# Construction of $C^{2}$ cubic splines on arbitrary triangulations

Tom Lyche111Tom Lyche, Dept. of Mathematics, University of Oslo, Norway,
email<EMAIL_ADDRESS>, Carla Manni222Carla Manni, Dept. of Mathematics,
University of Rome Tor Vergata, Italy, email<EMAIL_ADDRESS>, Hendrik
Speleers333Hendrik Speleers, Dept. of Mathematics, University of Rome Tor
Vergata, Italy, email<EMAIL_ADDRESS>
###### Abstract

In this paper, we address the problem of constructing $C^{2}$ cubic spline
functions on a given arbitrary triangulation ${\mathcal{T}}$. To this end, we
endow every triangle of ${\mathcal{T}}$ with a Wang–Shi macro-structure. The
$C^{2}$ cubic space on such a refined triangulation has a stable dimension and
optimal approximation power. Moreover, any spline function in such space can
be locally built on each of the macro-triangles independently via Hermite
interpolation. We provide a simplex spline basis for the space of $C^{2}$
cubics defined on a single macro-triangle which behaves like a
Bernstein/B-spline basis over the triangle. The basis functions inherit
recurrence relations and differentiation formulas from the simplex spline
construction, they form a nonnegative partition of unity, they admit simple
conditions for $C^{2}$ joins across the edges of neighboring triangles, and
they enjoy a Marsden-like identity. Also, there is a single control net to
facilitate control and early visualization of a spline function over the
macro-triangle. Thanks to these properties, the complex geometry of the
Wang–Shi macro-structure is transparent to the user. Stable global bases for
the full space of $C^{2}$ cubics on the Wang–Shi refined triangulation
${\mathcal{T}}$ are deduced from the local simplex spline basis by extending
the concept of minimal determining sets.

Keywords: B-splines, Simplex splines, Macro-elements, Triangulations

##  1 Introduction

Piecewise polynomial spaces defined over polygonal partitions, usually
triangulations, have applications in several branches of the sciences
including geometric modeling, signal processing, data analysis, visualization,
and numerical simulation; we refer the reader to [Cohen.Riensenfeld.Elber01,
Hughesbook, Lai.Schumaker07] and Section 5.2 for some examples. For many of
these applications, a smooth join between the different pieces is beneficial
or even required; $C^{2}$ smoothness is often preferred. Such spaces are
commonly referred to as (bivariate) spline spaces. According to
[Lai.Schumaker07, page 197], in general we would like to work with low degree
splines: they involve fewer coefficients, and have less tendency to oscillate.

An indispensable feature for a spline space to be useful in practice is having
a stable dimension that only depends on the degree ($d$), the order of
smoothness ($r$), and combinatorial — or other easy to check — properties of
the partition (${\mathcal{T}}$). When ${\mathcal{T}}$ is a triangulation, the
dimension can be expressed in terms of the above quantities for spline spaces
with $d\geq 3r+2$; see [IbrS91] and [Lai.Schumaker07, Chapter 9]. On the other
hand, instability in the dimension has been illustrated for $d=2r$ in [DD]. We
refer the reader to [Toshniwal19] for recent results on the dimension of
spline spaces on triangulations with nonuniform degrees. Similar results are
known for spline spaces over general rectilinear partitions; see
[Chui.Wang.83, Manni92] and references therein.

Spline spaces with too low degree compared to the smoothness are also exposed
to several other shortcomings. In particular, they might lack optimal
approximation power, a property strongly related to the possibility of
constructing stable bases with local support for the considered spaces
[Lai.Schumaker07]. In this perspective, the bound $d\geq 3r+2$ plays again an
important role in identifying the spline spaces with optimal approximation
power on a given triangulation [Lai.Schumaker07, Chapter 10]. Furthermore, the
possibility of constructing any function of the spline space locally on each
of the elements of ${\mathcal{T}}$ is often seen as a desirable, if not
imperative, property for practical purposes. On a triangulation, a degree
$d\geq 4r+1$ is necessary to admit such a local construction [Ciarlet.78,
Zenisek.74, Lai.Schumaker07].

The above lower bounds on the degree can be alleviated by considering so-
called macro-elements, where the partition ${\mathcal{T}}$ is further refined
in a specific manner (often referred to as splits). In case ${\mathcal{T}}$ is
a triangulation, the most famous examples are the Clough–Tocher (CT) split
[Clough.Tocher.65, Lai.Schumaker07, Sabl85, Ciarlet.78] and the Powell–Sabin
(PS) 6 and 12 splits [Alfeld.Schumaker02, Powell.Sabin77, Sabl85,
Lai.Schumaker07, Schumaker.Sorokina06]. They subdivide each triangle of
${\mathcal{T}}$ into $3$, $6$, and $12$ subtriangles, respectively. To achieve
global $C^{2}$ smoothness, polynomial pieces of at least degree $d=7$ are
necessary for the CT split, while at least degree $d=5$ is required for both
PS splits of ${\mathcal{T}}$. All these spline spaces have a stable dimension
and possess optimal approximation power [Lai.Schumaker01, Lai.Schumaker03,
Lai.Schumaker07]. Other common macro-elements also require at least degree
$d=5$ to realize $C^{2}$ splines with the above properties on a refined
partition only consisting of triangles [Lai.Schumaker07, Section 7.7].

The Bernstein polynomial basis is the most common tool for the construction
and analysis of splines on a given triangulation ${\mathcal{T}}$
[Lai.Schumaker07], as it helps in localizing imposition of smoothness
conditions across edges of (the refinement of) ${\mathcal{T}}$. Interesting
alternatives have been developed for CT and PS splits in
[Cohen.Lyche.Riesenfeld13, Lyche.Merrien18, Lyche.Merrien.Sauer.21,
Lyche.Muntingh17], where a simplex spline basis for the local spline space
over a triangle of ${\mathcal{T}}$ has been considered. Such a basis behaves
like a Bernstein polynomial basis for imposing smoothness across edges of
${\mathcal{T}}$ and like a B-spline basis internal to each triangle of
${\mathcal{T}}$. Neither the Bernstein polynomial basis nor the simplex spline
basis provide a global basis for the full spline space on (the refinement of)
${\mathcal{T}}$. To achieve a global basis, one may apply the general
framework of minimal determining sets via the local Bernstein basis; see
[Lai.Schumaker07]. Global B-spline bases have been constructed for $C^{1}$ PS
spline spaces on triangulations [Dierckx97, Groselj.Speleers17,
Groselj.Speleers21, Speleers15b], for PS spline spaces with higher smoothness
[Groselj16, Speleers10, Speleers13], and for CT spline spaces [Speleers10b].

While in the univariate case $C^{2}$ cubics are probably the best known and
most used splines, the above discussion shows that dealing with $C^{2}$ cubics
in the bivariate setting is an arduous task. In this paper, we address the
problem of building and handling $C^{2}$ cubic splines on a suitable
refinement of any given triangulation ${\mathcal{T}}$. Our wish list for the
spline space consists of stable dimension, optimal approximation power, and
local construction on any (refined) triangle of ${\mathcal{T}}$. Moreover, we
want a practical construction of a stable global basis for the space at our
disposal.

Despite the high smoothness and the minimum gap between degree and smoothness,
a $C^{2}$ cubic space can be obtained by splitting any triangle $\Delta$ in
${\mathcal{T}}$ according to a degree-dependent scheme introduced by Wang and
Shi [Wang.90]. Contrarily to the well-known splits mentioned above, the family
of Wang–Shi (WS) splits generates a very large number of polygonal pieces in
each $\Delta$; for cubics we get a set of 75 polygons which includes
triangles, quadrilaterals, and pentagons. In practice, this complex geometry
hampers a piecewise treatment — in terms of a local polynomial basis — of
spline functions on WS splits and discourages the use of such an interesting
space.

To overcome this issue, we propose a simplex spline basis for the local space
of $C^{2}$ cubics on the (cubic) WS split of any $\Delta$ in ${\mathcal{T}}$.
The basis functions enjoy the following properties:

  * •

they form a nonnegative partition of unity;

  * •

they inherit recurrence relations and differentiation formulas from the
simplex spline structure;

  * •

for each of them, the restriction to a boundary edge of $\Delta$ reduces to a
classical $C^{2}$ cubic univariate B-spline;

  * •

they admit simple conditions for $C^{2}$ joins to neighboring triangles in
${\mathcal{T}}$;

  * •

cubic polynomials can be represented through a Marsden-like identity;

  * •

they lead to well-conditioned collocation matrices for Lagrange and Hermite
interpolation using certain sites;

  * •

a control net can be formed that mimics the shape of the spline function and
exhibits distance $O(h^{2})$ to any one of its control points from its
surface, where $h$ is the length of the longest edge.

Thanks to the characteristics of the simplex spline basis, one can avoid to
consider separate polynomial representations on each of the polygonal
subelements of $\Delta$. Instead, there is a single control net to facilitate
control and early visualization of a spline function over each element
$\Delta$ in ${\mathcal{T}}$. This makes that the complex geometry of the WS
split is transparent to the user. However, the simplex spline basis is a local
basis and does not provide a global basis for the full space of $C^{2}$ cubics
on the (cubic) WS refinement of ${\mathcal{T}}$. To this end, we extend the
concept of minimal determining sets and use the simplex spline basis as a
stepping stone to the construction of a stable global basis for the full
space.

The remainder of this paper is divided into four sections. In Section 2, we
summarize the definition and some properties of simplex splines and describe
the family of WS splits. In Section 3, we present a local simplex spline basis
for the refinement of a single triangle and discuss some of its properties. To
simplify some computations, an alternative basis is also provided. Smoothness
conditions across the edges of the given triangulation and stable global bases
for the $C^{2}$ cubic space on the (cubic) WS refinement of any triangulation
are considered in Section 4. Section 5 collects some concluding remarks about
implementation aspects, possible application areas, and a higher-order
extension of the basis. Finally, the appendix aggregates data related to the
presented simplex spline bases that might be useful for practical
computations.

Throughout the paper, we use small boldface letters for vectors and capital
boldface letters for matrices. Calligraphic letters like ${\mathcal{B}}$
indicate sets, and we write $\\#{\mathcal{B}}$ for the cardinality of
${\mathcal{B}}$. Function spaces are denoted by symbols like ${\mathbb{S}}$.
In particular, ${\mathbb{P}}_{d}$ stands for the space of bivariate
polynomials with real coefficients of total degree $\leq d$. The partial
derivatives in $x$ and $y$ are denoted by $D_{x}$ and $D_{y}$, respectively.
Given a vector ${\boldsymbol{u}}$, the associated directional derivative is
denoted by $D_{\boldsymbol{u}}$. The directional derivative in the direction
of the vector from point ${\boldsymbol{p}}_{1}$ to ${\boldsymbol{p}}_{2}$ is
denoted by $D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}$.

##  2 Preliminaries

This section contains some preliminary material about simplex splines and the
splits of interest in the rest of the paper.

###  2.1 A short summary of simplex splines

For $e\in{\mathbb{N}}$, $d\in{\mathbb{N}}_{0}$, let $n:=d+e$ and
$\Xi:=\\{{\boldsymbol{\xi}}_{1},\ldots,{\boldsymbol{\xi}}_{n+1}\\}$ be a
sequence of possibly repeated points in ${\mathbb{R}}^{e}$ called knots. The
multiplicity of a knot is the number of times it occurs in the sequence. Let
$\langle\cdot\rangle$ denote the convex hull of a sequence of points. For the
sake of simplicity, we assume $\langle\Xi\rangle$ is nondegenerate, i.e.,
$\operatorname{vol}_{e}(\langle\Xi\rangle)>0$. Let
$\sigma=\langle\overline{\boldsymbol{\xi}}_{1},\ldots,\overline{\boldsymbol{\xi}}_{n+1}\rangle$
be any simplex in ${\mathbb{R}}^{n}$ with $\operatorname{vol}_{n}(\sigma)>0$,
whose projection $\pi:{\mathbb{R}}^{n}\to{\mathbb{R}}^{e}$ onto the first $e$
coordinates satisfies
$\pi(\overline{\boldsymbol{\xi}}_{i})={\boldsymbol{\xi}}_{i}$ for
$i=1,\ldots,n+1$.

The simplex spline $M_{\Xi}$ can be defined geometrically by

$M_{\Xi}:{\mathbb{R}}^{e}\to{\mathbb{R}},\quad
M_{\Xi}({\boldsymbol{x}}):=\dfrac{\operatorname{vol}_{n-e}\big{(}\sigma\cap\pi^{-1}({\boldsymbol{x}})\big{)}}{\operatorname{vol}_{n}(\sigma)}.$

For $d=0$ we have

$M_{\Xi}({\boldsymbol{x}})=\begin{cases}1/\operatorname{vol}_{n}(\langle\Xi\rangle),&{\boldsymbol{x}}\in\text{interior
of }\langle\Xi\rangle,\\\ 0,&\text{if
}{\boldsymbol{x}}\notin\langle\Xi\rangle,\end{cases}$

and the value of $M_{\Xi}$ on the boundary of $\langle\Xi\rangle$ has to be
dealt with separately. For properties of $M_{\Xi}$ and proofs, we refer the
reader to, e.g., [Micchelli79, Prautsch.Boehm.Paluszny02]. Here, we mention:

  * •

Knot dependence: $M_{\Xi}$ only depends on $\Xi$; in particular, it is
independent of the choice of $\sigma$ and the ordering of the knots.

  * •

Support: $M_{\Xi}$ has support $\langle\Xi\rangle$.

  * •

Normalization: $M_{\Xi}$ has unit integral.

  * •

Nonnegativity: $M_{\Xi}$ is a nonnegative piecewise polynomial of total degree
$d$.

  * •

Differentiation formula ($A$-recurrence): For any
${\boldsymbol{u}}\in{\mathbb{R}}^{e}$ and any $a_{1},\ldots,a_{d+e+1}$ such
that $\sum_{i}a_{i}{\boldsymbol{\xi}}_{i}={\boldsymbol{u}}$,
$\sum_{i}a_{i}=0$, we have

$D_{\boldsymbol{u}}M_{\Xi}=(d+e)\sum_{i=1}^{d+e+1}a_{i}M_{[\Xi\setminus{\boldsymbol{\xi}}_{i}]}.$

  * •

Recurrence relation ($B$-recurrence): For any
${\boldsymbol{x}}\in{\mathbb{R}}^{e}$ and any $b_{1},\ldots,b_{d+e+1}$ such
that $\sum_{i}b_{i}{\boldsymbol{\xi}}_{i}={\boldsymbol{x}}$,
$\sum_{i}b_{i}=1$, we have

$M_{\Xi}({\boldsymbol{x}})=\frac{d+e}{d}\sum_{i=1}^{d+e+1}b_{i}M_{[\Xi\setminus{\boldsymbol{\xi}}_{i}]}({\boldsymbol{x}}).$

  * •

Knot insertion formula ($C$-recurrence): For any
${\boldsymbol{y}}\in{\mathbb{R}}^{e}$ and any $c_{1},\ldots,c_{d+e+1}$ such
that $\sum_{i}c_{i}{\boldsymbol{\xi}}_{i}={\boldsymbol{y}}$,
$\sum_{i}c_{i}=1$, we have

$M_{\Xi}=\sum_{i=1}^{d+e+1}c_{i}M_{[\Xi\cup{\boldsymbol{y}}\setminus{\boldsymbol{\xi}}_{i}]}.$

If $e=1$ then $M_{\Xi}$ is the univariate B-spline of degree $d$ with knots
$\Xi$, normalized to have its integral equal to one.

In the bivariate case, $e=2$, the lines in the complete graph of $\Xi$ are
called knot lines. They provide a partition of $\langle\Xi\rangle$ into
polygonal elements. The simplex spline $M_{\Xi}$ is a polynomial of degree
$d=\\#\Xi-3$ in each region of this partition, and across a knot line

$M_{\Xi}\in C^{d+1-\mu},$

where $\mu$ is the number of knots on that knot line, including
multiplicities.

###  2.2 The Wang–Shi splits

Given three noncollinear points
${\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}$ in
${\mathbb{R}}^{2}$, the triangle
$\Delta:=\langle{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}\rangle$
with vertices ${\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}$
will serve as our macro-triangle. Given a degree $d\in{\mathbb{N}}$, we divide
each edge of $\Delta$ into $d$ equal segments, respectively, resulting into
$3d$ boundary points. Then, we refine $\Delta$ into a number of subelements
delineated by the complete graph connecting those boundary points. This is
called the ${{\rm WS}_{d}}$ split of $\Delta$ as it was originally proposed by
Wang and Shi [Wang.90]. We denote by $\Delta_{{{\rm WS}_{d}}}$ the obtained
mesh structure, and by ${\mathcal{P}}_{d}$ the set of polygons in
$\Delta_{{{\rm WS}_{d}}}$. All the possible intersections of the various lines
connecting the boundary points are called vertices of $\Delta_{{{\rm
WS}_{d}}}$. In particular, the boundary points are vertices of $\Delta_{{{\rm
WS}_{d}}}$. The cases $d=2,3,4$ are shown in Figure 1. For $d=2$ we obtain the
well-known PS-12 split [Powell.Sabin77], while for $d=1$ we have
${\mathcal{P}}_{1}=\\{\Delta\\}$. Note that for $d>2$ not all elements of
${\mathcal{P}}_{d}$ are triangles. We consider the space

${\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}}):=\\{s\in
C^{d-1}(\Delta):s_{|\tau}\in{\mathbb{P}}_{d},\ \forall\
\tau\in{\mathcal{P}}_{d}\\}.$ (1)

When the degree increases, the complexity of the mesh grows quickly. There are

  * •

$3d$ boundary points and $3d(d-1)$ interior lines in the complete graph;

  * •

the maximum number of lines intersecting at an interior vertex is
$3,3,4,5,6,7,6$ for $d=2,3,\ldots,8$;

  * •

the number of vertices of $\Delta_{{{\rm WS}_{d}}}$ is
$10,58,178,558,1255,2532,4786$ for $d=2,3,\ldots,8$.

Figure 1: WSd splits for $d=2,3,4$.

The dimension of ${\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}})$ can be
computed using the general dimension formula for spline spaces over cross-cut
partitions from [Chui.Wang.83, Theorem 3.1]. A partition ${\mathcal{T}}_{c}$
of a domain $\Omega$ is called a cross-cut partition if it is obtained by
drawing lines across $\Omega$. Let ${\mathbb{S}}^{r}_{d}({\mathcal{T}}_{c})$
be the space of functions in $C^{r}(\Omega)$ which belong to
${\mathbb{P}}_{d}$ when restricted to any polygon of ${\mathcal{T}}_{c}$.

######  Theorem 1.

Let $\Omega$ be a simply connected domain in ${\mathbb{R}}^{2}$. Let
${\mathcal{T}}_{c}$ be a cross-cut partition of $\Omega$, with $m$ cross-cuts,
$n$ interior vertices ${\boldsymbol{v}}_{1},\ldots,{\boldsymbol{v}}_{n}$, and
$m_{k}$ cross-cuts intersecting at ${\boldsymbol{v}}_{k}$, $k=1,\ldots,n$.
Then, the dimension of the spline space
${\mathbb{S}}^{r}_{d}({\mathcal{T}}_{c})$, $0\leq r\leq d-1$ is

$\dim({\mathbb{S}}^{r}_{d}({\mathcal{T}}_{c}))=\binom{d+2}{2}+m\binom{d-r+1}{2}+\sum_{k=1}^{n}\varsigma(m_{k}),$
(2)

where

$\varsigma(l):=\frac{1}{2}\left(d-r-\left\lfloor\frac{r+1}{l-1}\right\rfloor\right)_{+}\left((l-1)d-(l+1)r+(l-3)+(l-1)\left\lfloor\frac{r+1}{l-1}\right\rfloor\right).$

As usual, $\lfloor x\rfloor$ denotes the largest integer smaller than or equal
to $x$, and $(x)_{+}:=\max\\{x,0\\}$.

######  Theorem 2.

Assuming at most $d+1$ lines intersect at an interior vertex of $\Delta_{{{\rm
WS}_{d}}}$, we have

$\dim({\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm
WS}_{d}}}))=\dim{\mathbb{P}}_{d}+m,$

where $m=3d(d-1)$ is the number of interior lines in the complete graph.

###### Proof.

We make use of the dimension formula (2) in Theorem 1. In our case we have
$r=d-1$, and it is easy to check that

$\varsigma(l)=0,\quad l=1,\ldots,d+1.$

Since at most $d+1$ lines cross at each interior vertex, it immediately
follows from (2) that

$\dim({\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}}))=\binom{d+2}{2}+m,$

which completes the proof. ∎

${\boldsymbol{p}}_{1}$${\boldsymbol{p}}_{2}$${\boldsymbol{p}}_{3}$${\boldsymbol{p}}_{3,1}$${\boldsymbol{p}}_{3,2}$${\boldsymbol{p}}_{2,3}$${\boldsymbol{p}}_{2,1}$${\boldsymbol{p}}_{1,2}$${\boldsymbol{p}}_{1,3}$
Figure 2: Labeling of the knots on the boundary of the triangle $\Delta$.

##  3 Simplex spline bases for ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$

In this section, we focus on the case $d=3$, provide two (scaled) simplex
spline bases for the space ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ in
(1), and prove some properties of these bases. With a slight abuse of notation
we also refer to the corresponding basis functions as simplex splines.

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7

(h) 8 (i) 9 (j) 10 (k) 11 (l) 12 (m) 13 (n) 14

(o) 15 (p) 16 (q) 17 (r) 18 (s) 19 (t) 20 (u) 21

(v) 22 (w) 23 (x) 24 (y) 25 (z) 26 (aa) 27 (ab) 28

Figure 3: Sequences of knots for a set of simplex spline basis functions for
${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$. Each black disc shows the
position of a knot and the number inside indicates its multiplicity.

###  3.1 A simplex spline basis

For a given triangle
$\Delta=\langle{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}\rangle$,
the ${{\rm WS}_{3}}$ split is shown in the middle plot of Figure 1. From
Theorem 2 we know that the dimension of ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm
WS}_{3}}})$ is 28. In order to construct a basis for this space, we first
specify nine points along the boundary of the triangle (see Figure 2): the
three vertices
${\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}$ and

$\displaystyle{\boldsymbol{p}}_{1,2}$
$\displaystyle:=\frac{2}{3}{\boldsymbol{p}}_{2}+\frac{1}{3}{\boldsymbol{p}}_{3},$
$\displaystyle\quad{\boldsymbol{p}}_{1,3}$
$\displaystyle:=\frac{1}{3}{\boldsymbol{p}}_{2}+\frac{2}{3}{\boldsymbol{p}}_{3},$
(3) $\displaystyle{\boldsymbol{p}}_{2,1}$
$\displaystyle:=\frac{2}{3}{\boldsymbol{p}}_{1}+\frac{1}{3}{\boldsymbol{p}}_{3},$
$\displaystyle{\boldsymbol{p}}_{2,3}$
$\displaystyle:=\frac{1}{3}{\boldsymbol{p}}_{1}+\frac{2}{3}{\boldsymbol{p}}_{3},$
$\displaystyle{\boldsymbol{p}}_{3,1}$
$\displaystyle:=\frac{2}{3}{\boldsymbol{p}}_{1}+\frac{1}{3}{\boldsymbol{p}}_{2},$
$\displaystyle{\boldsymbol{p}}_{3,2}$
$\displaystyle:=\frac{1}{3}{\boldsymbol{p}}_{1}+\frac{2}{3}{\boldsymbol{p}}_{2}.$

Note that these points are part of the ${{\rm WS}_{3}}$ split. We then
consider the cubic simplex splines $M_{1},\ldots,M_{28}$ as schematically
illustrated in Figure 3, where each simplex spline has six (including
multiplicity) knots chosen among the nine points above. For instance, $M_{4}$
is defined by the sequence
$\\{{\boldsymbol{\xi}}_{1},\ldots,{\boldsymbol{\xi}}_{6}\\}=\\{{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{3,1},{\boldsymbol{p}}_{3,2},{\boldsymbol{p}}_{2,1}\\}$.
Each of them can be computed using the B-recurrence relation. We define the
following set of 28 (scaled) simplex splines:

${\mathcal{B}}:=\\{B_{i}:=w_{i}M_{i},\ i=1,\ldots,28\\},$ (4)

where, denoting by $|\Delta|$ the area of $\Delta$, the scaling factors are
given by

${\boldsymbol{w}}:=\frac{\lvert\Delta\rvert}{15}\left\\{\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{5}{6},\frac{5}{6},\frac{5}{6},\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},1\right\\}.$

Note that the scaling factors sum up to $|\Delta|$. There are seven different
types of simplex splines in ${\mathcal{B}}$. For each type, a representative
$B_{i}$ is depicted in Figures 10–16 in the appendix. Explicit expressions of
their polynomial pieces are given in Table 1 in the appendix; the remaining
ones can be obtained by symmetry.

On any edge of $\Delta$, there are six basis functions nonzero. Their
restrictions to that edge are nothing but the set of univariate $C^{2}$ cubic
B-splines defined on a uniform open-knot sequence with two interior knots. For
instance, for the edge ${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$, they
correspond to the univariate cubic B-splines on the knot sequence specified by
$\\{{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{3,1},{\boldsymbol{p}}_{3,2},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{2}\\}$.

######  Theorem 3.

The simplex splines $\\{B_{1},\ldots,B_{28}\\}$ in (4) form a nonnegative
partition of unity basis for the space ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm
WS}_{3}}})$.

###### Proof.

Let $B$ be one of the functions $B_{i}$. We first prove that
$B\in{\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$. Since $B$ has six knots,
it is a piecewise cubic polynomial. Moreover, the knots of $B$ are a subset of
the knots shown in Figure 2. Thus, the knot lines of $B$ are a subset of the
knot lines in the complete graph; see Figure 1. Since each interior knot line
contains exactly two knots, $B$ has $C^{2}$ smoothness according to the
smoothness property of simplex splines. It follows that
$B\in{\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$.

We now consider linear independence. Using the recurrence relation and
differentiation formula for simplex splines and the scaling factors
${\boldsymbol{w}}$, we compute values and derivatives of $B$ corresponding to
the following $28$ operators: $\rho_{1},\ldots,\rho_{18}$ are related to the
vertices,

$\displaystyle\rho_{1}(f)$ $\displaystyle:=f({\boldsymbol{p}}_{1}),$
$\displaystyle\quad\rho_{2}(f)$ $\displaystyle:=f({\boldsymbol{p}}_{2}),$
$\displaystyle\quad\rho_{3}(f)$ $\displaystyle:=f({\boldsymbol{p}}_{3}),$ (5)
$\displaystyle\rho_{4}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{1}),$
$\displaystyle\quad\rho_{5}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}f({\boldsymbol{p}}_{1}),$
$\displaystyle\quad\rho_{6}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2}{\boldsymbol{p}}_{3}}f({\boldsymbol{p}}_{2}),$
$\displaystyle\rho_{7}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2}{\boldsymbol{p}}_{1}}f({\boldsymbol{p}}_{2}),$
$\displaystyle\rho_{8}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3}{\boldsymbol{p}}_{1}}f({\boldsymbol{p}}_{3}),$
$\displaystyle\rho_{9}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{3}),$
$\displaystyle\rho_{10}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}^{2}f({\boldsymbol{p}}_{1}),$
$\displaystyle\quad\rho_{11}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}^{2}f({\boldsymbol{p}}_{1}),$
$\displaystyle\quad\rho_{12}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2}{\boldsymbol{p}}_{3}}^{2}f({\boldsymbol{p}}_{2}),$
$\displaystyle\rho_{13}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2}{\boldsymbol{p}}_{1}}^{2}f({\boldsymbol{p}}_{2}),$
$\displaystyle\rho_{14}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3}{\boldsymbol{p}}_{1}}^{2}f({\boldsymbol{p}}_{3}),$
$\displaystyle\rho_{15}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3}{\boldsymbol{p}}_{2}}^{2}f({\boldsymbol{p}}_{3}),$
$\displaystyle\rho_{16}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}f({\boldsymbol{p}}_{1}),$
$\displaystyle\rho_{17}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2}{\boldsymbol{p}}_{3}}D_{{\boldsymbol{p}}_{2}{\boldsymbol{p}}_{1}}f({\boldsymbol{p}}_{2}),$
$\displaystyle\rho_{18}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3}{\boldsymbol{p}}_{1}}D_{{\boldsymbol{p}}_{3}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{3});$

$\rho_{19},\ldots,\rho_{27}$ are related to the edges,

$\displaystyle\rho_{19}(f)$
$\displaystyle:=D_{{\boldsymbol{q}}_{3}{\boldsymbol{p}}_{3}}f({\boldsymbol{q}}_{3}),$
$\displaystyle\rho_{20}(f)$
$\displaystyle:=D_{{\boldsymbol{q}}_{1}{\boldsymbol{p}}_{1}}f({\boldsymbol{q}}_{1}),$
$\displaystyle\rho_{21}(f)$
$\displaystyle:=D_{{\boldsymbol{q}}_{2}{\boldsymbol{p}}_{2}}f({\boldsymbol{q}}_{2}),$
(6) $\displaystyle\rho_{22}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3,1}{\boldsymbol{p}}_{3}}^{2}f({\boldsymbol{p}}_{3,1}),$
$\displaystyle\rho_{23}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2,1}{\boldsymbol{p}}_{2}}^{2}f({\boldsymbol{p}}_{2,1}),$
$\displaystyle\rho_{24}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1,2}{\boldsymbol{p}}_{1}}^{2}f({\boldsymbol{p}}_{1,2}),$
$\displaystyle\rho_{25}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{3,2}{\boldsymbol{p}}_{3}}^{2}f({\boldsymbol{p}}_{3,2}),$
$\displaystyle\quad\rho_{26}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{2,3}{\boldsymbol{p}}_{2}}^{2}f({\boldsymbol{p}}_{2,3}),$
$\displaystyle\quad\rho_{27}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1,3}{\boldsymbol{p}}_{1}}^{2}f({\boldsymbol{p}}_{1,3}),$

where

${\boldsymbol{q}}_{1}:=\frac{1}{2}{\boldsymbol{p}}_{2}+\frac{1}{2}{\boldsymbol{p}}_{3},\quad{\boldsymbol{q}}_{2}:=\frac{1}{2}{\boldsymbol{p}}_{1}+\frac{1}{2}{\boldsymbol{p}}_{3},\quad{\boldsymbol{q}}_{3}:=\frac{1}{2}{\boldsymbol{p}}_{1}+\frac{1}{2}{\boldsymbol{p}}_{2};$
(7)

and the final $\rho_{28}$ is related to the triangle,

$\rho_{28}(f):=f({\boldsymbol{q}}),\quad{\boldsymbol{q}}:=\frac{1}{3}{\boldsymbol{p}}_{1}+\frac{1}{3}{\boldsymbol{p}}_{2}+\frac{1}{3}{\boldsymbol{p}}_{3}.$
(8)

The computed values are shown in Table 2 in the appendix. Since the matrix
$[\rho_{j}(B_{i})]\in{\mathbb{R}}^{28\times 28}$ is block upper triangular
with nonsingular $2\times 2$ blocks, linear independence of the set of
functions $\\{B_{1},\ldots,B_{28}\\}$ follows. From Theorem 2 we know that the
dimension of ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ is 28, and thus
the 28 linearly independent functions in (4) form a basis of this space. At
the same time, we may conclude linear independence of the set of operators
$\\{\rho_{1},\ldots,\rho_{28}\\}$ defined on
${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$.

Simplex splines are nonnegative, so it only remains to prove that the
functions in (4) sum up to one on $\Delta$. An inspection of Table 2 shows
that

$\rho_{j}\biggl{(}\sum_{i=1}^{28}B_{i}\biggr{)}=\sum_{i=1}^{28}\rho_{j}(B_{i})=\rho_{j}(1),\quad
j=1,\ldots,28.$

By linear independence of the operators $\rho_{j}$, $\sum_{i=1}^{28}B_{i}$
must be equal to the unity function which belongs to
${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$. ∎

From the proof of Theorem 3 it follows that we can formulate a Hermite
interpolation problem to characterize any spline in
${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$.

######  Corollary 4.

For given data $f_{k,\alpha,\beta}$, $g_{k}$, $g_{k,l}$, and $h_{0}$, there
exists a unique spline $s\in{\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$
such that

$\displaystyle D_{x}^{\alpha}D_{y}^{\beta}s({\boldsymbol{p}}_{k})$
$\displaystyle=f_{k,\alpha,\beta},\quad 0\leq\alpha+\beta\leq 2,\quad
k=1,2,3,$ $\displaystyle D_{{\boldsymbol{n}}_{k}}s({\boldsymbol{q}}_{k})$
$\displaystyle=g_{k},\quad
D_{{\boldsymbol{n}}_{k}}^{2}s({\boldsymbol{p}}_{k,l})=g_{k,l},\quad
k,l=1,2,3,\quad k\neq l,$ $\displaystyle s({\boldsymbol{q}})$
$\displaystyle=h_{0},$

where ${\boldsymbol{n}}_{k}$ is the normal direction of the edge opposite to
vertex ${\boldsymbol{p}}_{k}$, and the points ${\boldsymbol{p}}_{k,l}$,
${\boldsymbol{q}}_{k}$, and ${\boldsymbol{q}}$ are defined in (3), (7), and
(8), respectively.

The Hermite degrees of freedom specified in Corollary 4 are schematically
visualized in Figure 4 using graphical symbols that are common in finite
element literature; see, e.g., [Ciarlet.78].

Figure 4: Hermite degrees of freedom on the ${{\rm WS}_{3}}$ split.

###  3.2 Domain points and condition number

We now associate a special point in $\Delta$ with each basis function $B_{i}$
in (4), that plays an important role in geometric modeling. We solve the
system $\rho_{j}(\sum_{i=1}^{28}b^{*}_{i}(f)B_{i})=\rho_{j}(f)$ for the two
functions $f_{1}(x,y):=x$ and $f_{2}(x,y):=y$. The points

${\boldsymbol{b}}^{*}_{i}:=(b^{*}_{i}(f_{1}),b^{*}_{i}(f_{2})),\quad
i=1,\ldots,28,$ (9)

are called the domain points of the basis (4). Together with the partition of
unity the domain points provide an explicit representation of any affine
function with respect to the basis (4). The barycentric coordinates with
respect to the triangle $\Delta$ of the domain points (9) are given by

$\displaystyle{\boldsymbol{b}}^{*}_{1}$ $\displaystyle:\left(1,0,0\right),$
$\displaystyle\hskip 14.22636pt{\boldsymbol{b}}^{*}_{2}$
$\displaystyle:\left(0,1,0\right),$ $\displaystyle\hskip
14.22636pt{\boldsymbol{b}}^{*}_{3}$ $\displaystyle:\left(0,0,1\right),$
$\displaystyle\hskip 14.22636pt{\boldsymbol{b}}^{*}_{4}$
$\displaystyle:\left(\frac{8}{9},\frac{1}{9},0\right),$ (10)
$\displaystyle{\boldsymbol{b}}^{*}_{5}$
$\displaystyle:\left(\frac{8}{9},0,\frac{1}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{6}$
$\displaystyle:\left(0,\frac{8}{9},\frac{1}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{7}$
$\displaystyle:\left(\frac{1}{9},\frac{8}{9},0\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{8}$
$\displaystyle:\left(\frac{1}{9},0,\frac{8}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{9}$
$\displaystyle:\left(0,\frac{1}{9},\frac{8}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{10}$
$\displaystyle:\left(\frac{2}{3},\frac{1}{3},0\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{11}$
$\displaystyle:\left(\frac{2}{3},0,\frac{1}{3}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{12}$
$\displaystyle:\left(0,\frac{2}{3},\frac{1}{3}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{13}$
$\displaystyle:\left(\frac{1}{3},\frac{2}{3},0\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{14}$
$\displaystyle:\left(\frac{1}{3},0,\frac{2}{3}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{15}$
$\displaystyle:\left(0,\frac{1}{3},\frac{2}{3}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{16}$
$\displaystyle:\left(\frac{7}{9},\frac{1}{9},\frac{1}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{17}$
$\displaystyle:\left(\frac{1}{9},\frac{7}{9},\frac{1}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{18}$
$\displaystyle:\left(\frac{1}{9},\frac{1}{9},\frac{7}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{19}$
$\displaystyle:\left(\frac{7}{15},\frac{7}{15},\frac{1}{15}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{20}$
$\displaystyle:\left(\frac{1}{15},\frac{7}{15},\frac{7}{15}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{21}$
$\displaystyle:\left(\frac{7}{15},\frac{1}{15},\frac{7}{15}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{22}$
$\displaystyle:\left(\frac{5}{9},\frac{1}{3},\frac{1}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{23}$
$\displaystyle:\left(\frac{5}{9},\frac{1}{9},\frac{1}{3}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{24}$
$\displaystyle:\left(\frac{1}{9},\frac{5}{9},\frac{1}{3}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{25}$
$\displaystyle:\left(\frac{1}{3},\frac{5}{9},\frac{1}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{26}$
$\displaystyle:\left(\frac{1}{3},\frac{1}{9},\frac{5}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{27}$
$\displaystyle:\left(\frac{1}{9},\frac{1}{3},\frac{5}{9}\right),$
$\displaystyle{\boldsymbol{b}}^{*}_{28}$
$\displaystyle:\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right).$

These points are visualized in Figure 5 (left). When representing a spline
$s\in{\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ in the basis (4),

$s=\sum_{i=1}^{28}b_{i}B_{i},$ (11)

it is common to organize the coefficients in terms of control points
$({\boldsymbol{b}}^{*}_{i},b_{i})$, $i=1,\ldots,28$. There are several
possibilities to connect these points into a control net. Such a control net
forms a caricatural approximation for the graph of the function that is useful
for geometric modeling. A viable option for connecting these points is shown
in Figure 5 (left); the configuration consists of a small number of regions
but both triangles and quadrilaterals are involved. As shown in Section 4.1,
this choice allows for a geometric interpretation of $C^{1}$ smoothness
conditions analogous to the classical Bernstein representation for polynomial
triangular patches. An example spline and its corresponding control net is
illustrated in Figure 6.

Figure 5: A possible net configuration for ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm
WS}_{3}}})$. Left: the domain points (10). Right: the domain points (16).
Figure 6: A simplex spline surface and its control net for the domain points
(10).

In many applications it is of interest to have a bound on the condition number
of the basis we are dealing with. We consider the infinity norm, and we look
for constants $K^{-}_{\infty},K^{+}_{\infty}>0$ such that for all
${\boldsymbol{b}}:=(b_{1},\ldots,b_{28})^{T}\in{\mathbb{R}}^{28}$,

$K^{-}_{\infty}\|{\boldsymbol{b}}\|_{\infty}\leq\biggl{\|}\sum_{i=1}^{28}b_{i}B_{i}\biggr{\|}_{\infty}\leq
K^{+}_{\infty}\|{\boldsymbol{b}}\|_{\infty}.$ (12)

The condition number of the basis is then defined by

$\kappa_{\infty}({\mathcal{B}}):=\inf\\{K^{+}_{\infty}/K^{-}_{\infty}:K^{-}_{\infty}\text{
and }K^{+}_{\infty}\text{ satisfy \eqref{eq:local-stability}}\\}.$

######  Proposition 5.

The condition number of the simplex spline basis (4) is bounded by

$\kappa_{\infty}({\mathcal{B}})<37.$

###### Proof.

Since the simplex spline basis (4) forms a nonnegative partition of unity, it
is clear that $K^{+}_{\infty}=1$ satisfies (12). Let ${\boldsymbol{A}}$ be the
matrix for any unisolvent Lagrange interpolation problem with respect to the
basis (4). Then,

$\|{\boldsymbol{b}}\|_{\infty}\leq\|{\boldsymbol{A}}^{-1}\|_{\infty}\biggl{\|}\sum_{i=1}^{28}b_{i}B_{i}\biggr{\|}_{\infty}.$

Considering interpolation at the domain points (10), a direct computation
gives $\|{\boldsymbol{A}}^{-1}\|_{\infty}<37$. This implies that
$K^{-}_{\infty}=1/37$ satisfies (12). ∎

Note that the bound on the condition number in Proposition 5 is independent of
the shape of the triangle $\Delta$. From the proof we also deduce that the
condition number of ${\mathcal{B}}$ in (4) can be computed as

$\kappa_{\infty}({\mathcal{B}})=\inf\\{1/K^{-}_{\infty}:K^{-}_{\infty}\text{
satisfies \eqref{eq:local-stability}}\\}.$

By means of this number we can easily obtain the following distance result.

######  Proposition 6.

Let $s\in{\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ be given as in (11).
Then,

$|b_{i}-s({\boldsymbol{b}}^{*}_{i})|\leq
2\kappa_{\infty}({\mathcal{B}})\,h^{2}\max_{\alpha+\beta=2}\|D_{x}^{\alpha}D_{y}^{\beta}s\|_{\infty},\quad
i=1,\ldots,28,$

where $h$ is the length of the largest edge of $\Delta$.

###### Proof.

Let $\Pi_{i}\in{\mathbb{P}}_{1}$ be the linear Taylor approximation to $s$ at
the domain point ${\boldsymbol{b}}^{*}_{i}$. Note that
$\Pi_{i}({\boldsymbol{b}}^{*}_{i})=s({\boldsymbol{b}}^{*}_{i})$. Then, using
the definition of domain points, we have

$s-\Pi_{i}=\sum_{j=1}^{28}(b_{j}-\Pi_{i}({\boldsymbol{b}}^{*}_{j}))B_{j},$

so that from (12) we obtain

$|b_{i}-s({\boldsymbol{b}}^{*}_{i})|=|b_{i}-\Pi_{i}({\boldsymbol{b}}^{*}_{i})|\leq\max_{j=1,\ldots,28}|b_{j}-\Pi_{i}({\boldsymbol{b}}^{*}_{j})|\leq\kappa_{\infty}({\mathcal{B}})\|s-\Pi_{i}\|_{\infty}.$

Furthermore, since $s\in C^{2}(\Delta)$, Taylor approximation error analysis
tells us that

$\|s-\Pi_{i}\|_{\infty}\leq
2h^{2}\max_{\alpha+\beta=2}\|D_{x}^{\alpha}D_{y}^{\beta}s\|_{\infty},$

which completes the proof. ∎

Proposition 6 implies that the control points of the spline $s$ in (11)
converge like $O(h^{2})$ to the graph of $s$.

###  3.3 A Marsden-like identity

In Section 3.2, we have provided the representation of any affine function
with respect to the basis (4). We now extend this result by providing a
Marsden-like identity which allows us to represent any cubic polynomial. In
the univariate B-spline case, the Marsden identity is given by

$(z-x)^{d}=\sum_{j}\prod_{k=j+1}^{j+d}(z-\xi_{k})B_{j,d}(x),$

where $B_{j,d}$ is a normalized B-spline of degree $d$ defined by the knots
$\xi_{j},\xi_{j+1},\ldots,\xi_{j+d+1}$; see, e.g., [LycheMS08, Theorem 2].
Dividing both sides by $z^{d}$ and setting $y:=z^{-1}$ we obtain a form more
amenable to multivariate generalization

$(1-xy)^{d}=\sum_{j}\psi_{j,d}(y)B_{j,d}(x),\quad\psi_{j,d}(y):=\prod_{k=j+1}^{j+d}(1-y\xi_{k}).$
(13)

The functions $\psi_{j,d}$ are polynomials of degree $d$ and are called dual
polynomials. The following result is obtained by a direct computation.

######  Theorem 7.

We have

$(1+{\boldsymbol{y}}^{T}{\boldsymbol{x}})^{3}=\sum_{i=1}^{28}\psi_{i}({\boldsymbol{y}})B_{i}({\boldsymbol{x}}),\quad{\boldsymbol{y}}\in{\mathbb{R}}^{2},\quad{\boldsymbol{x}}\in\Delta,$

where the dual polynomials $\psi_{i}$, $i=1,\ldots,28$ are defined as follows.
Recalling the points in (3) and (8),

$\displaystyle\psi_{1}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1})^{3},\quad\psi_{2}({\boldsymbol{y}}):=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2})^{3},\quad\psi_{3}({\boldsymbol{y}}):=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3})^{3},$
$\displaystyle\psi_{4}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1})^{2}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1}),\quad\psi_{5}({\boldsymbol{y}}):=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1})^{2}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{6}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2})^{2}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2}),\quad\psi_{7}({\boldsymbol{y}}):=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2})^{2}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2}),$
$\displaystyle\psi_{8}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3})^{2}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3}),\quad\psi_{9}({\boldsymbol{y}}):=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3})^{2}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3}),$

and

$\displaystyle\psi_{10}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2}),$
$\displaystyle\ \psi_{11}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{12}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3}),$
$\displaystyle\psi_{13}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2}),$
$\displaystyle\psi_{14}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{15}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3}),$
$\displaystyle\psi_{16}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{17}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2}),$
$\displaystyle\psi_{18}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3}),$
$\displaystyle\psi_{19}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{m}}_{1}),$
$\displaystyle\psi_{20}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{m}}_{2}),$
$\displaystyle\ \psi_{21}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{m}}_{3}),$
$\displaystyle\psi_{22}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{23}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{24}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3}),$
$\displaystyle\psi_{25}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2}),$
$\displaystyle\psi_{26}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1}),$
$\displaystyle\psi_{27}({\boldsymbol{y}})$
$\displaystyle:=(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3}),$

and

$\displaystyle\psi_{28}({\boldsymbol{y}}):=(1+{\boldsymbol{y}}^{T}{\boldsymbol{q}})\bigl{[}2(1+{\boldsymbol{y}}^{T}{\boldsymbol{q}})^{2}$
$\displaystyle\quad-\frac{1}{3}\bigl{(}(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,3})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,3})+(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,2})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{1,2})+(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{3,1})(1+{\boldsymbol{y}}^{T}{\boldsymbol{p}}_{2,1})\bigr{)}\bigr{]},$

where

${\boldsymbol{m}}_{1}:=\frac{2}{5}{\boldsymbol{p}}_{1}+\frac{2}{5}{\boldsymbol{p}}_{2}+\frac{1}{5}{\boldsymbol{p}}_{3},\quad{\boldsymbol{m}}_{2}:=\frac{1}{5}{\boldsymbol{p}}_{1}+\frac{2}{5}{\boldsymbol{p}}_{2}+\frac{2}{5}{\boldsymbol{p}}_{3},\quad{\boldsymbol{m}}_{3}:=\frac{2}{5}{\boldsymbol{p}}_{1}+\frac{1}{5}{\boldsymbol{p}}_{2}+\frac{2}{5}{\boldsymbol{p}}_{3}.$

It is remarkable that the dual polynomials $\psi_{i}$, $i=1,\ldots,27$, can be
written as products of three linear polynomials and mimic the classical
univariate Marsden identity (13). It is worth noting that these functions are
the polar forms (or blossoms) of
$(1+{\boldsymbol{y}}^{T}{\boldsymbol{x}})^{3}$ evaluated at appropriate points
[Ramshaw89]. Similarly, polar forms were used in [Neamtu07] for the
representation of polynomials in terms of simplex splines whose knots are in
generic position. This is not the case for the knots in Figures 3 and 7.

(a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7

(h) 8 (i) 9 (j) 10 (k) 11 (l) 12 (m) 13 (n) 14

(o) 15 (p) 16 (q) 17 (r) 18 (s) 19 (t) 20 (u) 21

(v) 22 (w) 23 (x) 24 (y) 25 (z) 26 (aa) 27 (ab) 28

Figure 7: Sequences of knots for an alternative set of simplex spline basis
functions for ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$. Each black disc
shows the position of a knot and the number inside indicates its multiplicity.

###  3.4 An alternative simplex spline basis

An alternative basis for the space ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm
WS}_{3}}})$ is provided by the simplex splines identified by the knot
sequences in Figure 7 and scaled to form a partition of unity. We denote this
basis by

${\widetilde{\mathcal{B}}}=\\{{\widetilde{B}}_{1},\ldots,{\widetilde{B}}_{28}\\}.$
(14)

It can be checked that

$\displaystyle{\widetilde{B}}_{i}$ $\displaystyle=B_{i},\quad i=1,\ldots,21,$
(15) $\displaystyle{\widetilde{B}}_{22}$
$\displaystyle=2B_{22}-B_{23}+\frac{1}{3}B_{28},$
$\displaystyle\quad{\widetilde{B}}_{23}$
$\displaystyle=2B_{23}-B_{22}+\frac{1}{3}B_{28},$
$\displaystyle{\widetilde{B}}_{24}$
$\displaystyle=2B_{24}-B_{25}+\frac{1}{3}B_{28},$
$\displaystyle{\widetilde{B}}_{25}$
$\displaystyle=2B_{25}-B_{24}+\frac{1}{3}B_{28},$
$\displaystyle{\widetilde{B}}_{26}$
$\displaystyle=2B_{26}-B_{27}+\frac{1}{3}B_{28},$
$\displaystyle{\widetilde{B}}_{27}$
$\displaystyle=2B_{27}-B_{26}+\frac{1}{3}B_{28},$
$\displaystyle{\widetilde{B}}_{28}$ $\displaystyle=-B_{28}.$

From (15) we easily deduce that the domain points
$\tilde{\boldsymbol{b}}^{*}_{i}$ associated with the set of functions in (14)
are

$\displaystyle\tilde{\boldsymbol{b}}^{*}_{i}$
$\displaystyle={\boldsymbol{b}}^{*}_{i},\quad i=1,\ldots,21,$ (16)
$\displaystyle\tilde{\boldsymbol{b}}^{*}_{22}$
$\displaystyle=\frac{2}{3}{\boldsymbol{b}}^{*}_{22}+\frac{1}{3}{\boldsymbol{b}}^{*}_{23}:\left(\frac{15}{27},\frac{7}{27},\frac{5}{27}\right),$
$\displaystyle\quad\tilde{\boldsymbol{b}}^{*}_{23}$
$\displaystyle=\frac{2}{3}{\boldsymbol{b}}^{*}_{23}+\frac{1}{3}{\boldsymbol{b}}^{*}_{22}:\left(\frac{15}{27},\frac{5}{27},\frac{7}{27}\right),$
$\displaystyle\tilde{\boldsymbol{b}}^{*}_{24}$
$\displaystyle=\frac{2}{3}{\boldsymbol{b}}^{*}_{24}+\frac{1}{3}{\boldsymbol{b}}^{*}_{25}:\left(\frac{5}{27},\frac{15}{27},\frac{7}{27}\right),$
$\displaystyle\tilde{\boldsymbol{b}}^{*}_{25}$
$\displaystyle=\frac{2}{3}{\boldsymbol{b}}^{*}_{25}+\frac{1}{3}{\boldsymbol{b}}^{*}_{24}:\left(\frac{7}{27},\frac{15}{27},\frac{5}{27}\right),$
$\displaystyle\tilde{\boldsymbol{b}}^{*}_{26}$
$\displaystyle=\frac{2}{3}{\boldsymbol{b}}^{*}_{26}+\frac{1}{3}{\boldsymbol{b}}^{*}_{27}:\left(\frac{7}{27},\frac{5}{27},\frac{15}{27}\right),$
$\displaystyle\tilde{\boldsymbol{b}}^{*}_{27}$
$\displaystyle=\frac{2}{3}{\boldsymbol{b}}^{*}_{27}+\frac{1}{3}{\boldsymbol{b}}^{*}_{26}:\left(\frac{5}{27},\frac{7}{27},\frac{15}{27}\right),$
$\displaystyle\tilde{\boldsymbol{b}}^{*}_{28}$
$\displaystyle=\frac{1}{3}({\boldsymbol{b}}^{*}_{22}+\cdots+{\boldsymbol{b}}^{*}_{27})-{\boldsymbol{b}}^{*}_{28}:\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right).$

These points are depicted in Figure 5 (right). Note that Theorem 3 in
combination with (15) confirms that the functions
${\widetilde{B}}_{1},\ldots,{\widetilde{B}}_{28}$ are linearly independent and
form a partition of unity. On the other hand, they are not all nonnegative.
For subsequent use, some Hermite data of these basis functions are collected
in the appendix (Tables 2 and 3).

Any spline $s\in{\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ can be
represented in terms of this alternative basis, so

$s=\sum_{i=1}^{28}b_{i}B_{i}=\sum_{i=1}^{28}{\tilde{b}}_{i}{\widetilde{B}}_{i}.$
(17)

Note that ${\tilde{\boldsymbol{b}}}={{\boldsymbol{C}}}{\boldsymbol{b}}$, where
${\boldsymbol{C}}$ is the conversion matrix already used to obtain (16). It
can be easily checked that
$\|{\boldsymbol{C}}\|_{\infty}=\|{\boldsymbol{C}}^{-1}\|_{\infty}=3$.
Therefore, from (12) and (17) we immediately deduce

$\frac{1}{111}\|{\tilde{\boldsymbol{b}}}\|_{\infty}\leq\biggl{\|}\sum_{i=1}^{28}{\tilde{b}}_{i}{\widetilde{B}}_{i}\biggr{\|}_{\infty}\leq
3\|{\tilde{\boldsymbol{b}}}\|_{\infty},$ (18)

and the condition number can be bounded as
$\kappa_{\infty}({\widetilde{\mathcal{B}}})<333$. We can also formulate the
analogue of Proposition 6 as well as a Marsden-like identity for the basis
(14). We omit the details for the sake of brevity.

The simplex spline basis (4) forms a convex partition of unity and so it is
particularly useful for geometric modeling. On the other hand, as we will show
in Section 4.1, the simplex spline basis (14) is more suited to handle $C^{2}$
smoothness conditions between spline functions on adjacent macro-triangles. Of
course, there are many more alternative sets of simplex spline basis
functions. One could, for instance, take the 10 cubic Bernstein polynomials
defined on $\Delta$ (they are special simplex splines; see
[Prautsch.Boehm.Paluszny02]) and enrich them with 18 more simplex splines that
are linearly independent.

##  4 $C^{2}$ cubic splines on the ${{\rm WS}_{3}}$ refinement of a
triangulation

In the previous section, we have provided simplex spline bases for the spline
space ${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ of cubic $C^{2}$ splines
on the ${{\rm WS}_{3}}$ split of a given triangle $\Delta$. Let
${\mathcal{T}}$ be a triangulation of a polygonal domain $\Omega$ and let
${\mathcal{T}}_{{{\rm WS}_{3}}}$ denote its refinement obtained by taking the
${{\rm WS}_{3}}$ split of each of its triangles. In this section, we consider
the spline space of $C^{2}$ cubic splines on ${\mathcal{T}}_{{{\rm WS}_{3}}}$,
i.e.,

${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}}):=\\{s\in C^{2}(\Omega),\
s_{|\tau}\in{\mathbb{P}}_{3},\ \tau\text{ is polygon in }{\mathcal{T}}_{{{\rm
WS}_{3}}}\\}.$

The unisolvency of the Hermite interpolation problem stated in Corollary 4
implies that the dimension of the space only depends on combinatorial
properties of the triangulation, and so it is stable. From the corollary we
directly deduce that (see also [Wang.90])

$\dim({\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm
WS}_{3}}}))=6n_{V}+3n_{E}+n_{T},$ (19)

where $n_{V}$, $n_{E}$, $n_{T}$ are the number of vertices, edges, and
triangles of ${\mathcal{T}}$, respectively. Moreover, any spline function of
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ can be locally
constructed on each (macro-)triangle $\Delta$ of ${\mathcal{T}}$ via the
Hermite data, and the corresponding spline piece on $\Delta$ can be
represented in the form (17). Conversely, any function, which is represented
locally in the form (17) on each $\Delta$ of ${\mathcal{T}}$, is $C^{2}$
smooth over each $\Delta$ of ${\mathcal{T}}$ but not necessary $C^{2}$ smooth
across the edges of $\cal T$. First we derive conditions on the local spline
coefficients to ensure global $C^{2}$ smoothness, and then we describe a
stable global basis with local support for
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$.

###  4.1 Smoothness conditions

Let ${\mathcal{T}}$ be a triangulation of a polygonal domain
$\Omega\subset{\mathbb{R}}^{2}$. We seek conditions on the local spline
coefficients in (17) to guarantee $C^{r}$ smoothness across a common edge of
two adjacent triangles of ${\mathcal{T}}$ for $r=0,1,2$.

######  Theorem 8.

Suppose the triangles
$\Delta^{L}:=\langle{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}\rangle$
and
$\Delta^{R}:=\langle{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{4}\rangle$
share the common edge with vertices
${\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2}$, and let

${\boldsymbol{p}}_{4}=\eta_{1}{\boldsymbol{p}}_{1}+\eta_{2}{\boldsymbol{p}}_{2}+\eta_{3}{\boldsymbol{p}}_{3},\quad\eta_{1}+\eta_{2}+\eta_{3}=1.$
(20)

Let $\\{B^{L}_{i},\ i=1,\ldots,28\\}$ and $\\{B^{R}_{i},\ i=1,\ldots,28\\}$ be
the scaled simplex spline basis defined by the knot sequences in Figure 3 on
$\Delta^{L}$ and $\Delta^{R}$, respectively. We assume the numbering of the
basis functions in agreement with Figure 3. Let us consider the spline
functions

$s^{L}:=\sum_{i=1}^{28}b^{L}_{i}B^{L}_{i},\quad
s^{R}:=\sum_{i=1}^{28}b^{R}_{i}B^{R}_{i}.$

We have

  * •

$s^{L},s^{R}$ join $C^{0}$ across the common edge if and only if

$b^{R}_{i}=b^{L}_{i},\quad i=1,2,4,7,10,13;$ (21)

  * •

$s^{L},s^{R}$ join $C^{1}$ across the common edge if and only if they join
$C^{0}$ and in addition

$\displaystyle b^{R}_{5}$
$\displaystyle=\eta_{1}b^{L}_{1}+\eta_{2}b^{L}_{4}+\eta_{3}b^{L}_{5},$ (22)
$\displaystyle b^{R}_{16}$
$\displaystyle=\left(\eta_{1}+\frac{\eta_{2}}{2}\right)b^{L}_{4}+\frac{\eta_{2}}{2}b^{L}_{10}+\eta_{3}b^{L}_{16},$
$\displaystyle b^{R}_{19}$
$\displaystyle=\left(\frac{3}{5}\eta_{1}+\frac{2}{5}\eta_{2}\right)b^{L}_{10}+\left(\frac{2}{5}\eta_{1}+\frac{3}{5}\eta_{2}\right)b^{L}_{13}+\eta_{3}b^{L}_{19},$
$\displaystyle b^{R}_{17}$
$\displaystyle=\left(\frac{\eta_{1}}{2}+\eta_{2}\right)b^{L}_{7}+\frac{\eta_{1}}{2}b^{L}_{13}+\eta_{3}b^{L}_{17},$
$\displaystyle b^{R}_{6}$
$\displaystyle=\eta_{1}b^{L}_{7}+\eta_{2}b^{L}_{2}+\eta_{3}b^{L}_{6}.$

###### Proof.

Let us first discuss $C^{0}$ smoothness. Along the common edge
${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$ the two functions $s^{L}$ and
$s^{R}$ are univariate cubic $C^{2}$ splines with (interior) knots at the
points ${\boldsymbol{p}}_{3,1}$ and ${\boldsymbol{p}}_{3,2}$. Considering the
restriction onto the edge of the basis functions $\\{B_{i}^{L},\
i=1,\ldots,28\\}$ and $\\{B_{i}^{R},\ i=1,\ldots,28\\}$, we obtain that the
only nonzero elements are

$(B_{i}^{R})_{|{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}=(B_{i}^{L})_{|{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}},\quad
i=1,2,4,7,10,13.$

Since they are linearly independent, $C^{0}$ smoothness is equivalent to
agreement of the corresponding coefficients. This proves (21).

We now discuss $C^{1}$ smoothness across the common edge. It suffices to prove
that along the edge ${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$ the functions
$D_{{\boldsymbol{q}}_{3}{\boldsymbol{p}}_{3}}s^{L}$ and
$D_{{\boldsymbol{q}}_{3}{\boldsymbol{p}}_{3}}s^{R}$ agree. These functions are
univariate $C^{1}$ quadratic splines with (interior) knots at the points
${\boldsymbol{p}}_{3,1}$ and ${\boldsymbol{p}}_{3,2}$. Therefore, each of them
is uniquely determined by its value and first derivative at the two endpoints
of the edge and by the value at the midpoint ${\boldsymbol{q}}_{3}$. From (20)
we obtain

${\boldsymbol{p}}_{3}=\frac{1}{\eta_{3}}({\boldsymbol{p}}_{4}-\eta_{1}{\boldsymbol{p}}_{1}-\eta_{2}{\boldsymbol{p}}_{2}),$

and so

${\boldsymbol{q}}_{3}{\boldsymbol{p}}_{3}={\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}-\frac{1}{2}{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}=\frac{1}{\eta_{3}}{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{4}-\frac{\eta_{3}+2\eta_{2}}{2\eta_{3}}{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}.$

Then, by employing the $C^{0}$ smoothness conditions and the values in Table 2
we get

$\displaystyle
D_{{\boldsymbol{q}}_{3}{\boldsymbol{p}}_{3}}s^{L}({\boldsymbol{p}}_{1})$
$\displaystyle=-9b^{L}_{1}+9b^{L}_{5}-\frac{1}{2}(-9b^{L}_{1}+9b^{L}_{4}),$
$\displaystyle
D_{{\boldsymbol{q}}_{3}{\boldsymbol{p}}_{3}}s^{R}({\boldsymbol{p}}_{1})$
$\displaystyle=\frac{1}{\eta_{3}}(-9b^{L}_{1}+9b^{R}_{5})-\frac{\eta_{3}+2\eta_{2}}{2\eta_{3}}(-9b^{L}_{1}+9b^{L}_{4}).$

Equating the above expressions results in the first condition of (22). With
the same line of arguments we deduce the remaining four conditions. ∎

From the relations in (15) it is clear that the conditions in (21) and (22)
also ensure $C^{0}$ and $C^{1}$ smoothness, respectively, for local spline
representations in the alternative basis (14).

######  Corollary 9.

Consider the same assumptions as in Theorem 8. The $C^{0}$ smoothness
conditions for the control points can be written as

$({\boldsymbol{b}}^{*R}_{i},b^{R}_{i})=({\boldsymbol{b}}^{*L}_{i},b^{L}_{i}),\quad
i=1,2,4,7,10,13;$

and the $C^{1}$ smoothness conditions for the control points can be written as

$\displaystyle({\boldsymbol{b}}^{*R}_{5},b^{R}_{5})$
$\displaystyle=\eta_{1}({\boldsymbol{b}}^{*L}_{1},b^{L}_{1})+\eta_{2}({\boldsymbol{b}}^{*L}_{4},b^{L}_{4})+\eta_{3}({\boldsymbol{b}}^{*L}_{5},b^{L}_{5}),$
$\displaystyle({\boldsymbol{b}}^{*R}_{16},b^{R}_{16})$
$\displaystyle=\left(\eta_{1}+\frac{\eta_{2}}{2}\right)({\boldsymbol{b}}^{*L}_{4},b^{L}_{4})+\frac{\eta_{2}}{2}({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})+\eta_{3}({\boldsymbol{b}}^{*L}_{16},b^{L}_{16}),$
$\displaystyle({\boldsymbol{b}}^{*R}_{19},b^{R}_{19})$
$\displaystyle=\left(\frac{3}{5}\eta_{1}+\frac{2}{5}\eta_{2}\right)({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})+\left(\frac{2}{5}\eta_{1}+\frac{3}{5}\eta_{2}\right)({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})+\eta_{3}({\boldsymbol{b}}^{*L}_{19},b^{L}_{19}),$
$\displaystyle({\boldsymbol{b}}^{*R}_{17},b^{R}_{17})$
$\displaystyle=\left(\frac{\eta_{1}}{2}+\eta_{2}\right)({\boldsymbol{b}}^{*L}_{7},b^{L}_{7})+\frac{\eta_{1}}{2}({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})+\eta_{3}({\boldsymbol{b}}^{*L}_{17},b^{L}_{17}),$
$\displaystyle({\boldsymbol{b}}^{*R}_{6},b^{R}_{6})$
$\displaystyle=\eta_{1}({\boldsymbol{b}}^{*L}_{7},b^{L}_{7})+\eta_{2}({\boldsymbol{b}}^{*L}_{2},b^{L}_{2})+\eta_{3}({\boldsymbol{b}}^{*L}_{6},b^{L}_{6}).$

###### Proof.

The statements follow by direct computation from (22) and from the expressions
of the domain points in (10). ∎

The $C^{1}$ smoothness conditions in Corollary 9 have a nice geometric
interpretation. There are five sets of four control points that need to be
coplanar. In terms of our control net configuration in Figure 5 (left), that
means that the five triangles in both control nets along the common edge must
be all pairwise coplanar. This is illustrated in Figure 8.

Figure 8: A $C^{1}$ spline surface on two adjacent domain triangles. The five
pairs of triangles in the control nets that must be coplanar according to the
smoothness conditions are colored.

######  Theorem 10.

Consider the same assumptions as in Theorem 8. Let
$\\{{\widetilde{B}}^{L}_{i},\ i=1,\ldots,28\\}$ and
$\\{{\widetilde{B}}^{R}_{i},\ i=1,\ldots,28\\}$ be the spline bases defined by
(15) on $\Delta^{L}$ and $\Delta^{R}$, respectively. Then, the spline
functions

$s^{L}:=\sum_{i=1}^{28}b^{L}_{i}B^{L}_{i}=\sum_{i=1}^{21}b^{L}_{i}B^{L}_{i}+\sum_{i=22}^{27}{\tilde{b}}^{L}_{i}{\widetilde{B}}^{L}_{i}+{\tilde{b}}^{L}_{28}B^{L}_{28}$

and

$s^{R}:=\sum_{i=1}^{28}b^{R}_{i}B^{R}_{i}=\sum_{i=1}^{21}b^{R}_{i}B^{R}_{i}+\sum_{i=22}^{27}{\tilde{b}}^{R}_{i}{\widetilde{B}}^{R}_{i}+{\tilde{b}}^{R}_{28}B^{R}_{28}$

join $C^{2}$ across the common edge if and only if they join $C^{1}$ and in
addition

$\displaystyle b^{R}_{11}$
$\displaystyle=\eta_{1}(\eta_{1}-\eta_{2}-\eta_{3})b^{L}_{1}+\eta_{2}(3\eta_{1}-\eta_{3})b^{L}_{4}+\eta_{3}(3\eta_{1}-\eta_{2})b^{L}_{5}+\eta_{2}^{2}b^{L}_{10}+\eta_{3}^{2}b^{L}_{11}+4\eta_{2}\eta_{3}b^{L}_{16},$
(23) $\displaystyle b^{R}_{12}$
$\displaystyle=\eta_{2}(\eta_{2}-\eta_{1}-\eta_{3})b^{L}_{2}+\eta_{1}(3\eta_{2}-\eta_{3})b^{L}_{7}+\eta_{3}(3\eta_{2}-\eta_{1})b^{L}_{6}+\eta_{1}^{2}b^{L}_{13}+\eta_{3}^{2}b^{L}_{12}+4\eta_{1}\eta_{3}b^{L}_{17},$
$\displaystyle{\tilde{b}}^{R}_{22}$
$\displaystyle=\frac{1}{6}(\eta_{1}-\eta_{3})(2\eta_{1}+\eta_{2})b^{L}_{4}+\left({\frac{5}{18}}\eta_{2}+{\frac{7}{18}}{\eta_{{2}}}^{2}+\frac{2}{3}\eta_{{1}}+\frac{2}{3}\eta_{{2}}\eta_{{1}}\right)b^{L}_{10}$
$\displaystyle\quad+\frac{1}{9}\left(2\,\eta_{{1}}+3\,\eta_{{2}}\right)\left(\eta_{2}-2\eta_{3}\right)b^{L}_{13}+\frac{1}{3}\eta_{3}(3\eta_{1}+\eta_{2})b^{L}_{16}+\frac{10}{9}\eta_{3}(2\eta_{2}+\eta_{1})b^{L}_{19}+\eta_{3}^{2}{\tilde{b}}^{L}_{22},$
$\displaystyle{\tilde{b}}^{R}_{25}$
$\displaystyle=\frac{1}{6}(\eta_{2}-\eta_{3})(2\eta_{2}+\eta_{1})b^{L}_{7}+\left({\frac{5}{18}}\,\eta_{{1}}+{\frac{7}{18}}\,{\eta_{{1}}}^{2}+\frac{2}{3}\,\eta_{{2}}+\frac{2}{3}\eta_{{2}}\eta_{{1}}\right)b^{L}_{13}$
$\displaystyle\quad+\frac{1}{9}\left(3\eta_{1}+2\eta_{2}\right)\left(\eta_{{1}}-2\eta_{3}\right)b^{L}_{10}+\frac{1}{3}\eta_{3}(3\eta_{2}+\eta_{1})b^{L}_{17}+\frac{10}{9}\eta_{3}(2\eta_{1}+\eta_{2})b^{L}_{19}+\eta_{3}^{2}{\tilde{b}}^{L}_{25}.$

###### Proof.

Assume $s^{L}$ and $s^{R}$ join $C^{1}$ across the common edge
${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$. To prove $C^{2}$ smoothness across
the same edge, it suffices to prove that along
${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$ the functions
$D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}^{2}s^{L}$ and
$D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}^{2}s^{R}$ agree. Along the edge
${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$, these functions are univariate
$C^{0}$ linear splines with (interior) knots at the points
${\boldsymbol{p}}_{3,1}$ and ${\boldsymbol{p}}_{3,2}$. Therefore, each of them
is uniquely determined by its value at the two endpoints of the edge and by
the value at the points ${\boldsymbol{p}}_{3,1}$ and ${\boldsymbol{p}}_{3,2}$.
From (20) we know that

${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}=\frac{1}{\eta_{3}}{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{4}-\frac{\eta_{2}}{\eta_{3}}{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2},$

so that

$D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}^{2}=\left(\frac{1}{\eta_{3}}D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{4}}-\frac{\eta_{2}}{\eta_{3}}D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}\right)^{2}.$

Then, by employing the values in Table 2 we get

$\displaystyle
D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}^{2}s^{L}({\boldsymbol{p}}_{1})$
$\displaystyle=54b^{L}_{1}-81b^{L}_{5}+27b^{L}_{11},$ $\displaystyle
D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}^{2}s^{R}({\boldsymbol{p}}_{1})$
$\displaystyle=\frac{1}{\eta_{3}^{2}}(54b^{R}_{1}-81b^{R}_{5}+27b^{R}_{11})-\frac{2\eta_{2}}{\eta_{3}^{2}}(54b^{R}_{1}-54b^{R}_{4}-54b^{R}_{5}+54b^{R}_{16})$
$\displaystyle\quad+\frac{\eta_{2}^{2}}{\eta_{3}^{2}}(54b^{R}_{1}-81b^{R}_{5}+27b^{R}_{10}).$

By equating the above expressions and taking into account the $C^{1}$
smoothness conditions, we obtain the first condition of (23). With the same
line of arguments, taking into account (15) and the additional values in Table
3, we deduce the remaining three conditions. ∎

######  Corollary 11.

Consider the same assumptions as in Theorem 10. The $C^{2}$ smoothness
conditions for the control points can be written as

$\displaystyle({\boldsymbol{b}}^{*R}_{11},b^{R}_{11})$
$\displaystyle=\eta_{1}(\eta_{1}-\eta_{2}-\eta_{3})({\boldsymbol{b}}^{*L}_{1},b^{L}_{1})+\eta_{2}(3\eta_{1}-\eta_{3})({\boldsymbol{b}}^{*L}_{4},b^{L}_{4})$
$\displaystyle\quad+\eta_{3}(3\eta_{1}-\eta_{2})({\boldsymbol{b}}^{*L}_{5},b^{L}_{5})+\eta_{2}^{2}({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})+\eta_{3}^{2}({\boldsymbol{b}}^{*L}_{11},b^{L}_{11})+4\eta_{2}\eta_{3}({\boldsymbol{b}}^{*L}_{16},b^{L}_{16}),$
$\displaystyle({\boldsymbol{b}}^{*R}_{12},b^{R}_{12})$
$\displaystyle=\eta_{2}(\eta_{2}-\eta_{1}-\eta_{3})({\boldsymbol{b}}^{*L}_{2},b^{L}_{2})+\eta_{3}(3\eta_{2}-\eta_{1})({\boldsymbol{b}}^{*L}_{6},b^{L}_{6})$
$\displaystyle\quad+\eta_{1}(3\eta_{2}-\eta_{3})({\boldsymbol{b}}^{*L}_{7},b^{L}_{7})+\eta_{1}^{2}({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})+\eta_{3}^{2}({\boldsymbol{b}}^{*L}_{12},b^{L}_{12})+4\eta_{1}\eta_{3}({\boldsymbol{b}}^{*L}_{17},b^{L}_{17}),$
$\displaystyle(\tilde{\boldsymbol{b}}^{*R}_{22},{\tilde{b}}^{R}_{22})$
$\displaystyle=\frac{1}{6}(\eta_{1}-\eta_{3})(2\eta_{1}+\eta_{2})({\boldsymbol{b}}^{*L}_{4},b^{L}_{4})+\left({\frac{5}{18}}\eta_{2}+{\frac{7}{18}}{\eta_{2}}^{2}+\frac{2}{3}\eta_{1}+\frac{2}{3}\eta_{2}\eta_{1}\right)({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})$
$\displaystyle\quad+\frac{1}{9}\left(2\eta_{1}+3\eta_{2}\right)\left(\eta_{2}-2\eta_{3}\right)({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})+\frac{1}{3}\eta_{3}(3\eta_{1}+\eta_{2})({\boldsymbol{b}}^{*L}_{16},b^{L}_{16})$
$\displaystyle\quad+\frac{10}{9}\eta_{3}(2\eta_{2}+\eta_{1})({\boldsymbol{b}}^{*L}_{19},b^{L}_{19})+\eta_{3}^{2}(\tilde{\boldsymbol{b}}^{*L}_{22},{\tilde{b}}^{L}_{22}),$
$\displaystyle(\tilde{\boldsymbol{b}}^{*R}_{25},{\tilde{b}}^{R}_{25})$
$\displaystyle=\frac{1}{6}(\eta_{2}-\eta_{3})(2\eta_{2}+\eta_{1})({\boldsymbol{b}}^{*L}_{7},b^{L}_{7})+\left({\frac{5}{18}}\eta_{{1}}+{\frac{7}{18}}{\eta_{1}}^{2}+\frac{2}{3}\eta_{2}+\frac{2}{3}\eta_{2}\eta_{1}\right)({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})$
$\displaystyle\quad+\frac{1}{9}\left(3\eta_{1}+2\eta_{2}\right)\left(\eta_{1}-2\eta_{3}\right)({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})+\frac{1}{3}\eta_{3}(3\eta_{2}+\eta_{1})({\boldsymbol{b}}^{*L}_{17},b^{L}_{17})$
$\displaystyle\quad+\frac{10}{9}\eta_{3}(2\eta_{1}+\eta_{2})({\boldsymbol{b}}^{*L}_{19},b^{L}_{19})+\eta_{3}^{2}(\tilde{\boldsymbol{b}}^{*L}_{25},{\tilde{b}}^{L}_{25}).$

###### Proof.

The statements follow by direct computation from (23) and from the expressions
of the domain points in (10) and (16). ∎

Using the relations between the domain points ${\boldsymbol{b}}^{*}_{i}$ and
$\tilde{\boldsymbol{b}}^{*}_{i}$ in (16), we can immediately rewrite the
$C^{2}$ smoothness conditions solely in terms of the control points of the
basis (4). For instance, the third condition in Corollary 11 reads as

$\displaystyle\frac{2}{3}({\boldsymbol{b}}^{*R}_{22},b^{R}_{22})$
$\displaystyle+\frac{1}{3}({\boldsymbol{b}}^{*R}_{23},b^{R}_{23})$
$\displaystyle=\frac{1}{6}(\eta_{1}-\eta_{3})(2\eta_{1}+\eta_{2})({\boldsymbol{b}}^{*L}_{4},b^{L}_{4})+\left({\frac{5}{18}}\eta_{2}+{\frac{7}{18}}{\eta_{2}}^{2}+\frac{2}{3}\eta_{1}+\frac{2}{3}\eta_{2}\eta_{1}\right)({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})$
$\displaystyle\quad+\frac{1}{9}\left(2\eta_{1}+3\eta_{2}\right)\left(\eta_{2}-2\eta_{3}\right)({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})+\frac{1}{3}\eta_{3}(3\eta_{1}+\eta_{2})({\boldsymbol{b}}^{*L}_{16},b^{L}_{16})$
$\displaystyle\quad+\frac{10}{9}\eta_{3}(2\eta_{2}+\eta_{1})({\boldsymbol{b}}^{*L}_{19},b^{L}_{19})+\eta_{3}^{2}\left(\frac{2}{3}({\boldsymbol{b}}^{*L}_{22},b^{L}_{22})+\frac{1}{3}({\boldsymbol{b}}^{*L}_{23},b^{L}_{23})\right),$

and the fourth condition as

$\displaystyle\frac{2}{3}({\boldsymbol{b}}^{*R}_{25},b^{R}_{25})$
$\displaystyle+\frac{1}{3}({\boldsymbol{b}}^{*R}_{24},b^{R}_{24})$
$\displaystyle=\frac{1}{6}(\eta_{2}-\eta_{3})(2\eta_{2}+\eta_{1})({\boldsymbol{b}}^{*L}_{7},b^{L}_{7})+\left({\frac{5}{18}}\eta_{1}+{\frac{7}{18}}{\eta_{1}}^{2}+\frac{2}{3}\eta_{2}+\frac{2}{3}\eta_{2}\eta_{1}\right)({\boldsymbol{b}}^{*L}_{13},b^{L}_{13})$
$\displaystyle\quad+\frac{1}{9}\left(3\eta_{1}+2\eta_{2}\right)\left(\eta_{1}-2\eta_{3}\right)({\boldsymbol{b}}^{*L}_{10},b^{L}_{10})+\frac{1}{3}\eta_{3}(3\eta_{2}+\eta_{1})({\boldsymbol{b}}^{*L}_{17},b^{L}_{17})$
$\displaystyle\quad+\frac{10}{9}\eta_{3}(2\eta_{1}+\eta_{2})({\boldsymbol{b}}^{*L}_{19},b^{L}_{19})+\eta_{3}^{2}\left(\frac{2}{3}({\boldsymbol{b}}^{*L}_{25},b^{L}_{25})+\frac{1}{3}({\boldsymbol{b}}^{*L}_{24},b^{L}_{24})\right).$

The smoothness conditions in Corollary 11 show a structural similarity with
the $C^{2}$ join of two adjacent triangular Bernstein–Bézier patches. We refer
to [Lai97, Example 2] for a geometric interpretation.

###  4.2 Stable bases for ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm
WS}_{3}}})$

In Sections 3.1 and 3.4, we have constructed two simplex spline bases for the
space of $C^{2}$ cubic splines on the ${{\rm WS}_{3}}$ split of a given
triangle $\Delta$. We have also shown that these bases enjoy similar
properties to the Bernstein polynomial basis defined on a triangle. Here, we
consider the space ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ and
we take the similarity between these bases one step further: we extend the
concept of minimal determining sets developed for the Bernstein polynomial
basis [Lai.Schumaker07] to our simplex spline bases, with the aim of
constructing stable bases with local support for the space
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. For the sake of
simplicity we focus on the simplex spline basis (14).

To stress their dependence on a specific triangle $\Delta$, from now on we
denote the simplex spline basis (14) by $\\{{\widetilde{B}}_{i,\Delta},\
i=1,\ldots,28\\}$. Any spline $s\in{\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm
WS}_{3}}})$ can be identified by its coefficients $\\{{\tilde{b}}_{i,\Delta},\
i=1,\ldots,28\\}$ with respect to the above basis over any triangle $\Delta$
of $\cal T$, i.e.,

$s_{|\Delta}=\sum_{i=1}^{28}{\tilde{b}}_{i,\Delta}{\widetilde{B}}_{i,\Delta},$
(24)

where the coefficients in (24) must satisfy the smoothness conditions derived
in Section 4.1 to ensure the $C^{2}$ joins across the edges of
${\mathcal{T}}$. Following [Lai.Schumaker07, Chapter 5], for any triangle
$\Delta$ of ${\mathcal{T}}$ we denote by ${\widetilde{\mathcal{D}}}_{\Delta}$
the set of domain points specified in (16). Then, we define a (minimal)
determining set as follows.

######  Definition 12.

Assume a set
${\widetilde{\mathcal{D}}}\subseteq(\cup_{\Delta\in{\mathcal{T}}}{\widetilde{\mathcal{D}}}_{\Delta})$
is such that if $s\in{\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ has
all the coefficients corresponding to elements in ${\widetilde{\mathcal{D}}}$
equal to zero then $s\equiv 0$. Then, ${\widetilde{\mathcal{D}}}$ is a
determining set for ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. A
determining set is a minimal determining set in case it has the smallest
possible cardinality.

By using the same line of arguments as the proof of [Lai.Schumaker07, Theorem
5.13] we infer that the cardinality of a minimal determining set for
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ equals the dimension of
the space. With the aim of specifying such a minimal determining set, let us
first introduce some terminology regarding the domain points in a triangle
$\Delta=\langle{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}\rangle$
of ${\mathcal{T}}$:

  * •

the domain points associated with the vertex ${\boldsymbol{p}}_{i}$ are the
six points in (16) with the $i$-th barycentric coordinate $\geq 2/3$,
$i=1,2,3$ (for instance, the domain points associated with
${\boldsymbol{p}}_{1}$ are
$\tilde{\boldsymbol{b}}^{*}_{1},\tilde{\boldsymbol{b}}^{*}_{4},\tilde{\boldsymbol{b}}^{*}_{5},\tilde{\boldsymbol{b}}^{*}_{10},\tilde{\boldsymbol{b}}^{*}_{11},\tilde{\boldsymbol{b}}^{*}_{16}$);

  * •

the domain points associated with the edge
${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}$ are the three points
$\tilde{\boldsymbol{b}}^{*}_{19},\tilde{\boldsymbol{b}}^{*}_{22},\tilde{\boldsymbol{b}}^{*}_{25}$
in (16);

  * •

the domain points associated with the edge
${\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}$ are the three points
$\tilde{\boldsymbol{b}}^{*}_{21},\tilde{\boldsymbol{b}}^{*}_{23},\tilde{\boldsymbol{b}}^{*}_{26}$
in (16);

  * •

the domain points associated with the edge
${\boldsymbol{p}}_{2}{\boldsymbol{p}}_{3}$ are the three points
$\tilde{\boldsymbol{b}}^{*}_{20},\tilde{\boldsymbol{b}}^{*}_{24},\tilde{\boldsymbol{b}}^{*}_{27}$
in (16);

  * •

the domain point associated with the triangle $\Delta$ is the point
$\tilde{\boldsymbol{b}}^{*}_{28}$ in (16).

We can construct a minimal determining set for
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ as follows; see also
Figure 9.

Figure 9: A minimal determining set for
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. Filled circles: the
domain points associated with a vertex. Filled squares: the domain points
associated with an edge. Filled triangle: the domain points associated with a
triangle. Empty circles: the domain points associated with coefficients
obtained from those in the minimal determining set by imposing the smoothness
conditions.

######  Theorem 13.

For a given triangulation $\cal T$, let ${\widetilde{\mathcal{M}}}$ be the set
consisting of the following domain points:

  * •

for each vertex ${\boldsymbol{p}}$ of ${\mathcal{T}}$, choose a triangle
$\Delta$ of ${\mathcal{T}}$ such that ${\boldsymbol{p}}$ is a vertex of
$\Delta$ and select the six domain points in $\Delta$ associated with
${\boldsymbol{p}}$;

  * •

for each edge of $\cal T$, choose a triangle $\Delta$ of ${\mathcal{T}}$
sharing this edge and select the three domain points in $\Delta$ associated
with the edge;

  * •

for each triangle $\Delta$ of ${\mathcal{T}}$, select the domain point
associated with it.

Then, ${\widetilde{\mathcal{M}}}$ is a minimal determining set for
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$.

###### Proof.

Let $s\in{\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. Assume that
all its coefficients associated with the domain points in
${\widetilde{\mathcal{M}}}$ are set equal to $0$. Let ${\boldsymbol{p}}$ be
any vertex of ${\mathcal{T}}$. All the coefficients of $s$ corresponding to
the domain points associated with ${\boldsymbol{p}}$ (in any triangle of
${\mathcal{T}}$ surrounding ${\boldsymbol{p}}$) are zero because either they
belong to ${\widetilde{\mathcal{M}}}$ or they are uniquely determined by the
conditions in (21), the first two conditions in (22) and the first condition
in (23) for $C^{2}$ smoothness across the edges emanating from
${\boldsymbol{p}}$. Let ${\boldsymbol{p}}{\boldsymbol{q}}$ be any edge of
${\mathcal{T}}$. All the coefficients of $s$ corresponding to the domain
points associated with the edge (in any of the two triangles of
${\mathcal{T}}$ sharing the edge ${\boldsymbol{p}}{\boldsymbol{q}}$) are zero
because either they belong to ${\widetilde{\mathcal{M}}}$ or they are uniquely
determined by the third condition in (22) and the third and fourth conditions
in (23) for $C^{2}$ smoothness across the edge. Finally, the domain point
associated with any triangle in ${\mathcal{T}}$ belongs to
${\widetilde{\mathcal{M}}}$ and so the corresponding coefficient is $0$.
Hence, for any triangle $\Delta$ of ${\mathcal{T}}$ all the coefficients in
(24) are $0$ and so $s\equiv 0$, i.e., ${\widetilde{\mathcal{M}}}$ is a
determining set. Moreover, the cardinality of ${\widetilde{\mathcal{M}}}$
clearly equals the dimension of the space, see (19), and so
${\widetilde{\mathcal{M}}}$ is a minimal determining set. ∎

Given a minimal determining set ${\widetilde{\mathcal{M}}}$ for
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$, suppose we assign
values to all the coefficients corresponding to the domain points in it. The
proof of Theorem 13 shows that these coefficients uniquely identify a spline
function of ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ because all
the remaining coefficients in the representation (24) can be deduced from the
smoothness conditions. Therefore, any minimal determining set enables us to
built a basis for ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. Let
us consider the set of functions

$\\{B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}},\
\tilde{\boldsymbol{b}}^{*}\in{\widetilde{\mathcal{M}}}\\},$ (25)

where $B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}$ is the unique function of
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ obtained by zeroing all
the coefficients corresponding to the domain points in
${\widetilde{\mathcal{M}}}$ except the one related to
$\tilde{\boldsymbol{b}}^{*}$ which is set equal to $1$. By construction, the
functions in (25) are clearly linearly independent (see also [Lai.Schumaker07,
Theorem 5.20]) and their number agrees with the dimension of the space because
${\widetilde{\mathcal{M}}}$ is a minimal determining set. Thus, (25) is a
basis for ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. The
following proposition ensures that the elements of the above basis are
uniformly bounded and have local support.

######  Proposition 14.

For a given triangulation $\cal T$, let ${\widetilde{\mathcal{M}}}$ be a
minimal determining set for ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm
WS}_{3}}})$ as specified in Theorem 13. For all
$\tilde{\boldsymbol{b}}^{*}\in{\widetilde{\mathcal{M}}}$, the support of
$B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}$ is contained in

  1. a)

the union of the triangles of ${\mathcal{T}}$ sharing the vertex
${\boldsymbol{p}}$, if $\tilde{\boldsymbol{b}}^{*}$ is a domain point in
${\widetilde{\mathcal{M}}}$ associated with the vertex ${\boldsymbol{p}}$;

  2. b)

the union of the two triangles of ${\mathcal{T}}$ sharing the edge
${\boldsymbol{p}}{\boldsymbol{q}}$, if $\tilde{\boldsymbol{b}}^{*}$ is a
domain point in ${\widetilde{\mathcal{M}}}$ associated with the edge
${\boldsymbol{p}}{\boldsymbol{q}}$;

  3. c)

the triangle $\Delta$ of ${\mathcal{T}}$, if $\tilde{\boldsymbol{b}}^{*}$ is
the domain point in ${\widetilde{\mathcal{M}}}$ associated with the triangle
$\Delta$.

Moreover, there exists a constant ${\widetilde{K}}$ only depending on the
minimal angle of ${\mathcal{T}}$ such that

$\|B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}\|_{\infty}\leq{\widetilde{K}},\quad\tilde{\boldsymbol{b}}^{*}\in{\widetilde{\mathcal{M}}}.$
(26)

###### Proof.

Let $\tilde{\boldsymbol{b}}^{*}$ be fixed. A direct inspection of the
smoothness conditions in Theorems 8 and 10 immediately gives that the
coefficients in (24) for $B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}$ are
$0$ whenever $\Delta$ is not a triangle listed in the items a)-b)-c) and so
$B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}$ vanishes outside the union of
those triangles. In order to prove (26), we first note that the number of
triangles surrounding a vertex of ${\mathcal{T}}$ and the (absolute value of
the) barycentric coordinates of a point of a triangle with respect to an
adjacent triangle are bounded in terms of the minimum angle of ${\mathcal{T}}$
(see [Lai.Schumaker07, proof of Lemma 2.29]). Let $\Delta$ be a triangle of
${\mathcal{T}}$ belonging to the support of
$B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}$. We have that
$(B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}})_{|\Delta}$ can be represented
in the form (24) where the coefficients ${\tilde{b}}_{i,\Delta}$ are obtained
from the value $1$ corresponding to the domain point
$\tilde{\boldsymbol{b}}^{*}\in{\widetilde{\mathcal{M}}}$ by applying the
smoothness conditions in Theorems 8 and 10; these conditions consist of linear
or quadratic relations involving barycentric coordinates of points in adjacent
triangles. Therefore, denoting by ${\tilde{\boldsymbol{b}}}_{\Delta}$ the
vector of these $28$ coefficients, we get
$\|{\tilde{\boldsymbol{b}}}_{\Delta}\|_{\infty}\leq K^{\prime}$, where
$K^{\prime}$ is a constant only depending on the minimum angle of
${\mathcal{T}}$. Hence, from (18) we get

$\|(B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}})_{|\Delta}\|_{\infty}\leq
3\|{\tilde{\boldsymbol{b}}}_{\Delta}\|_{\infty}\leq 3K^{\prime}.$

Taking the maximum over all the triangles of ${\mathcal{T}}$ we arrive at (26)
with ${\widetilde{K}}=3K^{\prime}$. ∎

A basis with properties as in Proposition 14 is called a stable basis with
local support. For such a basis, using the same line of arguments as the proof
of [Lai.Schumaker07, Theorem 5.22], we can show that for all
${\tilde{\boldsymbol{c}}}:=({\tilde{c}}_{\tilde{\boldsymbol{b}}^{*}}\in{\mathbb{R}}:\tilde{\boldsymbol{b}}^{*}\in{\widetilde{\mathcal{M}}})^{T}$,

$K^{-}\|{\tilde{\boldsymbol{c}}}\|_{\infty}\leq\biggl{\|}\sum_{\tilde{\boldsymbol{b}}^{*}\in{\widetilde{\mathcal{M}}}}{\tilde{c}}_{\tilde{\boldsymbol{b}}^{*}}B_{\tilde{\boldsymbol{b}}^{*},{\mathcal{T}}}\biggr{\|}_{\infty}\leq
K^{+}\|{\tilde{\boldsymbol{c}}}\|_{\infty},$ (27)

where $K^{-}$ and $K^{+}$ are positive constants depending only on the
smallest angle of ${\mathcal{T}}$. The inequalities in (27) extend the local
stability result in (18) to the full spline space
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$. Similar stability
results in any $L_{q}$-norm for (a properly scaled version of) the basis can
also be achieved; see again the proof of [Lai.Schumaker07, Theorem 5.22].
Furthermore, from the partition of unity property of the local basis in (14),
we directly deduce that the global basis in (25) forms a partition of unity as
well.

Note that the determining set in Theorem 13 is a stable local determining set
in the sense of [Lai.Schumaker07, Definition 5.16]. This feature is, roughly
speaking, the key ingredient in the proof of Proposition 14. Similarly to
[Lai.Schumaker07, Section 5.7], it also ensures that the full spline space
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ has optimal
approximation power. More precisely, Theorems 5.18 and 5.19 in
[Lai.Schumaker07] hold true for ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm
WS}_{3}}})$.

The global basis in (25) can be easily expressed over any triangle $\Delta$
with respect to the local basis (4) through the conversion (15). Contrarily to
this local basis, the functions in (25) are in general not nonnegative.
However, paraphrasing [Lai.Schumaker07, Section 5.8], we observe that the
explicit basis in (25) has mainly a theoretical interest. For computation with
splines belonging to ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ it
is more convenient to work directly with the local representations provided by
the bases (4) or (14), rather than with the basis for the full spline space.

##  5 Concluding remarks

In [Cohen.Lyche.Riesenfeld13], a simplex spline basis was described for the
$C^{1}$ quadratic spline space on the Powell–Sabin 12 split, which is the
quadratic member of the Wang–Shi split family. In this paper, we have
addressed the $C^{2}$ cubic case and constructed two simplex spline bases for
the ${{\rm WS}_{3}}$ split. The characteristics of the $C^{2}$ cubic simplex
spline bases make it unnecessary to consider separate polynomial
representations on each of the numerous polygonal regions of the partitioned
macro-triangle. This paves the path for a practical construction of globally
$C^{2}$ cubic splines on any triangulation by extending the concept of minimal
determining sets.

In the following, we outline some implementation aspects and we identify few
problems where the provided simplex spline bases may be prosperous, in order
to complement the theoretical interest of our investigation with an
application-oriented perspective. We end with a discussion on a higher-order
extension of the construction.

###  5.1 Implementation aspects

For computation with splines belonging to
${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$, it is convenient to
work directly with the local representations provided by the bases (4) or
(14). On the one hand, evaluation of the simplex spline basis functions can be
achieved by applying the recurrence relation (B-recurrence) of simplex
splines; see Section 2.1. On the other hand, it might be more convenient to
use the explicit expressions of the simplex spline basis functions (4) given
in Table 1 in the appendix. The alternative basis functions (14) can be
immediately deduced from the previous ones by means of the linear relation
(15).

Having at our disposal such a table, evaluation of any spline in
${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$ can be efficiently performed
by combining a lookup-table process with a search algorithm based on boolean
vectors. Given (the barycentric coordinates of) any point ${\boldsymbol{p}}$
in $\Delta$, the values of the simplex spline basis functions (4) at
${\boldsymbol{p}}$ can be directly obtained from Table 1 once we have figured
out which polygonal region of the macro-triangle the evaluation point belongs
to. Since the ${{\rm WS}_{3}}$ split is a cross-cut partition, any polygonal
region in $\Delta_{{{\rm WS}_{3}}}$ is uniquely identified by the sign of the
linear expressions of the 18 interior lines in the split. These signs can be
interpreted as binary digits of an integer belonging to $[0,2^{18}-1]$.
Therefore, in order to detect which polygonal region of the macro-triangle a
given point ${\boldsymbol{p}}$ belongs to, it suffices to evaluate all the 18
interior lines at ${\boldsymbol{p}}$, to collect the resulting signs in a
boolean vector, and to interpret such a vector as binary digits of an integer.
A similar search algorithm has been described in [Cohen.Lyche.Riesenfeld13,
Algorithm 1.1].

It is important to remark that the selection of the different polynomial
pieces is just an implementation aspect. Thanks to the characteristics of the
simplex spline representation, there is a single control net to facilitate
control and early visualization of a spline function over each element
$\Delta$ in ${\mathcal{T}}$. This single control net makes that the complex
geometry of the ${{\rm WS}_{3}}$ split (consisting of $75$ polygons including
triangles, quadrilaterals, and pentagons) is transparent to the user. In this
perspective, an interesting topic of possible future research is to
investigate whether the control net introduced in the paper can give rise to a
de Casteljau/de Boor-type algorithm for evaluation of splines in
${\mathbb{S}}_{3}^{2}(\Delta_{{{\rm WS}_{3}}})$.

###  5.2 Application areas

Splines on (refined) triangulations are valuable in several application areas.
When dealing with bivariate/multivariate problems, the straightforward
approach is to rely on tensor-product structures, and in particular tensor-
product splines. Tensor-product structures offer several advantages, mainly
the simplicity of their use and the inheritance of univariate properties.
Major drawbacks, however, are the lack of adequate local refinement and the
struggle to represent geometries with complicated shapes. Although there are
several appealing extensions of tensor-product splines towards local
refinement (see, e.g., [Dokken13, Giannelli12, Sederberg03]) and complex
geometries (see, e.g., [Bercovier17, Peters08, Reif97]), splines on
triangulations emerge as the natural tool to efficiently deal with problems
where local features has to be detected, modeled, or simulated.

As mentioned in the introduction, low degree splines are preferable due to
their stable behavior and their low computational complexity. In particular,
univariate $C^{2}$ cubic splines are one of the most used tools in modeling,
approximation, and simulation. Constructing $C^{2}$ splines of low degree on
triangulations is a difficult task, but their interest remains unquestionable
in the bivariate setting. We limit ourselves to mention two important
application areas: computer aided surface modeling and numerical simulation.

In computer aided design/manufacturing (CAD/CAM) high quality free-form
surfaces are of utmost importance. The quality of the surfaces can be checked
by different techniques, such as the well-established isophotes [Poeschl84],
to detect irregularities of intrinsic measures of surface smoothness like the
Gaussian curvature or the distribution of the surface normals. For milling
surfaces by five axis machines, second derivatives should not jump too much
across edges and $C^{2}$ smoothness is desirable. In this context, our $C^{2}$
cubic simplex spline representations on triangulations could be beneficial.
The spline surfaces could be constructed by direct (interactive) modeling via
the control net or by data fitting using quasi-interpolation schemes based on
the Marsden-like identity, similar to [Speleers15a]. In CAD/CAM systems it is
common to rely on general parametric surfaces; in our case such surfaces are
specified on each macro-triangle by a control net consisting of triangles and
quadrilaterals. See also [Zhang.17] for the use of simplex splines in the
context of parametric surface reconstruction. As a possible future work, it is
of interest to investigate the interplay with tensor-product (piecewise)
bicubic parametric surfaces in Bernstein–Bézier (or B-spline) form, which are
ubiquitous in industrial applications. In particular, an important question is
whether one can blend standard bicubic Bernstein–Bézier patches with
parametric triangular patches whose components are $C^{2}$ cubic splines
represented in terms of the simplex spline bases introduced in the paper.

Isogeometric analyis (IgA) is a numerical simulation paradigm that extends
finite element analysis (FEA) by providing a true design-through-analysis
methodology [Hughesbook]. The isogeometric paradigm has some important
advantages over traditional FEA. The geometry of the physical domain is
exactly described, so the interaction with the CAD system during any further
refinement process in the analysis phase is eliminated. Moreover, the
discretization spaces possess an inherent higher smoothness (with respect to
the polynomial degree) than classical FEA spaces, leading to a higher accuracy
per degree of freedom [Bressan19, Sande20]. The success of IgA roots in the
above two properties, the latter being even more relevant. Besides the use of
spline spaces based on (local) tensor-product structures and rather involved
multipatch constructions (see, e.g., [Blidia20, Chan18, Kapl19, Sangalli16,
Toshniwal17]), a powerful IgA formulation has been obtained by considering
spline spaces on triangulations (see, e.g., [Cao19, Jaxon14, Speleers12,
Speleers15c, Wang18]). In particular, spline representations obtained from
local Bernstein representations by means of minimal determining sets have been
profitably applied and efficiently implemented via Bézier-extraction
[Jaxon14]. In this context, the space of $C^{2}$ cubic splines defined on the
${{\rm WS}_{3}}$ refinement of a given triangulation is appealing because it
combines low degree and high smoothness. Our simplex spline bases are the
natural counterpart of Bernstein polynomials to define stable global bases by
means of minimal determining sets (see Section 4), and allow for a
straightforward extension of the Bézier-extraction procedure for practical
implementation. Of course, in order to efficiently exploit the potential of
the space ${\mathbb{S}}_{3}^{2}({\mathcal{T}}_{{{\rm WS}_{3}}})$ and its local
representations in terms of simplex spline bases in the context of IgA,
several steps are still missing, for instance, the need for tailored
quadrature rules.

###  5.3 Higher-order extension of the basis

Besides the application-oriented investigations mentioned in the previous
subsection, an interesting follow-up work would be the generalization of the
simplex spline construction to $C^{d-1}$ spline spaces
${\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}})$ on the general ${{\rm
WS}_{d}}$ split of a triangle
$\Delta:=\langle{\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3}\rangle$
for degree $d>3$. Under the assumption of Theorem 2, the dimension can be
written as

$\displaystyle\dim({\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}}))$
$\displaystyle=\frac{(d+2)(d+1)}{2}+3d(d-1)$
$\displaystyle=\frac{3(d+1)d}{2}+\frac{3d(d-1)}{2}+\frac{(d-1)(d-2)}{2}.$

Then, similar to Corollary 4, we may formulate the following Hermite
interpolation problem to characterize the space
${\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}})$: for given data
$f_{k,\alpha,\beta}$, $g_{k,\alpha,l}$, and $h_{\alpha,\beta}$, there is a
unique spline $s\in{\mathbb{S}}_{d}^{d-1}(\Delta_{{{\rm WS}_{d}}})$ such that

$\displaystyle D_{x}^{\alpha}D_{y}^{\beta}s({\boldsymbol{p}}_{k})$
$\displaystyle=f_{k,\alpha,\beta},$ $\displaystyle 0\leq\alpha+\beta\leq
d-1,\quad k=1,2,3,$ $\displaystyle
D_{{\boldsymbol{n}}_{k}}^{\alpha}s({\boldsymbol{q}}_{k,\alpha,l})$
$\displaystyle=g_{k,\alpha,l},$ $\displaystyle\alpha=1,\ldots,d-1,\quad
l=1,\ldots,\alpha,\quad k=1,2,3,$ $\displaystyle
D_{x}^{\alpha}D_{y}^{\beta}s({\boldsymbol{q}})$
$\displaystyle=h_{\alpha,\beta},$ $\displaystyle 0\leq\alpha+\beta\leq d-3,$

where

${\boldsymbol{q}}:=\frac{{\boldsymbol{p}}_{1}+{\boldsymbol{p}}_{2}+{\boldsymbol{p}}_{3}}{3},\quad{\boldsymbol{q}}_{k,\alpha,l}:=\frac{l{\boldsymbol{p}}_{(k\text{
mod }3)+1}+(\alpha-l+1){\boldsymbol{p}}_{((k+1)\text{ mod }3)+1}}{\alpha+1},$

and ${\boldsymbol{n}}_{k}$ is the normal direction of the edge opposite to
vertex ${\boldsymbol{p}}_{k}$. Given a general triangulation ${\mathcal{T}}$,
this scheme can be used to construct a globally $C^{d-1}$ spline of degree $d$
on ${\mathcal{T}}$ where every triangle is refined with the ${{\rm WS}_{d}}$
split. Such a construction is local, in the sense that the spline can be built
on each macro-triangle $\Delta$ of ${\mathcal{T}}$ separately, and the simplex
spline basis would then be useful to represent the corresponding spline piece
on $\Delta$, without considering explicitly the complicated geometry in the
${{\rm WS}_{d}}$ split.

## Acknowledgements

This work was supported by the Beyond Borders Programme of the University of
Rome Tor Vergata through the project ASTRID (CUP E84I19002250005) and by the
MIUR Excellence Department Project awarded to the Department of Mathematics,
University of Rome Tor Vergata (CUP E83C18000100006). The authors are grateful
to the Mathematisches Forschungsinstitut Oberwolfach for the Research in Pairs
support (R1926). C. Manni and H. Speleers are members of Gruppo Nazionale per
il Calcolo Scientifico, Istituto Nazionale di Alta Matematica.

## References

##  Appendix A Appendix

In this appendix, we collect data related to the spline basis functions
$B_{1},\ldots,B_{28}$ in (4) that might be useful for practical computations.
We also provide few data for the alternative spline basis functions
${\widetilde{B}}_{1},\ldots,{\widetilde{B}}_{28}$ in (14).

Figure 10: The simplex spline basis function $B_{1}$ and its support.

Figure 11: The simplex spline basis function $B_{4}$ and its support.

Figure 12: The simplex spline basis function $B_{10}$ and its support.

Figure 13: The simplex spline basis function $B_{16}$ and its support.

Figure 14: The simplex spline basis function $B_{19}$ and its support.

Figure 15: The simplex spline basis function $B_{22}$ and its support.

Figure 16: The simplex spline basis function $B_{28}$ and its support.

###  A.1 Visualization of basis functions

Each simplex spline basis function $B_{i}$ is a piecewise polynomial of degree
three on the partition formed by the complete graph of its knots. The
different types of basis functions are depicted in Figures 10–16. For each
basis function, its support is indicated in the figure and all polynomial
pieces are marked by different colors.

###  A.2 Explicit expressions of basis functions

Figure 17: Numbering of the 58 intersection points ${\boldsymbol{v}}_{k}$ in
the ${{\rm WS}_{3}}$ split.

Here we provide the polynomial expressions of the basis functions $B_{i}$,
$i=1,\ldots,28$, up to symmetries. These polynomials can be expressed in terms
of the barycentric coordinates $(\beta_{1},\beta_{2},\beta_{3})$ with respect
to the macro-triangle $\Delta$. To this end, we define the following
polynomials:

$\displaystyle l_{1,2}$ $\displaystyle:=\beta_{3}^{3},$ $\displaystyle\quad
l_{1,3}$ $\displaystyle:=\beta_{2}^{3},$ $\displaystyle\quad l_{1,6}$
$\displaystyle:=(2\beta_{3}-\beta_{2})^{3},$ (28) $\displaystyle l_{2,3}$
$\displaystyle:=\beta_{1}^{3},$ $\displaystyle l_{2,9}$
$\displaystyle:=(\beta_{1}-2\beta_{3})^{3},$ $\displaystyle l_{4,6}$
$\displaystyle:=(3\beta_{2}-3\beta_{3}-1)^{3},$ $\displaystyle l_{4,8}$
$\displaystyle:=(3\beta_{1}-3\beta_{2}-1)^{3},$ $\displaystyle\quad l_{4,9}$
$\displaystyle:=(3\beta_{1}-2)^{3},$ $\displaystyle\quad l_{5,6}$
$\displaystyle:=(3\beta_{2}-2)^{3},$ $\displaystyle l_{5,7}$
$\displaystyle:=(3\beta_{2}-3\beta_{1}-1)^{3},$ $\displaystyle l_{5,8}$
$\displaystyle:=(3\beta_{1}-1)^{3},$ $\displaystyle l_{5,9}$
$\displaystyle:=(3\beta_{1}-3\beta_{3}-1)^{3},$ $\displaystyle l_{6,8}$
$\displaystyle:=(3\beta_{3}-3\beta_{1}-1)^{3},$ $\displaystyle l_{6,9}$
$\displaystyle:=(3\beta_{3}-1)^{3},$ $\displaystyle l_{7,8}$
$\displaystyle:=(3\beta_{3}-2)^{3},$ $\displaystyle l_{7,9}$
$\displaystyle:=(3\beta_{3}-3\beta_{2}-1)^{3}.$

Let ${\boldsymbol{v}}_{k}$ be the vertices of $\Delta_{{{\rm WS}_{3}}}$
visualized in Figure 17. The equation $l_{i,j}=0$ represents the cubic power
of the straight line connecting the points ${\boldsymbol{v}}_{i}$ and
${\boldsymbol{v}}_{j}$. Then, the polynomial pieces of the basis functions are
described in Table 1, up to symmetries. The corresponding regions are
specified as the convex hull of the points ${\boldsymbol{v}}_{k}$.

###  A.3 Hermite data of basis functions

With the aim of showing linear independence of the spline functions
$B_{1},\ldots,B_{28}$, we have set up a Hermite interpolation problem in the
proof of Theorem 3. The Hermite data is computed through the operators
$\rho_{1},\ldots,\rho_{28}$; they are defined in (5), (6), and (8). The values
of these operators applied to the $B_{i}$’s are collected in Table 2. The
table also provides those values for the spline functions
${\widetilde{B}}_{22},\ldots,{\widetilde{B}}_{28}$. The other values are
obtained through the identity ${\widetilde{B}}_{i}=B_{i}$, $i=1,\ldots,21$.
Finally, we collect some additional second derivative values of the
${\widetilde{B}}_{i}$’s in Table 3, where

$\displaystyle\rho_{29}(f)$
$\displaystyle:=D^{2}_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{3,1}),$
$\displaystyle\quad\rho_{30}(f)$
$\displaystyle:=D^{2}_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}f({\boldsymbol{p}}_{3,1}),$
$\displaystyle\quad\rho_{31}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{3,1}),$
(29) $\displaystyle\rho_{32}(f)$
$\displaystyle:=D^{2}_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{3,2}),$
$\displaystyle\rho_{33}(f)$
$\displaystyle:=D^{2}_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}f({\boldsymbol{p}}_{3,2}),$
$\displaystyle\rho_{34}(f)$
$\displaystyle:=D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{3}}D_{{\boldsymbol{p}}_{1}{\boldsymbol{p}}_{2}}f({\boldsymbol{p}}_{3,2}).$

These are useful to prove Theorem 10.

Basis | Region | Expression
---|---|---
$B_{1}$ | 1,4,9 | $l_{4,9}$
$B_{4}$ | 4,5,9 | $\frac{1}{4}l_{5,9}$
| 1,4,9 | $\frac{1}{4}l_{5,9}-2l_{4,9}$
$B_{16}$ | 5,8,53 | $\frac{1}{2}l_{5,8}$
| 4,5,53 | $\frac{1}{2}l_{5,8}-\frac{1}{2}l_{5,9}$
| 8,9,53 | $\frac{1}{2}l_{5,8}-\frac{1}{2}l_{4,8}$
| 1,4,9 | $\frac{27}{2}\beta_{2}\beta_{3}(9\beta_{1}-5)$
| 9,4,53 | $\frac{27}{2}\beta_{2}\beta_{3}(9\beta_{1}-5)-4l_{4,9}$
$B_{28}$ | 4,5,55 | $\frac{81}{2}l_{1,2}$
| 6,7,57 | $\frac{81}{2}l_{2,3}$
| 8,9,54 | $\frac{81}{2}l_{1,3}$
| 5,6,56 | $-\frac{3}{2}l_{5,6}$
| 7,8,58 | $-\frac{3}{2}l_{7,8}$
| 4,53,9 | $-\frac{3}{2}l_{4,9}$
| 4,55,23 | $\frac{81}{2}l_{1,2}+\frac{1}{2}l_{4,6}$
| 5,32,55 | $\frac{81}{2}l_{1,2}+\frac{1}{2}l_{5,9}$
| 5,56,32 | $-\frac{3}{2}l_{5,6}+\frac{1}{2}l_{5,7}$
| 6,57,40 | $\frac{81}{2}l_{2,3}+\frac{1}{2}l_{6,8}$
| 8,54,50 | $\frac{81}{2}l_{1,3}+\frac{1}{2}l_{4,8}$
| 9,14,54 | $\frac{81}{2}l_{1,3}+\frac{1}{2}l_{7,9}$
| 4,23,53 | $-\frac{3}{2}l_{4,9}+\frac{1}{2}l_{4,8}$
| 6,40,56 | $-\frac{3}{2}l_{5,6}+\frac{1}{2}l_{4,6}$
| 7,58,45 | $-\frac{3}{2}l_{7,8}+\frac{1}{2}l_{7,9}$
| 8,50,58 | $-\frac{3}{2}l_{7,8}+\frac{1}{2}l_{6,8}$
| 7,45,57 | $\frac{81}{2}l_{2,3}+\frac{1}{2}l_{5,7}$
| 9,53,14 | $-\frac{3}{2}l_{4,9}+\frac{1}{2}l_{5,9}$
| 14,53,23,10 | $-\frac{3}{2}l_{4,9}+\frac{1}{2}(l_{4,8}+l_{5,9})$
| 23,55,32,10 | $\frac{81}{2}l_{1,2}+\frac{1}{2}(l_{4,6}+l_{5,9})$
| 32,56,40,10 | $-\frac{3}{2}l_{5,6}+\frac{1}{2}(l_{5,7}+l_{4,6})$
| 40,57,45,10 | $\frac{81}{2}l_{2,3}+\frac{1}{2}(l_{6,8}+l_{5,7})$
| 45,58,50,10 | $-\frac{3}{2}l_{7,8}+\frac{1}{2}(l_{6,8}+l_{7,9})$
| 50,54,14,10 | $\frac{81}{2}l_{1,3}+\frac{1}{2}(l_{4,8}+l_{7,9})$
|  |

Basis | Region | Expression
---|---|---
$B_{10}$ | 1,4,9 | $\frac{9}{4}\beta_{2}^{2}(6\beta_{1}-5\beta_{2}-12\beta_{3})$
| 4,5,9 | $\frac{9}{2}l_{2,9}-\frac{3}{4}l_{5,9}$
| 5,2,9 | $\frac{9}{2}l_{2,9}$
$B_{19}$ | 1,20,9 | $\frac{45}{4}l_{1,3}$
| 2,6,35 | $\frac{45}{4}l_{2,3}$
| 6,9,24 | $-\frac{5}{2}l_{6,9}$
| 1,4,20 | $\frac{45}{4}(l_{1,3}+l_{1,6})$
| 5,2,35 | $\frac{45}{4}(l_{2,3}-l_{2,9})$
| 6,24,32 | $-\frac{5}{2}l_{6,9}+\frac{45}{4}l_{1,6}$
| 9,23,24 | $-\frac{5}{2}l_{6,9}-\frac{45}{4}l_{2,9}$
| 9,20,23 | $\frac{45}{4}l_{1,3}+\frac{5}{4}l_{4,9}$
| 6,32,35 | $\frac{45}{4}l_{2,3}+\frac{5}{4}l_{5,6}$
| 4,55,23,20 | $\frac{45}{4}(l_{1,3}+l_{1,6})+\frac{5}{4}l_{4,9}$
| 5,35,32,55 | $\frac{45}{4}(l_{2,3}-l_{2,9})+\frac{5}{4}l_{5,6}$
| 4,5,55 | $\frac{45}{2}\beta_{3}(\beta_{3}^{2}+6\beta_{1}\beta_{2}-1)$
| 55,32,24,23 | $-\frac{5}{2}l_{6,9}+\frac{45}{4}(l_{1,6}-l_{2,9})$
$B_{22}$ | 5,6,32 | $-\frac{1}{2}l_{5,6}$
| 5,32,55 | $-\frac{1}{2}l_{5,6}-l_{5,8}$
| 4,5,55 | $27\beta_{3}^{2}(3\beta_{1}-1)$
| 4,55,23,21 | $27\beta_{3}^{2}(3\beta_{1}-1)+\frac{2}{3}l_{4,6}$
| 1,4,20 | $27\beta_{3}^{2}(3\beta_{2}-2\beta_{3})$
| 4,21,20 | $27\beta_{3}^{2}(3\beta_{2}-2\beta_{3})+2l_{4,9}$
| 6,32,25 | $-\frac{1}{2}l_{5,6}+\frac{2}{3}l_{4,6}$
| 55,32,25,23 | $-\frac{1}{2}(3\beta_{2}-2)^{3}+\frac{2}{3}l_{4,6}-l_{5,8}$
| 6,8,10 | $-\frac{1}{12}l_{6,8}$
| 8,14,10 | $-\frac{1}{12}l_{6,8}-l_{5,8}$
| 8,9,14 | $\frac{27}{4}\beta_{2}^{2}(7\beta_{1}+\beta_{3}-3)$
| 9,53,14 | $\frac{27}{4}\beta_{2}^{2}(7\beta_{1}+\beta_{3}-3)+2l_{6,9}$
| 1,20,9 | $\frac{27}{4}\beta_{2}^{2}(6\beta_{3}-\beta_{2})$
| 9,20,21,53,9 | $\frac{27}{4}\beta_{2}^{2}(6\beta_{3}-\beta_{2})+2l_{4,9}$
| 6,10,25 | $-\frac{1}{12}l_{6,8}+2l_{6,9}$
| 23,25,10,14,53 | $-\frac{1}{12}l_{6,8}+2l_{6,9}-l_{5,8}$
| 21,23,53 | $\frac{27}{4}\beta_{2}^{2}(6\beta_{3}-\beta_{2})+2l_{4,9}-\frac{1}{3}l_{4,8}$

Table 1: Explicit expressions of the polynomial pieces of the basis functions $B_{i}$ in terms of the functions defined in (28). The regions of the polynomial pieces are specified as the convex hull of the points ${\boldsymbol{v}}_{k}$ numbered in Figure 17. | $\rho_{1}$ | $\rho_{2}$ | $\rho_{3}$ | $\rho_{4}$ | $\rho_{5}$ | $\rho_{6}$ | $\rho_{7}$ | $\rho_{8}$ | $\rho_{9}$ | $\rho_{10}$ | $\rho_{11}$ | $\rho_{12}$ | $\rho_{13}$ | $\rho_{14}$ | $\rho_{15}$ | $\rho_{16}$ | $\rho_{17}$ | $\rho_{18}$ | $\rho_{19}$ | $\rho_{20}$ | $\rho_{21}$ | $\rho_{22}$ | $\rho_{23}$ | $\rho_{24}$ | $\rho_{25}$ | $\rho_{26}$ | $\rho_{27}$ | $\rho_{28}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$B_{1}$ | 1 | 0 | 0 | -9 | -9 | 0 | 0 | 0 | 0 | 54 | 54 | 0 | 0 | 0 | 0 | 54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$B_{2}$ | 0 | 1 | 0 | 0 | 0 | -9 | -9 | 0 | 0 | 0 | 0 | 54 | 54 | 0 | 0 | 0 | 54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$B_{3}$ | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -9 | -9 | 0 | 0 | 0 | 0 | 54 | 54 | 0 | 0 | 54 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
$B_{4}$ | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | -81 | 0 | 0 | 0 | 0 | 0 | -54 | 0 | 0 |  -$\frac{27}{32}$ | 0 | 0 | $\frac{75}{2}$ | 0 | 0 | 0 | 0 | 0 | 0
$B_{5}$ | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | -81 | 0 | 0 | 0 | 0 | -54 | 0 | 0 | 0 | 0 |  -$\frac{27}{32}$ | 0 | $\frac{75}{2}$ | 0 | 0 | 0 | 0 | 0
$B_{6}$ | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | -81 | 0 | 0 | 0 | 0 | -54 | 0 | 0 |  -$\frac{27}{32}$ | 0 | 0 | 0 | $\frac{75}{2}$ | 0 | 0 | 0 | 0
$B_{7}$ | 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | -81 | 0 | 0 | 0 | -54 | 0 |  -$\frac{27}{32}$ | 0 | 0 | 0 | 0 | 0 | $\frac{75}{2}$ | 0 | 0 | 0
$B_{8}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | -81 | 0 | 0 | 0 | -54 | 0 | 0 |  -$\frac{27}{32}$ | 0 | 0 | 0 | 0 | $\frac{75}{2}$ | 0 | 0
$B_{9}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 0 | 0 | -81 | 0 | 0 | -54 | 0 |  -$\frac{27}{32}$ | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{75}{2}$ | 0
$B_{10}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  -$\frac{189}{32}$ | 0 | 0 | $\frac{31}{2}$ | 0 | 0 | 49 | 0 | 0 | 0
$B_{11}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  -$\frac{189}{32}$ | 0 | $\frac{31}{2}$ | 0 | 0 | 49 | 0 | 0
$B_{12}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  -$\frac{189}{32}$ | 0 | 0 | 0 | $\frac{31}{2}$ | 0 | 0 | 49 | 0
$B_{13}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |  -$\frac{189}{32}$ | 0 | 0 | 49 | 0 | 0 | $\frac{31}{2}$ | 0 | 0 | 0
$B_{14}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |  -$\frac{189}{32}$ | 0 | 49 | 0 | 0 | $\frac{31}{2}$ | 0 | 0
$B_{15}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 |  -$\frac{189}{32}$ | 0 | 0 | 0 | 49 | 0 | 0 | $\frac{31}{2}$ | 0
$B_{16}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 0 | 0 | $\frac{9}{8}$ | 0 | $\frac{9}{8}$ | -63 | -63 | 0 | 0 | 0 | 0 | 0
$B_{17}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 0 | $\frac{9}{8}$ | $\frac{9}{8}$ | 0 | 0 | 0 | -63 | -63 | 0 | 0 | 0
$B_{18}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 0 | $\frac{9}{8}$ | $\frac{9}{8}$ | 0 | 0 | 0 | 0 | -63 | -63 | 0
$B_{19}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{45}{4}$ | 0 | 0 | -120 | 0 | 0 | -120 | 0 | 0 | 0
$B_{20}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{45}{4}$ | 0 | 0 | 0 | -120 | 0 | 0 | -120 | 0
$B_{21}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{45}{4}$ | 0 | -120 | 0 | 0 | -120 | 0 | 0
$B_{22}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 27 | 0 | 0 | 0 | 0 | $\frac{1}{12}$
$B_{23}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 54 | 0 | 0 | 0 | 0 | $\frac{1}{12}$
$B_{24}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 27 | 0 | 0 | $\frac{1}{12}$
$B_{25}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 54 | 0 | 0 | $\frac{1}{12}$
$B_{26}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 54 | 27 | $\frac{1}{12}$
$B_{27}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 | 54 | $\frac{1}{12}$
$B_{28}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | $\frac{1}{2}$
${\widetilde{B}}_{22}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 81 | 0 | 0 | 0 | 0 | 0 | $\frac{1}{4}$
${\widetilde{B}}_{23}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 81 | 0 | 0 | 0 | 0 | $\frac{1}{4}$
${\widetilde{B}}_{24}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 81 | 0 | 0 | 0 | $\frac{1}{4}$
${\widetilde{B}}_{25}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 81 | 0 | 0 | $\frac{1}{4}$
${\widetilde{B}}_{26}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 81 | 0 | $\frac{1}{4}$
${\widetilde{B}}_{27}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 81 | $\frac{1}{4}$
${\widetilde{B}}_{28}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |  -$\frac{1}{2}$
Table 2: Values of $\rho_{j}(B_{i})$ and $\rho_{j}({\widetilde{B}}_{i})$ for $i,j=1,\ldots,28$, where $\rho_{j}$ is defined in (5), (6), and (8). Note that ${\widetilde{B}}_{i}=B_{i}$ for $i=1,\ldots,21$. | $\rho_{29}$ | $\rho_{30}$ | $\rho_{31}$ | $\rho_{32}$ | $\rho_{33}$ | $\rho_{34}$
---|---|---|---|---|---|---
${\widetilde{B}}_{1}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{2}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{3}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{4}$ | $\frac{27}{2}$ | 54 | 27 | 0 | 0 | 0
${\widetilde{B}}_{5}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{6}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{7}$ | 0 | 0 | 0 | $\frac{27}{2}$ | $\frac{27}{2}$ |  -$\frac{27}{2}$
${\widetilde{B}}_{8}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{9}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{10}$ |  -$\frac{45}{2}$ | 0 | -27 | 9 | 81 | 27
${\widetilde{B}}_{11}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{12}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{13}$ | 9 | 36 | -18 |  -$\frac{45}{2}$ | $\frac{63}{2}$ | $\frac{9}{2}$
${\widetilde{B}}_{14}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{15}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{16}$ | 0 | -81 | -27 | 0 | 0 | 0
${\widetilde{B}}_{17}$ | 0 | 0 | 0 | 0 | -27 | 27
${\widetilde{B}}_{18}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{19}$ | 0 | -90 | 45 | 0 | -180 | -45
${\widetilde{B}}_{20}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{21}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{22}$ | 0 | 81 | 0 | 0 | 0 | 0
${\widetilde{B}}_{23}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{24}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{25}$ | 0 | 0 | 0 | 0 | 81 | 0
${\widetilde{B}}_{26}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{27}$ | 0 | 0 | 0 | 0 | 0 | 0
${\widetilde{B}}_{28}$ | 0 | 0 | 0 | 0 | 0 | 0
Table 3: Values of $\rho_{j}({\widetilde{B}}_{i})$ for $i=1,\ldots,28$ and
$j=29,\ldots,34$, where $\rho_{j}$ is defined in (29).