Lai and 1,∗ Schumaker (2007). Let S d ( B ) be the restriction of S d1 () on B, i.e., the subspace of S d1 ( B ) with the supersmoothness described in Theorem 4.1. It then follows from Theorem 9.25 in Lai and Schumaker (2007) that 1,∗
dim S d ( B ) =
n d d ( j − ri ) + 3 + ( j + 1 − j )+ , i =1 j =2
j =2
where r i is the smoothness across the edge [ v 0 , v i ],
mi , j :=
0, 0, j − ri ,
j :=
(4)
n
i =1 mi , j ,
and
if there exists l with αi = αl and rl < r i , if there exists l > i with αi = αl and rl = r i , otherwise.
We next simplify (4) further by considering each case of Section 2. 1,∗
Lemma 5.2. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of S d1 () on B. In the generic case,
1,∗ dim S 2 ( B )
= 6, dim
1,∗ S 3 ( B )
⎧ d−1 ⎪ ⎪ ⎨ 9 + 4 2 , if m = 2 ∧ d ≥ 4, 1,∗ 1 dim S d ( B ) = 7 + n d− , if m = 3 ∧ d ≥ 3, 2 ⎪ ⎪ ⎩ 6 + n d−1, if m ≥ 4. 2
= 12, if m = 2 and for all remaining cases
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Proof. In this case, Theorem 4.1 yields r i = 2, for all i = 1, . . . , n, and j and ri in (4), we obtain 1,∗
dim S d ( B ) =
7
j = m( j − 2) for all j ≥ 2. Substituting the values for
d n d ( j − 2) + 3 + ( j + 1 − m( j − 2))+ i =1 j =3
=n
d−1 2
j =2
+6+
d
(2m + 1 − j (m − 1))+ .
j =3
Since m ≥ 2, we have
2m + 1 − j (m − 1) ≤ 0 ⇐⇒ 2m + 1 ≤ j (m − 1) ⇐⇒ j ≥ 2 +
3 m−1
.
Since j starts at 3, once m ≥ 4, the inequality is always true. Therefore, (2m + 1 − j (m − 1))+ = 0 if m ≥ 4, and the d cases m = 2 and m = 3 must be computed separately. If m = 3, then j = 3 is the only positive term and j =3 (2m + 1 − d j (m − 1))+ = 1. If m = 2, we have two non-negative terms j = 3, and j = 4, and j =3 (2m + 1 − j (m − 1))+ = 3.
n
2
2
For d = 2, the dimension formula yields j =2 ( j − r i ) + 3 + j =2 ( j + 1 − m( j − 2))+ . Since r i = 2 for all i, and i =1 j = 2, the first sum is a summation of zeros, and the second summation has only one term, namely (2 + 1 − m(2 − 2))+ = 3. Therefore, the dimension is 6. 3 n 3 For d = 3, the dimension formula gives j =2 ( j − r i ) + 3 + j =2 ( j + 1 − m( j − 2))+ . Again, since r i = 2 for all i, i =1
n
the double summation becomes i =1 (3 − 2) = n. The second sum is equal to 3 + (4 − m)+ , so the value depends on m. Considering the cases m = 2, m = 3, and m ≥ 4 completes the proof. 2 1,∗
Lemma 5.3. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of
S d1 ()
on B. In the first coplanar case I, we have
dim
1,∗ S d ( B )
=
d
2
d−1 + (n − 1) + 5. 2
Proof. In this case, the plane containing v n+1 , v n+2 , and v 0 contains exactly one boundary vertex of B. Without loss of generality, assume that it is v 1 . Then Theorem 4.1 yields
r i := Moreover,
1, if i = 1, 2, if 2 ≤ i ≤ n.
j = ( j − 1) + (m − 1)( j − 2), for all j ≥ 2. We substitute the values for j and ri in (4) 1,∗
dim S d ( B ) =
d d n d ( j − 1) + ( j − 2) + 3 + (2 − (m − 1)( j − 2))+ j =2
i =2 j =3
j =2
d d d−1 + (n − 1) +5+ = (2m − j (m − 1))+ . 2
2
j =3
Since m ≥ 3, we have
2m − j (m − 1) ≤ 0 ⇐⇒ 2m ≤ j (m − 1) ⇐⇒ j ≥ 2 +
2 m−1
.
Since j starts at 3 and m ≥ 3, the inequality is always true. Therefore, (2m − j (m − 1))+ = 0, and the result follows.
2
1,∗
Lemma 5.4. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of
S d1 ()
on B. In the second coplanar case II,
1,∗ dim S 2 ( B )
= 7, and for d ≥ 3, we have
d d−1 6, if m = 2, 1,∗ + (n − 2) + dim S d ( B ) = 2 5, if m ≥ 3. 2 2
Proof. In this case, the plane containing v n+1 , v n+2 , and v 0 contains exactly two boundary vertices of B. Without loss of generality, assume they are v 1 and v 2 , and they are collinear with v 0 . Then Theorem 4.1 yields
r i :=
1, if i = 1 or i = 2, 2, if 3 ≤ i ≤ n.
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Moreover, j = ( j − 1) + (m − 1)( j − 2), for all j ≥ 2. Note that j does not change between coplanar cases I and II, but the difference in r i must be taken into account for the first part of the formula. Another difference with the previous case is that the value of r2 = 1. Thus 1,∗
dim S d ( B ) = 2
d d n d ( j − 1) + ( j − 2) + 3 + (2 − (m − 1)( j − 2))+ j =2
i =3 j =3
j =2
d d d−1 + (n − 2) +5+ =2 (2m − j (m − 1))+ , 2
2
j =3 1,∗
and the result follows for d ≥ 3. When d = 2, the double summation has only two non-zero terms of 1, so dim S 2 ( B ) = 2 + 3 + 2 = 7. 2 In the collinear case, there is no supersmoothness associated with the base cell and one can use Lemma 5.1 to compute the dimension. 1,∗
Lemma 5.5. Let B be a bivariate cell with n edges and m slopes in the base B of a bipyramid cell, and let S d ( B ) be the restriction of
S d1 ()
on B. In the collinear case, we have
dim
1,∗ S d ( B )
=n
d
2
+
4, 3,
if m = 2, if m ≥ 3.
6. New upper bounds and the dimension of splines on bipyramids In this section we obtain a new upper bound using the results from Section 5 with the help of Bernstein–Bézier methods (see the survey of Alfeld, 2015 for the terminology). The advantage of this upper bound over the upper bound of Mourrain and Villamizar (2014) is that our upper bound does not require knowing a priori that the module of C r splines over bipyramids is free. We start by defining levels in a bipyramid. Definition 6.1. Let B := [ v 1 , . . . , v n ] be the base, v 0 be the interior vertex, and v n+1 , v n+2 be the two remaining vertices of a bipyramid. The k-th level of (denoted k ) is the cross-section of by the plane parallel to B and passing through the point
(d−k) v
0 +kv n+1
d
, if 0 ≤ k ≤ d,
d
, if − d ≤ k < 0.
(d+k) v 0 −kv n+2
Each k is a bivariate cell of the same geometry as the base cell B =: 0 . To explain the method, we first note that all domain points of s ∈ S dr () can be placed on the 2d + 1 levels parallel to the base B. Clearly, if we simply ignore all smoothness conditions involving domain points from different levels and consider each level separately, the dimension of S d1 () can only decrease. Thus, we obtain the following upper bound on the dimension of splines over bipyramids
dim S d1 () ≤
d k=−d
dim S d1−|k| (k ),
(5)
where S d1−|k| (k ) is the space of bivariate splines of degree d − |k| and smoothness 1 over k . We then note that the bound (5) can be immediately improved. First, if we set all coefficients corresponding to the domain points in the levels 0 , and 1 , then all coefficients associated with the level −1 are defined by C 1 smoothness conditions across the base B. Since k and −k are simply shifts of the same bivariate cell B , we obtain
dim S d1−|k| (k ) = dim S d1−|k| (−k ) = dim S d1−|k| ( B ), ∀k, and dim S 01 ( B ) = 1, dim S 11 ( B ) = 3. Additionally, Theorem 4.1 combined with the results in Section 5 implies that 1,∗
dim S d1 ( B ) = dim S d ( B ), 1,∗
where S d ( B ) is defined in Section 5. Thus, for d ≥ 2, the bound (5) improves to
dim
S d1 ()
≤
1,∗
dim S 2 ( B ) + 5, dim
1,∗ S d ( B ) + dim
S d1−1 ( B ) + 2
d−2 k =2
if d = 2, dim
S k1 ( B ) + 8,
if d ≥ 3.
(6)
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To further simplify (6), we note that d −2
dim
k d −2 4 + 4 2 , if m = 2, = k 3 + n 2 , if m ≥ 3. k =2 d d−1 4(d − 3) + 4 3 − 4 2 , if m = 2, = d d−1 3(d − 3) + n 3 − n 2 , if m ≥ 3.
S k1 ( B )
k =2
Thus, for d ≥ 3, the bound (6) simplifies to 1,∗
dim S d1 () ≤ dim S d ( B ) + dim S d1−1 ( B )
+
− 8 d−2 1 , if m = 2, d d−1 6d − 10 + 2n 3 − 2n 2 , if m ≥ 3. 8d − 16 + 8
d 3
(7)
We are now ready to prove the main results of our paper. Theorem 6.2. Let be a bipyramid cell. In the collinear case, for d ≥ 2, the following holds
d
+ 12(d − 1) + (3 − d)+ , if m = 2, d 2n 3 + n(d − 1) + 6d − 4, if m ≥ 3. 8
dim S d1 () =
3
Proof. In this case, the upper bound (6) is equal to the dimension since the only C 1 smoothness condition involving coefficients associated with different levels has been taken into account. As a result, for d = 2, using Lemma 5.1 in the bound (6), we have
dim S 21 () =
13, 8 + n,
if m = 2, if m ≥ 3.
For d ≥ 3, combining Lemma 5.5 and Lemma 5.1 in the bound (7), we obtain
dim
S d1 ()
=
=
d
+ 4 d−2 1 , if m = 2, 2 d d−1 6 + n 2 + n 2 , if m ≥ 3. d d−1 8d − 16 + 8 3 − 8 2 , if m = 2, + d d−1 6d − 10 + 2n 3 − 2n 2 , if m ≥ 3. d 8 3 + 12(d − 1), if m = 2, d 2n 3 + n(d − 1) + 6d − 4, if m ≥ 3. 8+4
Combining the formulae for d = 2, and d ≥ 3 completes the proof.
2
Remark 7.2 shows how to extend this result to higher values of r. Theorem 6.3. Let be a bipyramid cell. In the coplanar case I, for d ≥ 2, the following holds
dim S d1 () = 2n
d
3
+ 7d − 3.
Proof. For d = 2, using the bound (6) and Lemma 5.3 we obtain the following upper bound
dim S 21 () ≤ 11, which is equal to the lower bound of Lemma 3.4. We now consider d ≥ 3. Recall that m = 2 in this case. Using Lemma 5.3 and Lemma 5.1 the bound (7), we obtain the following upper bound
d−1 d−1 d d−1 + (n − 1) +n + 6d − 2 + 2n − 2n 2 2 2 3 2 d . = 7d − 3 + 2n
dim S d1 () ≤
d
3
Combining the results for d = 2, and d ≥ 3, and comparing them with the corresponding lower bound in Lemma 3.4 completes the proof. 2
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Theorem 6.4. Let be a bipyramid cell. In the coplanar case II, for d ≥ 2, the following holds
d S d1 ()
dim
=
+ 10d − 8, if m = 2, d 2n 3 + 8d − 4, if m ≥ 3. 8
3
Proof. For d = 2, using the bound (6) and Lemma 5.4 we obtain the following upper bound
dim S 21 () ≤ 12, which is equal to the lower bound of Lemma 3.4 if m ≥ 3. We now consider d ≥ 3. Using Lemma 5.3 and Lemma 5.1 the bound (7), we obtain the following upper bound
d−1 10 + 4 2 , if m = 2, d−1 + (n − 2) + ≤2 d−1 2 2 8+n 2 , if m ≥ 3, d d−1 8d − 16 + 8 3 − 8 2 , if m = 2, + d d−1 6d − 10 + 2n 3 − 2n 2 , if m ≥ 3. d 8 3 + 10d − 8, if m = 2, = d 2n 3 + 8d − 4, if m ≥ 3.
S d1 ()
dim
d
Combining the results for d = 2, and d ≥ 3, and comparing them with the corresponding lower bound in Lemma 3.4 completes the proof. 2 Theorem 6.5. Let be a bipyramid cell. In the generic case, for d ≥ 2, and m ≥ 3, the following bounds hold
d
2n
+ 6d − 2 ≤ dim
3
S d1 ()
d
≤ 2n
3
+ 6d − 1 + (4 − m)+ .
Moreover, for m = 2, dim S 21 () = 11, and for d ≥ 3, the following bounds hold
d
8
3
+ 8d − 5 ≤ dim
S d1 ()
≤8
d
+ 8d − 3 − (4 − d)+ .
3
Proof. In the generic case, there are more subcases to consider following the structure in Lemma 5.2. We start with d = 2. Using the bound (6), and Lemma 5.2 for the upper bound and Lemma 3.4 for the lower bound we obtain the following estimate
11 ≥ dim S 21 () ≥
11, if m = 2, 10, if m ≥ 3.
Similarly, in the case d = 3, and m = 2, we have
27 ≤ dim S 31 () ≤ 28. We now consider m = 2, and d ≥ 4. Combining Lemma 3.4 and (6) we obtain
8
d
3
+ 8d − 5 ≤ dim S d1 () ≤ 8
d
3
For m = 3, and d ≥ 3, we obtain
2n
d
3
+ 8d − 3.
+ 6d − 2 ≤ dim S d1 () ≤ 2n
d
3
+ 6d.
We finally consider m ≥ 4, and d ≥ 3. Using (7), Lemma 5.2 and Lemma 3.4 we obtain the following tight bounds
2n
d
3
+ 6d − 2 ≤ dim
S d1 ()
≤ 2n
d
3
+ 6d − 1.
Combining all these subcases completes the proof.
2
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7. Remarks and conclusions Through the combination of homological and Bernstein–Bézier methods, we have been able to provide exact dimensions of trivariate splines on bipyramid cells in most cases, and tight bounds in one subcase. While limited to the case r = 1 and a specific type of partition, our hope is that the methods contained in this paper can be refined and extended to other classes of trivariate splines. We conclude the paper with several remarks. Remark 7.1. Theorem 5.1 in Mourrain and Villamizar (2014) applied directly to the bipyramid case yields a weaker lower bound than the one in Shan (2014). However, working through the proof and making necessary adjustments improves the result and matches the bound in Theorem 3.3. Indeed, in the proof of Theorem 5.1 in Mourrain and Villamizar (2014) we find 0
dim
S dr
f0 2 i J (γi )k , ≥ dim Rk + (−1) dim J (β)k + dim i =1
(8)
i =1
β∈03−i
where the notation matches the one in Section 4 of Schenck (2015). In the case of a cell partition, f 00 = 1, and J (γ ) = J (γ ), where γ is the single interior vertex. Moreover, for r = 1, and particularly in the bipyramid case, the values of J (γ )k can be explicitly computed. This is exactly what is done in Shan (2014), and stated in Lemma 3.1. Using Lemma 3.1 in bound (8) in the proof of Theorem 5.1 in Mourrain and Villamizar (2014), yields the lower bound of Corollary 3.4. Remark 7.2. In the collinear case, one can easily generalize the result to an arbitrary value of r ≤ d. Indeed, using the addition of the dimension associated with the bivariate layers k , k = −d, . . . , 0, . . . , d and smoothness r across the base 0 we obtain
dim S dr () =
r k =0
dim S dr −k (k ) + 2
d k=r +1
dim S dr −k ,
where each dimension can be computed using Theorem 9.3 in Lai and Schumaker (2007). For r = 1, Theorem 6.2 provides a closed-form expression for the sum. For higher values of r, the closed-form expression can be obtained as well; however, it does not look as elegant as the one of Theorem 6.2. Remark 7.3. In all examples of the generic case of Theorem 6.5 that we have checked using software, the dimension of the spline space has matched our upper bound. Per Shan (2014), when the dimension does not match the lower bound used in d this paper, the implication is that k=0 dim H 2 ( R / J )k = 0. Remark 7.4. In comparison to the upper bound of Mourrain and Villamizar (2014), if the module of splines is not known to be free, our upper bound of Section 6 is a significant improvement, though it is limited to the case of bipyramid cells. The upper bound of Mourrain and Villamizar (2014) is a powerful general tool because it can be applied in many more situations than ours. References Alfeld, P., MDS software. www.math.utah.edu/~pa. Alfeld, P., 1996. Upper and lower bounds on the dimension of multivariate spline spaces. SIAM J. Numer. Anal. 33 (2), 571–588. Alfeld, P., 2015. Multivariate splines and the Bernstein–Bézier form of a polynomial. Comput. Aided Geom. Des. http://dx.doi.org/10.1016/j.cagd.2015.11.007. In this issue. Alfeld, P., Schumaker, L.L., 2008. Bounds on the dimension of trivariate spline spaces. Adv. Comput. Math. 29, 315–335. Billera, L., 1988. Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. Am. Math. Soc. 310, 325–340. Lai, M.-J., Schumaker, L.L., 2007. Spline Functions on Triangulations. Cambridge University Press, Cambridge. Lau, W., 2005. A lower bound for the dimension of trivariate spline spaces. Constr. Approx. 23 (1), 23–31. Mourrain, B., Villamizar, N., 2014. Dimension of spline spaces on tetrahedral partitions: a homological approach. Math. Comput. Sci. 8 (2), 157–174. Schenck, H., 2015. Algebraic methods in approximation theory. Comput. Aided Geom. Des. http://dx.doi.org/10.1016/j.cagd.2015.11.001. In this issue. Schenck, H., Stilman, M., 1997. Local cohomology of bivariate splines. J. Pure Appl. Algebra 117–118, 535–548. Shan, J.J., 2014. Lower bound on the dimension of trivariate splines on cells. In: Fasshauer, Gregory E., Schumaker, Larry L. (Eds.), Approximation Theory XIV. San Antonio 2013. In: Springer Proceedings in Mathematics, vol. 83. Springer, pp. 309–333. Schumaker, L.L., 1979. On the dimension of spaces of piecewise polynomials in two variables. In: Schempp, W., Zeller, K. (Eds.), Multivariate Approximation Theory. Birkhäuser, Basel, pp. 396–412. Schumaker, L.L., Sorokina, T., 2004. C 1 quintic splines on type-4 tetrahedral partitions. Adv. Comput. Math. 21, 421–444. Sorokina, T., 2010. Intrinsic supersmoothness of multivariate splines. Numer. Math. 116, 421–434.