Simplex--Splines on the Clough-Tocher Split with Arbitrary Smoothness.

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Simplex–Splines on the Clough-Tocher Split with Arbitrary Smoothness.

Jean-Louis Merrien, Tom Lyche, Tomas Sauer

To cite this version:

Jean-Louis Merrien, Tom Lyche, Tomas Sauer. Simplex–Splines on the Clough-Tocher Split with

Arbitrary Smoothness.. 2021. �hal-03138780�

Simplex–Splines on the Clough–Tocher Split with Arbitrary Smoothness.

Tom Lyche, Jean-Louis Merrien, and Tomas Sauer

Abstract The space of piecewise polynomials of smoothnessrand degree 3ris considered on the Clough-Tocher split of a triangle. For anyr ≥1 we give a basis of simplex splines for this space, then a Marsden-like identity, which is proved explicitly forr ≤ 3 and symbolically for 4≤r ≤6. In addition, generalizing results forr =1, we prove forr =2,3 a geometry independent bound for the condition number in the infinity norm of this basis, and conditions to connect two triangles with smoothnessr. For odd values ofr, and especiallyr =3, using the previous constructions on each triangle, and the connecting conditions, we obtain globalCr continuity on any triangulation.

Keywords: Triangle Mesh, Piecewise polynomials, Interpolation, Simplex Splines, Marsden-like Identity.

1.1 Introduction

Splines over triangulations have applications in several branches of the sciences ranging from finite element analysis, surfaces in computer aided design and other engineering problems, see for example [6, 9, 17]. For many of these applications, piecewise linearC⁰surfaces do not suffice. In some cases, we need smoother elements for modeling, or higher polynomial degrees to increase the approximation order.

In this paper, we are interested in the space of polynomial splines over a triangulation∆of a polygonal domain ΩofR²,

S^r

d(∆):={f ∈ C^r(Ω): f|T ∈P_d, for allT ∈∆},

whered >r >0 are given integers, andP_dis the space of bivariate polynomials of total degree≤d. The dimension of this finite dimensional vector space is difficult to determine in general [17], but with the restrictiond ≥3r+2 its dimension can be expressed solely in terms ofdandr, see [14].

We can use lower degrees if we are willing to split each triangle into a number of subtriangles. The most well known examples are the Clough-Tocher split [5], and the Powell-Sabin 6 and 12 splits [23]. For these splits each triangle is divided into 3,6 and 12 subtriangles, respectively. For material on these splits and B-spline like bases for splines on triangulations see [2, 3, 7, 8, 10, 12, 13, 15–20, 25–37].

Here we consider the Clough-Tocher triangulation∆CT, see [4, 5], where each triangle in the original triangu- lation∆is split into 3 subtriangles by connecting the vertices of each triangle to its barycenter , see Figure 1.1. In [18], Hermite interpolation problems were considered for super-spline subspaces ofS^r

3r+1(∆CT)andS^r

3r(∆CT)for reven and odd, respectively. It was also stated that these degrees are minimal for globalC^r. See also [25]. In [15]

stable local bases were constructed for even smaller super-spline subspaces ofS^r

3r+1(∆CT)andS^r

3r(∆CT).

In this paper, we consider for anyr ∈N, the spacesS^r

3r( )on a single triangleT = in∆CT for which we construct a B-spline like basis made out of simplex splines. This extends and generalizes the caser=1 that was considered in [19]. For more on the Clough-Tocher split see [1, 4, 11, 13, 15, 17, 18, 21, 28, 32]. Looking in more Tom Lyche

Dept. of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316 Oslo, Norway e-mail:[email protected] Jean-Louis Merrien

Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France e-mail:[email protected] Tomas Sauer

Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung & FORWISS, Universität Passau, Innstr. 43, D-94032 Passau, Germany e-mail:[email protected]

1

detail at the casesr =2,3, corresponding to degreesd =6,9, we moreover give explicit formulas for connecting two neighboring triangles in a C^r fashion across an edge using Bernstein-Bézier techniques, and give an upper bound for theL_∞condition number of the basis. This upper bound is independent of the shape of the triangleT. We also give a Marsden-like identity for the reproduction of polynomials which is proved forr ≤ 3 and shown symbolically forr ≤6. We conjecture that it holds for anyr.

It was shown in [15, 18] that globalC²continuity cannot be achieved ford =6 for a general triangulation refined by Clough-Tocher splits into ∆CT. However, forr odd it follows, again from [15, 18], that globalC^r continuity holds ford=3r. This means in particular that the formulas forC³continuity across an edge in Section 1.5.2 can be used to compute with elements inS³

9(∆CT), using the simplex spline basis on each triangle inS³

9(∆)in the usual Bernstein-Bézier fashion.

The paper is organized as follows. Since it depends heavily on properties of Bernstein polynomials and simplex splines, we recall some well known facts about these functions in the next section. Section 1.3 introduces the Clough-Tocher split on a triangle and gives some basic facts about the dimension of the associated spline spaces.

In Section 1.4, we derive a local basis for the spline space and prove its basis property, mainly by a combinatorial argument. In Section 1.5, we consider the cases of globalC²andC³regularity in more detail and state and prove a Marsden-like identity.

1.2 Preliminaries

In this section we recall some properties of Bernstein polynomials on a triangle and bivariate simplex splines. Here we use the notationd ∈N₀:=N∪ {0}, and lethSidenote theconvex hullof the setS ⊂R².

1.2.1 Bernstein Polynomials

For a given nondegenerate triangleT := h{p₁,p₂,p₃}i ∈R², andi,j,k ∈ N₀, the Bernstein polynomial B_{i jk}^d : R² →Rof degreed :=i+j+k∈N₀, is defined by

B_{i jk}^d (x,y)=B^d_{i jk}(β₁, β₂, β₃):= d!

i!j!k!βⁱ₁β₂^jβ^k₃, (1.1) where β=(β1, β₂, β₃)are thebarycentric coordinatesofx =(x,y) ∈R²with respect toT, i. e.,

x =β₁p₁+β₂p₂+β₃p₃, β₁+β₂+β₃=1. (1.2) The barycentric form ofMarsden’s identityfor Bernstein polynomials then is simply the multinomial expansion

(u1β₁+u₂β₂+u₃β₃)^d = Õ

i+j+k=d

uⁱ₁u₂^ju^k₃B_{i jk}^d (β1, β₂, β₃), (u1,u₂,u₃) ∈R³, β₁+β₂+β₃=1, (1.3) where in the expressionÍ

i+j+k=d it is understood thati,j,k ∈N₀, which is consistent with the convention that B^d

i jk =0 if one of the indices becomes negative.

Taking partial derivatives of order ν, µ, κ ∈ N₀ with respect to u₁,u₂,u₃, respectively, in (1.3) and setting u=(1,1,1)we obtain

β^ν₁β₂^µβ^κ₃ = Õ

i+j+k=d

γ_{i jk}(β^ν₁β^µ₂β₃^κ)B_{i jk}^d (β₁, β₂, β₃), ν+µ+κ≤d, γ_{i jk}(β^ν₁β^µ₂β₃^κ) ∈R. (1.4) We find

1= Õ

i+j+k=d

B_{i jk}^d (β₁, β₂, β₃) (β1, β₂, β₃)=

Õ

i+j+k=d

b^∗_{i jk}B_{i jk}^d (β1, β₂, β₃), b^∗_{i jk} = i

d, j d,k

d

.

(1.5)

The vector b^∗_{i jk} is called thebarycentric formof thedomain pointofB_{i jk}^d . From (1.4) and (1.5) it follows that the elements in the set

B^d :={B_{i jk}^d :i,j,k≥0,i+j+k=d} (1.6) form apartition of unity basisforP_d. Indeed, the number of elements #B^dofB^dequals ^d+2₂

, the dimension of P_d. We refer to [17] for further properties ofB^d

i jk.

1.2.2 Bivariate Simplex Splines

For our purpose it is convenient to work witharea normalized bivariate simplex splines[20] of degreed ≥0 withknots

K:={k1, . . . ,kd+3}, kj ∈R²\ {0}, j=1, . . . ,d+3.

We can considerKeither as a multiset or as a matrixK ∈R^2×(d+3).

The simplex splineQ[K]:R² →R, is now defined asQ[K](x)=0 for allx ∈R²if rankK<2, and otherwise Q[K]:= area(T )

d+2 2

M[K], (1.7)

where area(T )is the positive area of a fixed reference triangle in the original triangulation∆, andM[K]is aunit integral bivariate normalized simplex spline, defined as a linear functionalM[K]:C(R²) →Rgiven by

M[K](ϕ):=(d+2)!

∫

Sd+2

ϕ^d+3Õ

j=1

kjt_j

dt1· · ·dtd+2, ϕ∈C(R²), (1.8)

with Sn := {(t1, . . . ,t_n+1) ∈ Rⁿ⁺¹ : t_i ≥ 0,Ín+1

i=1 t_i = 1},the unit simplex in Rⁿ,n ∈ N. If K has full rank, rank(K)=2, thenM[K]can be identified with a functionM[K]:R²→R, and we write (1.8) in the form

∫

R^s

M[K](x)ϕ(x)dx =(d+2)!

∫

Sd+2

ϕÕ^d+3

j=1

k_jt_j

dt1· · ·dtd+2,, ϕ∈C(R²). (1.9) We mention the following well-known properties ofM[K][22, 24] andQ[K].

1. Q[K]andM[K]arepiecewise polynomialsof degreed=#K−3 witht supporthKi.

2. Local smoothness: Across aknot line, which is a line in the complete graph associated withK, we have that M[K],Q[K] ∈ C^d+1−µ, whereµis the number of knots on that knot line, including multiplicities.

3. Differentiation formula: For u = (u₁,u₂) ∈ R² and any choice of a₁, . . . ,a_d+3 such that Í

ja_jk_j = u, Í

ja_j =0, one has

D_uM[K]=(d+2)

d+3Õ

j=1

a_j M[K\kj], D_uQ[K]=d Õd+3

j=1

a_jQ[K\kj], (1.10) whereD_u:=u₁D₁+u₂D₂andD₁,D₂denotes partial derivatives. (A-recurrence)

4. Recurrence relation: For anyx ∈R²and anyb1, . . . ,bd+3such thatÍ

jb_jk_j =x,Í

jb_j =1, one has M[K](x)= d+2

d Õd+3

j=1

b_jM[K\kj](x), Q[K](x)=

d+3Õ

j=1

b_jQ[K\kj](x). (1.11) (B-recurrence)

5. Knot insertion formula: For anyy∈R²and anyc1, . . . ,cd+3such thatÍ

jc_jk_j =y,Í

jc_j =1, one has M[K]=

Õd+3 j=1

c_jM[K∪y\kj], Q[K]=

d+3Õ

j=1

c_jQ[K∪y\kj]. (1.12) (C-recurrence)

6. Degree zero: ForK={k1,k2,k3}

p1

pT

p3

p2

Fig. 1.1 The Clough–Tocher split,T=hp1,p₂,p₃i,T1:=hpT,p₂,p₃i,T2:=hpT,p₃,p₁iandT3:=hpT,p₁,p₂i

M[K](x):= 1

area(hKi), Q[K](x) := area(T )

area(hKi),x∈ hKi^o, M[K](x):=Q[K](x)=0,x <hKi,

(1.13)

whereS^o is the interior of the setS. The values of M[K]andQ[K]on the boundary of hKi has to be dealt with separately, see below.

We refer to [22, 24] for further properties ofM[K].

1.3 Clough–Tocher split, dimension and smoothness

Given a nondegenerate triangleTinR², we connect the verticesp₁,p₂,p₃to the barycenterp_T :=(p₁+p₂+p₃)/3.

With this construction, known as the Clough–Tocher split , we obtain three subtrianglesT1 := hp_T,p₂,p₃i, T2:=hp_T,p₃,p₁iandT3:=hp_T,p₁,p₂i, see Figure 1.1.

We consider the spline spaceS^r

d( )with respect to these three subtriangles onT. To obtain a unique function value at each point inT we associate the half open edges

hp_i,p_T):={(1−t)p_i+tp_T : 0≤t<1}, i=1,2,3, to the three subtriangles ofT as follows

hp₁,p_Ti ∈ T₂, hp₂,p_Ti ∈ T₃, hp₃,p_Ti ∈ T₁, (1.14) and we somewhat arbitrarily associate the pointp_T toT₂.

We first give the dimension ofS^r

d( ), and a set of conditionscs^r_d,iinduced by requesting smoothnessC^racross an edge hp_i,p_i+1i of two neighbouring triangles in∆. For 1 ≤ m ≤ d ands ∈ S^m

d( )letD_nis(pⁱ_j,ℓ)denote a derivative across the edgehp_i,p_i+1iat the pointpⁱ_j,ℓ, where

pⁱ_j,ℓ := (j−ℓ+1)p_i+ℓp_i+1

j+1 , forℓ=1, . . . ,j, j ∈N,andi=1,2,3, p₄ :=p₁. Fori=1,2,3 we let, form∈N,

V_m,i(s):={D₁^αD₂^βs(p_i): 0≤α+β≤m},

E_m,i^even(s):={D_n^j−1_i s(pⁱ_j,ℓ):(j, ℓ) ∈I_m}, E_m,i^odd(s):={D_n^j_is(pⁱ_j,ℓ):(j, ℓ) ∈ I_m}, I_m:={(j, ℓ):ℓ=1, . . . ,jandj=1, . . . ,m}.

(1.15)

We define fors∈S^r

d( )andi=1,2,3 the following independent degrees of freedom (interpolation conditions), cs^r_d,i :=

(V_q−1,i(s) ∪E^even

r+1,i(s), ifd =2qis even,

V_q,i(s) ∪E_r^odd_,i (s), ifd =2q+1 is odd. (1.16) These conditions guarantee smoothnessC^racross the edgehp_i,p_i+1i.

Proposition 1We have

dimS^r

d( )= r+2

2

+3

d−r+1 2

+

Õd−r j=1

(r+1−2j)₊ (1.17)

and

#cs^r_2q,i = q+1

2

+ r+2

2

, #cs_2q+1,i^r = q+2

2

+ r+1

2

. (1.18)

Moreover,dimS^r

3r( ) ≥#cs^r_3r :=Í3

i=1#cs_3r,i^r for allr∈N. Proof The formula (1.17) for dimS^r

d( )follows from Theorem 9.3 in [17] withn=m_v=3. SinceV_m,i(s)andI_m given by (1.15) contain ^m+2₂

and ^m+1₂

linearly independent, hence nonredundant, elements, respectively, (1.18) follows.

In the even and odd case (1.17) and (1.18) imply that dimS^2s

6s( )=27s²+9s+1, #cs^2s_6s =3

2(13s²+11s) dimS^2s+1

6s+3( )=3(9s²+12s+4), #cs_6s+3^2s+1 = 3

2(13s²+21s+8),

(1.19)

and it easily follows that dimS^r

3r( ) ≥#cs^r_3r for allr∈N. From (1.19) we have the following table

r 1 2 3 4 5 6

dim 12 37 75 127 192 271

#cs_3r^r 12 36 63 111 153 225 . In Figure 1.2 we display the degrees freedomcs^r_3r forr=1,2,3.

Fig. 1.2 The degrees of freedomcs_3r^r forC¹,C²andC³. For the order of a cross boundary derivativeDn^α_iwe use a dot forα=0, and colors red, blue and green forα=1,2,3.

1.4 A basis for S

3r

We now focus on a special collection of simplex splines on the Clough–Tocher split of one single triangle in the following way: for integersi,j,k, ℓ, we consider the simplex splineQh

p₁^{i},p₂^{j},p₃^{k},p^{ℓ}_T i

, wherep^{i_j ^}, means that the vertex p_j has multiplicityi, i. e., is repeateditimes. For brevity, we will use the alternative notations, a more compact and a more illustrative one, for simplex splines of degreed=i+j+k+ℓ−3 from now on, namely,

Qh

p₁^{i},p₂^{j},p₃^{k},p^{ℓ_T^}i

=: [i,j,k;ℓ]=: i jkℓ=:

k

ℓ i j

, i,j,k∈N, ℓ∈N₀:=N∪ {0}. (1.20) By the local smoothness property for simplex splines we note that i jkℓhas smoothnessd+1−i−ℓacross the knotlinehp₁,p_Tiforℓ >0. In this notation, the Bernstein polynomialsB_{i jk}(u,v,w):= ^(i+j+k)!_i!j!k! uⁱv^jw^k of degree

d=i+j+khave the form

B_{i jk} = i+1,j+1,k+1,0=

k+1

0 j+1 i+1

, i+j+k=d. (1.21)

We will use the more graphic form on the right hand side of (1.20) whenever possible to make the basic ideas more accessible, but it is convenient to use the more compact notations in computations. The well–knownknot insertion formula for the insertion of a knot at the barycenter, as a special case of the C-recurrence (1.12), for example, akes the form

k

ℓ

i j =1

3

k

ℓ+1

i−1 j +

k

ℓ+1 j−1

i +

k−1

ℓ+1 i j

!

(1.22) in the graphical notation.

Definition 1The numberµ:=d+1−ris called the maximummultiplicityof an interior knot line of the simplex spline i jkℓ.

Some obvious properties for simplex splines to be inS^r

d( )are listed in the following lemma.

Lemma 1If _{i jkℓ} ∈S^r

d( )then 1.i+j+k+ℓ=d+3=µ+r+2,

2. if, in addition,ℓ,0then regularityC^rimpliesmax{i,j,k}+ℓ≤µ.

Next, let us focus on specific elements ofS^r

3r( )that will turn out to be useful. The three types are categorized by whether max{i,j,k}+ℓequalsµ, exceeds it or is strictly less thanµ, where the second case occurs only forℓ=0.

Definition 2LetΣ^r_3r denote the set of elements i jkℓ ofS^r

3r( )that consists of the elements of the following three types:

Type (1): max{i,j,k}+ℓ=µand min{i,j,k} ≥1, Type (2): ℓ=0 and max{i,j,k}> µ,

Type (3): max{i,j,k}+ℓ < µand min{i,j,k}=1.

Remark 1The types are symmetric with respect toi,j,k, i. e., all subclasses are closed under permutation of the indices.

Forr=1,2,3,4, Definition 2 results in the following set of splines.

Example 1Forr=1,d =3,µ=3, dimS¹

3( )=12 we have 1. Type (1): 9 elements

1

1 2 2

1

0 2 3

and symmetries 2. Type (2): 3 elements

1

0 1 4

and symmetries, 3. Type (3): 0 elements.

Example 2Forr=2,d =6,µ=5, dimS²

6( )=37 the different types look as follows:

1. Type (1): 25 elements

2

3 2 2

2

2 2 3

1

2 3 3

2

1 2 4

1

1 3 4

2

0 2 5

1

0 3 5

and symmetries,

2. Type (2) 9 elements

1

0 2 6

1

0 1 7

and symmetries 3. Type (3): 3 elements

1

0 4 4

and symmetries.

Example 3In the caser=3,d=9,µ=7, dimS³

9( )=75 we get 1. Type (1): 48 elements

2

4 3 3

2

3 3 4

1

3 4 4

2

2 3 5

1

2 4 5

2

1 3 6

1

1 4 6

2

0 3 7

1

0 4 7

and symmetries,

2. Type (2): 18 elements

2

0 2 8

1

0 3 8

1

0 2 9

1

0 1 10

and symmetries 3. Type (3): 9 elements

1

0 5 6

1

1 5 5

and symmetries Example 4Forr=4,d =12, dimS⁴

12( )=127, the elements look as follows:

1. Type (1): 79 elements

3

6 3 3

3

5 3 4

2

5 4 4

3

4 3 5

2

4 4 5

1

4 5 5

3

3 3 6

2

3 4 6

1

3 5 6

3

2 3 7 2

2 4 7

1

2 5 7

3

1 3 8

2

1 4 8

1

1 5 8

3

0 3 9

2

0 4 9

1

0 5 9

and symmetries, 2. Type (2): 30 elements

2

0 3 10

1

0 10 4

2

0 2 11

1

0 11 3

1

0 2 12

1

0 13 1

and symmetries 3. Type (3): 18 elements

1

2 6 6

1

1 6 7

1

0 6 8

1

0 7 7

and symmetries

Remark 2Some Bernstein polynomials arenotof any of the three types and thus are not in the basis. A Bernstein polynomial in the basis can be of Type (1), (2) or (3), see Figure 1.3 and the previous examples.

One central result of this paper, stated in Theorem 1, is thatΣ^r_3r is in fact abasisforS^r

3r( ). The proof of this fact will consist of showing thatΣ^r_3r is a subset of dim(S^r

3r( ))linearly indenpendent elements of the space S^r

3r( ).

To count the number of elements ofΣ_3r^r , we start with some bounds ofℓwith respect to the different types of functions inΣ^r_3r.

Removed Type (1) Type (2)

Type (1) Type (2) Type (3) Removed

Type (1) Type (2) Type (3) Removed

Fig. 1.3 Domain points of the Bernstein polynomials used in the CTS basis, left:r = 1,d = 3, middle:r = 2,d = 6, right:

r=3,d=9. “Removed” means that the respective Bernstein polynomials are of none of the Types (1), (2) or (3).

Lemma 2For Type (1) we haveℓ ≤d−3r/2while for Type (3)ℓ ≤d−2r−2holds.

Proof For a Type (1) element, assume thati+ℓ= µ, thenj+ℓ ≤ µandk+ℓ ≤ µ, hencei+j+k+3ℓ ≤3µ. Sincei+j+k+ℓ=d+3 andµ=d−r+1, we thus deduce thatd+3+2ℓ≤3d+3−3rorℓ≤d−3r/2.

For Type (3), we assume thatk=1, hencei+ℓ≤µ−1 andj+ℓ≤µ−1, theni+j+k+2ℓ≤2µ−1. Since i+j+k+ℓ=d+3 andµ=d−r+1, it follows thatd+3+ℓ≤2d−2r+2−1 orℓ≤d−2r−2.

Next we prove that the sum of the numbers of elements of the three types is exactly the dimension of dimS^r

3r( ).

Proposition 2Forr≥1andd=3r, we have the following table according to the parity ofr.

r 2s,s>0 2s+1,s≥0

d 6s 6s+3

dimS^r

d( ) 27s²+9s+1 3(9s²+12s+4)

#Type(1)3s(5s+3)+1 3(s+1)(5s+3)

#Type(2) 3s(2s+1) 3(s+1)(2s+1)

#Type(3) 3s(2s−1) 3s(2s+1)

(1.23)

Hence,

#Type(1)+#Type(2)+#Type(3)=dimS^r

3r( ), r≥1. (1.24)

Proof We begin withrofevenparity, i. e.,r=2s,d =3r=6s,µ=4s+1,i+j+k+ℓ=6s+3 and count the basis elements of different types.

1. Type (1): According to Lemma 2 we haveℓ≤d−3r/2=3sand a generic element of Type (1) is of the form [i=s+p+1,j,k=2s+2−j; 3s−p], 1≤ j ≤i,1≤k≤i, (1.25) from which it follows thats−p+1 ≤ j ≤2s+1 and p=0, . . . ,3s. We count the number of elements with respect top:

a. Forp=0: 1 element.

The only possible choice is [s+1,s+1,s+1; 3s].

b. For 1≤p≤s: 6pelements.

The elements [i,j,k; 3s−p]are of the form

[s+p+1,s−p+1+q,s+p+1−q; 3s−p], q=0, . . . ,2p−1, (1.26) i. e.,i=s+p+1, j =s−p+1+q,s+p+1−q, and with the permutations[i,j,k],[j,k,i]and[k,j,i], which gives 3 elements for anyqin (1.26). We notice that in two cases we obtain the same three elements up to the permutation, namely, forq=0 the valuesi=k=s+p+1, j =s−p+1, and forq=2pthe values i= j=s+p+1,k=s−p+1. In the other cases, 0<q <2p, we have j=s−p+1+q <i=s+p+1 andk=s+p+1−q<i=s+p+1, so that the elements are different.

c. Fors+1≤p≤3s: 6s+3 elements consisting of

[s+p+1,q+1,2s−q+1; 3s−p], q=0, . . . ,2s, (1.27) and again the permutations[i,j,k],[j,k,i],[k,j,i].

Hence the total number of elements of Type (1) is 1+

Õs p=1

6p+ Õ3s p=s+1

(6s+3)=1+3s(s+1)+2s(6s+3)=1+3s(5s+3) as listed in (1.23).

2. Type (2) requiresℓ=0 by definition and the generic elements of the form

[i=4s+2+p,j,k=s+1−j−p; 0], 1 ≤j ≤i,i≤k≤i, (1.28) with the additional constrainti ≥ µ+1=4s+2 that leads to 1≤ j ≤2s−p,p=0, . . . ,2s−1. This gives 3(2s−p)elements

[4s+2+p,q+1,2s−p−q; 0], q=0, . . . ,2s−p−1, (1.29) and the respective permutations so that the total number of elements of Type (2) is

2s−1Õ

p=1

3(2s−p)= Õ2s q=1

q=3s(2s+1) in this case.

3. Type (3): up to symmetry we can assume thatk=1 and, by Lemma 2 thatℓ≤d−2r−2=2s−2. The generic element is of the form

[i,j =4s+4+p−i,k=1;ℓ=2s−2−p], i+ℓ≤µ−1=4s, j+ℓ ≤4sp=0, . . . ,2s−2, (1.30) sincei+j+k+ℓ=d+3=6s+3. Hence, for 0≤p≤2s−2 we get 3(p+1)elements

[2s+2+p−q,2s+2+q,1; 2s−2−p], q=0, . . . ,p, (1.31) and the permutations[i,j,k],[j,k,i],[k,j,i], leading to a total of

2s−2Õ

p=0

3(p+1)=3s(2s−1) elements of Type (3).

In the case of odd parity, i. e.,r=2s+1,d =3r=6s+3,µ=4s+3,i+j+k+ℓ=6s+6, we proceed in the same way and distinguish by types.

1. For Type (1) we have the boundℓ ≤3s+1 and the generic element

[i=s+p+2,j,k=2s+3−j; 3s+1−p], 1≤ j ≤i1≤k≤i,p=0, . . . ,3s+1. (1.32) Again, Type (1) request the distinction of several cases according top.

a. For 0 ≤p≤s: 6p+3 elements The generic elements are

[s+p+2,s+1−p+q,s+2+p−q; 3s+1−p], q=0, . . . ,2p, (1.33) and the permutations[i,j,k],[j,k,i],[k,j,i]. Again, we notice that we obtain the same three elements up to the permutations forq=0, namelyi=k=s+p+2,j=s−p+1, and forq=2p+1, namelyi= j=s+p+2, k=s−p+1, respectively. For 0<q<2p, on the other hand, we havej=s+1−p+q<i=s+p+2 and k=s+2+p−q <i=s+p+2 so that all the elements are different again, just like in the case of evenr.

b. Fors+1≤p≤3s+1: 6(s+1)elements based on the generic element

[s+p+2,q+1,2s−q+2; 3s+1−p], q=0, . . . ,2s+1, (1.34) and its permutations.

Therefore, the total number of elements of Type (1) is

Õs p=0

(6p+3)+

3s+1Õ

p=s+1

6(s+1)=3(s+1)(5s+3).

2. Type (2) again requestsℓ=0 and leads to 3(2s+1−p)elements based on the generic element

[4s+4+p,q+1,2s+1−p−q; 0], q=0, . . . ,2s−p, (1.35) and its permutations, so that the total number of elements of Type (2) is

Õ2s p=0

3(2s+1−p)=3(s+1)(2s+1).

3. For Type (3) we again assume thatk=1, noteℓ ≤d−2r−2=2s−1 and obtain 3(p+1)elements from the generic element

[2s+3+p−q,2s+3+q,1; 2s−1−p], q=0, . . . ,p (1.36) and its permutations totalling up to

2s−1Õ

p=0

3(p+1)=3s(2s+1) elements of Type (3).

Having completed the table in (1.23) the claim (1.24) follows from summing up the columns of the table.

Theorem 1Σ^r_3r is a basis ofS^r

3r.

We prove Theorem 1 by verifying in Proposition 3 that the elements ofΣ^r_3r are linearly independent. Since we already know from Proposition 2 that #Σ^r_3r =dimS^r

3r, this indeed shows that they are a basis of the spline space.

Consequently,Σ_3r^r spans the space of all simplex splines containedS^r

3r. We give an independent proof of this fact in Proposition 8 in the appendix as it may be of independent interest and motivates the classification of the simplex splines according to the three types.

To prove linear independence, we need the following technical tool concerning particular derivatives of simplex splines.

Lemma 3Fori>0We have that

D_p1−x [i,j,k;ℓ](x)=d

[i−1,j,k;ℓ](x) − [i,j,k;ℓ](x)

, (1.37)

while

D_p1−x [0,j,k;ℓ](x)=d

3 [0,j,k;ℓ−1](x) − [0,j−1,k;ℓ](x) − [0,j,k−1;ℓ](x) − [0,j,k;ℓ](x) . (1.38) Since fori,j,k,l∈N₀, andx ∈ T with barycentric coordinatesβ₁, β₂, β₃with respect to p₁,p₂,p₃, we have

[0,j+1,k+1;ℓ+1](x)=(j+k+ℓ)!

j!k!ℓ! (β₂−β₁)^j(β₃−β₁)^k(3β₁)^ℓ, x∈ T₁, [i+1,0,k+1;ℓ+1](x)= (i+k+ℓ)!

i!k!ℓ! (β₁−β₂)ⁱ(β₃−β₂)^k(3β₂)^ℓ, x∈ T₂, [i+1,j+1,0;ℓ+1]x)=(i+j+ℓ)!

i!j!ℓ! (β1−β₃)ⁱ(β2−β₃)^j(3β₃)^ℓ, x∈ T3, [i+1,j+1,k+1; 0](x)= (i+j+k)!

i!j!k! βⁱ₁β₂^jβ₃^k =B_{i jk}^d (x), x∈ T,

(1.39)

we observe that the formula (1.38) in fact corresponds to taking a partial derivative with respect toβ₁in (1.39).

Proof Writex=Íα_jp_j, withÍα_j =1. Then the derivative formula and the recurrence yield that

D_p1−x [i,j,k;ℓ](x)=d

(1−α₁) [i−1,j,k;ℓ](x) −α₂ [i,j−1,k;ℓ](x) −α₃ [i,j,k−1;ℓ](x)

=d [i−1,j,k;ℓ](x) −d

α₁ [i−1,j,k;ℓ](x)+α₂ [i,j−1,k;ℓ](x)+α₃ [i,j,k−1;ℓ](x)

=d

[i−1,j,k;ℓ](x) − [i,j,k;ℓ](x) ,

which is (1.37). For the second identity we note thatp_T =¹₃(p₁+p₂+p₃)impliesp₁ =3p_T−p₂−p₃and hence, writingx=α_Tp_T +α₂p₂+α₃p₃withÍα_j =1, we obtain that

D_p1−x [0,j,k;ℓ](x)

=d

(3−α_T) [0,j,k;ℓ−1](x) − (1+α₂) [0,j−1,k;ℓ](x) − (1+α₃) [0,j,k−1;ℓ](x) which can be recombined as above to yield (1.38).

Proposition 3The elements ofΣ^r_3rare linearly independent.

Proof Denote by

I^r:=I(Σ^r_3r)=

(i,j,k, ℓ): [i,j,k;ℓ] ∈Σ_3r^r (1.40) the set of all knot multiplicities of splines inΣ^r_3r. Now assume that there exist coefficientsa_{i jkℓ}such that

s:= Õ

(i,j,k,ℓ)∈I^r

a_{i jkℓ} [i,j,k;ℓ]=0.

On the boundary p₂,p₃

we have [i,j,k;ℓ],0 if and only ifi =1 andℓ =0 and the splines [1,j,k; 0]

reduce to univariate Bernstein polynomials that can be classified as follows:

1. Type (1): [1, µ−ℓ, ℓ+r+1; 0],

2. Type (2): [1, µ+1,r+1; 0], . . . , [1, µ+r+1,1; 0], 3. Type (3): [1, µ−ℓ−m, ℓ+r+1+m; 0], 1≤m≤ ^µ−r−1₂ −ℓ,

together with their symmetric elements where jandkare interchanged. Recall that if these symmetries coincide they are considered as only one element inΣ^r_3r. These Bernstein polynomials are linearly independent within the same type by construction and between types since the maximal multiplicity is= µ−ℓ in Type (1), > µ−ℓ in Type (2) and< µ−ℓin Type (3). Therefore, restrictingsto the boundary

p₂,p₃

, it follows thata_{i jkℓ} =0 for(i,j,k, ℓ)=(1,j,k,0). Considering the other boundaries ofT, we can thus conclude thata_{i jkℓ} =0 whenever min{i,j,k}=1 andℓ=0.

Starting from this observation, we prove by induction onm=1,2, . . . that

a_{i jkℓ} =0, min{i,j,k}+ℓ=m, (1.41)

where the casem=1 has been treated in the first part of this proof. We will treat the casem=2 explicitly as the general procedure will become clear by then. We assume thatiis the minimal value, consider the identity

0=D_p1−xs(x), x∈ p₂,p₃

and find by Lemma 3 that there are only two types of splines which are nonzero on the boundary. The first is D_p1−x [2,j,k; 0](x)= [1,j,k; 0](x) − [2,j,k; 0](x)

which coincides with [1,j,k; 0](x)on p₂,p₃

. The second is

D_p1−x [1,j,k; 1](x)= [0,j,k; 1](x) − [1,j,k; 1](x), which by (1.39) coincides with [0,j,k; 1]= [1,j,k; 0]on

p₂,p₃

. Again, these splines are linearly indepen- dent within the types by construction and across the types by the different values of the maximal multiplicity, and (1.41) form=2 follows by considering all three faces ofT by a symmetry argument.

The general induction step proceeds in exactly this way by assuming thati=min{i,j,k}andi+ℓ=m+1 and applying (1.37)itimes we find from (1.39) withx=(β1, β₂, β₃)andℓ=m+1−i

Dⁱ_p

1−x [i,j,k;ℓ](x)=K [0,j,k;ℓ](x)=(j+k+ℓ)!

j!k!ℓ! (β₂−β₁)^j(β₃−β₁)^k(3β₁)^ℓ

for some connstantK. Differentiatingℓtimes with respect toβ₁and settingβ₁ =0 we find

∂^ℓ

∂ β^ℓ

1

Dⁱ_p

1−x [i,j,k;ℓ](x)=Kβ₂^jβ₃^k,

where again K is some nonzero constant. where the admissible values for j,k lead to linearly independent polynomials on the boundary. This completes the proof of Proposition 3.

Example 5We illustrate the elimination procedure of Proposition 3 for the caser=3.

The elimination proceudre starts by considering the Bernstein polynomials

1

0 4 7

1

0 3 8

1

0 2 9

1

0 1 10

1

0 5 6

on the “lower” edge of the triangle where alle other simplex splines from our list vanish. Note that the three types are then distinguished by whether the maximum equals µ=7 (the element on the left), exceeds this values (the three elements in the middle) or is strictly smaller (the element on the right). Therefore, the coefficients of these splines and all their symmetries have to vanish which deals with the first “ring” of coefficients on the boundary.

Applying the differential operatorD_p1−xonce, gives us the Bernstein polynomials

1

0 3 7

1

0 2 8

of Type (1) and (2). As the list forr =4 shows, Type (3) elements are not excluded in principle. In addition, we get the splines

0

1 4 6

0

1 5 5

which also reduce to nonvanishing univariate Bernstein polynomials on the boundary. This finishes off all elements with min{i,j,k}+ℓ=2 and especially all elements of the Types (2) and (3) forr=2.

The application ofD²_p

1−xgives us

D²_p

1−x 1

2 4

5 =3

0

1 4 5

−

0

2 4 4

−

0

2 3 5

→3

0

1 4 5

since the two Berstein polynomias which are subtracted vanish on the boundary. The other nonvanishing element

is

0

1 3 6

.

Now it should be clear how the process is completed:D³_p

1−xextracts the nonzero polynomials

0

1 4 4

0

1 3 5

together with their symmetries and applyingD⁴_p

1−xandD⁵_p

1−xwe get

0

1 4 3

and

1

0 3 3

,

respectively. Observe that in each step of the process always Bernstein polynomials of the same fixed degree are considered on the boundary.

1.5 Marsden-like Identity, Domain points, Stability and C

–connection

A Marsden-like identity allows us to deriveexplicitformulas for the representation of polynomials of degree up to 3r in the simplex spline basis ofS^r

3r( ). Even if we have identified a basis ofS^r

3r( ), it turns out that for the partition of unity and the Marsden-like identity alone, i.e., for the generation of polynomials and espcially the constant function, we do not need all the elements of that basis. Hence, instead of having redundancy or a null component in the sums, we remove one further element according to the following definition.

Definition 3Depending on the parity ofr, we define the following sets and spaces:

1. Ifr=2s+1 withs≥0, then

Σ¯^r_3r :=Σ_3r^r , S¯^r

3r( )=S^r

3r( ).

2. Ifr=2sthen

Σ¯^r_3r :=Σ^r_3r\

(s+1,s+1,s+1; 3s) , S¯^r

3r( ):=span ¯Σ_3r^r .

In analogy with (1.40), the set of indices(i,j,k, ℓ)of the basis ¯Σ^r_3rwill be written as ¯I^r. Moreover, for the Bernstein polynomials in ¯Σ^r_3r (cf. Figure 1.3), we define the index sets

I_removed^r :={(i,j,k):i+j+k=3r,max{i,j,k} ≤2r−1,min{i,j,k} ≥1},

I_Bernstein^r :={(i,j,k):i,j,k∈N₀,i+j+k=3r} \ I_removed^r . (1.42) It immediately follows that

#I_Bernstein^r =3

2r(r+5), #I_removed^r =3r(r−1)+1 #I_Bernstein^r +#I_removed^r = 3r+2

2

=dimP_3r. (1.43) We now consider the casesr =2,3 in more detail. We prove, using knot insertion, a Marsden-like identity for r=2, and state it in a form valid for anyr≥1. It is verified symbolically forr ≥6.Shouldn’t it ber≤6?

1.5.1 TheC²elements, ¯Σ₆²

According to Definition 3, we use only 36 elements of ¯Σ²₆ with indices from the set ¯I², which corresponds to removing 2223fromΣ₆². In Definition 4 we list these elements, normalized to ensure partition of unity. The set Σ¯₆²consists of Bernstein polynomials as depicted in Figure 1.3, and 15 other simplex splines with at least one knot at the barycenter of the triangle.

Definition 4 (The functionsΣ¯₆²)

The functions ¯Σ²₆consists of the 21 Bernstein polynomials

S_i+1,⁶ _j+1,k+1,0 := i+1,j+1,k+1;0=B_{i jk}⁶ , (i,j,k) ∈ I_Bernstein² , (1.44) and the additional simplex splines

S₂₂⁶ :=S₄₃₁₁⁶ :=1

3 ⁴³¹¹, S₂₃⁶ :=S₃₄₁₁⁶ :=1

3 ³⁴¹¹, S⁶₂₄:=S₁₄₃₁⁶ :=1 3 ¹⁴³¹, S⁶

25:=S⁶

1341:=1

3 ¹³⁴¹, S⁶

26:=S⁶

3141:=1

3 ³¹⁴¹, S⁶

27:=S⁶

4131:=1 3 ⁴¹³¹, S₂₈⁶ :=S₄₂₂₁⁶ :=2

3 ⁴²²¹, S₃₀⁶ :=S₂₄₂₁⁶ :=2

3 ²⁴²¹, S⁶₃₂:=S₂₂₄₁⁶ :=2 3 ²²⁴¹, S₂₉⁶ :=S₃₃₁₂⁶ :=1

3 ³³¹², S₃₁⁶ :=S₁₃₃₂⁶ :=1

3 ¹³³², S⁶₃₃:=S₃₁₃₂⁶ :=1 3 ³¹³², S⁶

34:=S⁶

3222:=2

3 ³²²², S⁶

35:=S⁶

2322:=2

3 ²³²², S⁶

36:=S⁶

2232:=2 3 ²²³².

(1.45)