On Hermitian interpolation of first order data with

locally generated C1-splines over triangular meshes

L.L. STACHÓ

Abstract. Given a system of triangles in the plane $\mathbb{R}^{2}$ along with
given data of function and gradient values at the vertices, we describe the
general pattern of local linear methods invoving only four smooth standard
shape functions which results in a spline function fitting the given value and
gradient data value with ${\cal C}^{1}$-coupling along the edges of the
triangles. We characterize their invariance properties with relavance for the
construction of interpolation surfaces over triangularizations of scanned 3D
data. The numerically simplest procedures among them leaving invarant all
polynomials of 2-variables with degree 0 resp 1 involve only polynomials of
5-th resp. 6-th degree, but the characteizations give rise to a huge variety
of procedures with non-polynomial shape functions.

000Received March 29, 2023. AMS 2010 Subject Classification: 65D07; 41A15;
65D15. Key words: ${\cal C}^{1}$-spline, triangular mesh, 2D Hermite
interpolation, reduced side derivatives (RSD), shape function, range shift
property, affinity invariance.

1\. Introduction

Recently [1] we published a ${\cal C}^{1}$-spline interpolation method over
2-dimensional triangular meshes with polynomials of 5-th degree with low
operational costs: using first order data (value and gradient) at the mesh
vertices for input. The result relies on the following basic tool: given a
non-degenerate triangle with vertices ${\bf p}_{1},{\bf p}_{2},{\bf
p}_{3}\in\mathbb{R}^{2}$ along with three values
$f_{1},f_{2},f_{2}\in\mathbb{R}$ respectively three linear functionals
$A_{1},A_{2},A_{3}\in{\cal L}(\mathbb{R}^{2},\mathbb{R})$ and three vectors
${\bf u}_{1},{\bf u}_{2},{\bf u}_{3}\in\mathbb{R}^{2}$ such that ${\bf
u}_{k}\not\parallel{\bf p}_{i}-{\bf p}_{j}$, we can construct a polynomial
$F:\mathbb{R}^{2}\to\mathbb{R}$ of 5-th degree of the form $F=F_{0}-H$ where

$\displaystyle F_{0}({\bf x})=\sum_{i=1}^{3}\Big{[}\Phi\big{(}\lambda_{i}({\bf
x})\big{)}f_{i}+\Theta\big{(}\lambda_{i}({\bf x})\big{)}A_{i}({\bf x}-{\bf
p}_{1})\Big{]},$ $None$ $\displaystyle H({\bf x})=30\lambda_{1}({\bf
x})^{2}\lambda_{2}({\bf x})^{2}\lambda_{3}^{2}({\bf
x})^{2}\sum_{k=1}^{3}\lambda_{k}({\bf x})^{-1}{M_{k}{\bf u}_{k}\over G_{k}{\bf
u}_{k}}$

in terms of the barycentric weights
$\lambda_{1},\lambda_{2},\lambda_{3}:\mathbb{R}^{2}\to\mathbb{R}$ of the
triangle ${\bf T}={\rm Conv}\\{{\bf p}_{1},{\bf p}_{2},{\bf p}_{3}\\}$
associated with the vertices and the shape functions

$\Phi(t)=t^{3}(10-15t+6t^{2}),\qquad\Theta(t)=t^{3}(4-3t)$ $None$

for a Hermite interpolation $F_{0}$ on ${\bf T}$ with the data $f_{i},A_{i}$
$(i=1,2,3)$. The correction term $H$ is defined by means of the linear
functionals $\mathbb{R}^{2}\to\mathbb{R}$

$\displaystyle G_{k}{\bf u}=\lambda_{k}^{\prime}{\bf u}={d\over
dt}\Big{|}_{t=0}\lambda_{k}({\bf x}+t{\bf u})=\lambda_{k}({\bf x}+{\bf
u})-\lambda_{k}({\bf x}),$ $\displaystyle M_{k}{\bf
u}=\sum_{\\{i,j\\}\\!=\\{1,2,3\\}\\!\setminus\\!\\{k\\}}[G_{i}{\bf
u}]\big{(}30f_{i}+12A_{i}({\bf p}_{j}-{\bf p}_{i})\big{)}.$

It is worth to notice that the function $F_{0}$ in $(1.1)$ restricted to the
edge $[{\bf p}_{i},{\bf p}_{k}]$ depends only on the terms
$f_{i},f_{j},A_{i},A_{j}$ and the impact of the correction by adding $H$
results in the reduced side derivatives (RSD)

$F^{\prime}({\bf x}_{t}){\bf u_{k}}=\Theta(t)A_{i}{\bf
u}_{k}+\Theta(1-t)A_{j}{\bf u}_{k}\quad\hbox{for}\ \ {\bf x}_{t}=t{\bf
p}_{i}+(1-t){\bf p}_{j}.$

As a consequence, if we choose any family $\big{\\{}{\bf u}_{{\bf E}}:{\bf E}$
being a mesh edge$\big{\\}}$ of vectors such that ${\bf u}_{\\{\bf
E\\}}\not\parallel{\bf E}$, by applying the construction $(1.1)$ with the
associated data over every triangle of the mesh, we obtain a ${\cal
C}^{1}$-smooth (continuously differentable) function on the union of the mesh
triangles.

The shape functions $\Phi,\Theta$ and also the RSD method described above
appeared in [1] without heuristic introduction. Actually they arose from our
earlier study [2] with somewhat restrictive postulates on the possible
polynomial ${\cal C}^{1}$-interpolations over triangular meshes based on
computer algebraic analysis of technics developed in [3],[4],[5],[6].

Our goal here is to characterize all RSD methods with shape functions
$\Psi_{0},\Psi_{1}\in{\cal C}^{1}\big{(}[0,1]\big{)}$ instead of $(1.2)$. This
requires a completely different approach as that in [2] relying heavily upon
polynomial identities. From our main result Theorem $3.5$ it turns out that
there is a very general pattern behind $(1.1)$ in the form

$\displaystyle
F_{0}=\sum_{k=1}^{3}\Big{[}\Psi_{0}\big{(}\lambda_{k}\big{)}f_{k}+\Psi_{1}\big{(}\lambda_{k}\big{)}A_{k}({\bf
x}-{\bf p}_{k})\Big{]},$ $None$ $\displaystyle
H=\hskip-8.53581pt\sum_{\\{\ell,m,n\\}=\\{1,2,3\\}}\biggl{\\{}\\!f_{\ell}{G_{\ell}{\bf
u}_{n}\over G_{n}{\bf u}_{n}}\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})\
+$ $\displaystyle\hskip 42.67912pt+A_{\ell}({\bf p}_{m}\\!\\!-\\!{\bf
p}_{\ell}){G_{\ell}{\bf u}_{n}\over G_{n}{\bf
u}_{n}}\chi_{1}(\lambda_{\ell},\lambda_{m},\lambda_{n})\\!\biggr{\\}}.$

If $\Psi_{0},\Psi_{1}$ fulfill the minimal necessary condition $(2.1)$ for
giving rise to a Hermite interpolation of the type $F_{0}$ in $(1.1)$ of first
order then we can find plenty of suitable functions $\chi_{0},\chi_{1}\in{\cal
C}^{1}_{0}(\mathbb{R}_{+}^{3})$, namely those satisfying only the not too
restrictive conditions $(3.6),\\!(3.7)$, in order for $H$ being an RSD
correction for $F_{0}$. This degree of freedom enables various canonical
constructions for the modifiers $\chi_{0},\chi_{1}$ in terms of suitable shape
functions $\Psi_{0},\Psi_{1}$ to ensure specific geometric properties of the
interpolation operator $\hbox{{\Goth F}}:\big{\\{}({\bf
p}_{k},f_{k},A_{k}):k=1,2,3\big{\\}}\mapsto\big{[}F_{0}+H$ by $(1.3)\big{]}$.

We complete the paper with the investigation of two essential properties of
the interpolation F having obvious importance in applications used to
construct smooth surfaces in the form of the graph of a function
$\mathbb{R}^{2}\to\mathbb{R}$ passing through a finite family of points in
$\mathbb{R}^{3}$ (obtained as vertices of a tringularization from scanned
data):

(i) The range shift property $\hbox{{\Goth F}}\big{\\{}({\bf
p}_{k},1,0):k=1,2,3\big{\\}}\equiv 1$, that is all constant functions remain
invariant when interpolated with F. In this case the graph of $\hbox{{\Goth
F}}\big{\\{}({\bf p}_{k},f_{k}+c,f^{\prime}({\bf p}_{k})):k=1,2,3\big{\\}}$ is
just a shifted form of the graph of $\hbox{{\Goth F}}\big{\\{}({\bf
p}_{k},f_{k},({\bf p}_{k})):k=1,2,3\big{\\}}$ for any function $f\in{\cal
C}^{1}(\mathbb{R}^{2})$.

(ii) Affinity invariance: $\hbox{{\Goth F}}\big{\\{}({\bf p}_{k},A{\bf
x}+b,A):k=1,2,3\big{\\}}\equiv A{\bf x}+b$. That is all affine
(constant+linear) functions remain invariant when interpolated with F. In
particular if the graph of a function ${\rm Conv}\\{{\bf p}_{1},{\bf
p}_{2},{\bf p}_{3}\\}\to\mathbb{R}$ is a 3D plane triangle then the function
obtained with interpolation by F has the same graph.

In Section 4 we give a parametric classification of the procedures with range
shift property. Concerning affinity invariance, we have no complete results
yet: in Section 5 we provide a several necessary and sufficient algebraic
conditions. As a by no means obvious fact, it turns out that the procedure
$(1.1)$ with the polynomials $(1.2)$ has range shift property but fails to be
affinity invariant. We also provide an affinity invariant procedure with
polynomials of 6-th degree: namely that of the form $(1.3)$ with
$\Psi_{0}=\Psi_{1}=\Phi$ and
$\chi_{0}(t_{1},t_{2},t_{3})=30t_{1}^{2}t_{2}^{2}t_{3}$,
$\chi_{1}(t_{1},t_{2},t_{3})=30t_{1}^{2}t_{2}^{3}t_{3}+15t_{1}^{2}t_{2}^{2}t_{3}^{2}$.

2\. Preliminaries

For standard terminology, $\mathbb{R}^{N}$ resp. $\mathbb{R}^{N}_{+}$ stand
for the set of all real resp. non-negative $N$-tuples
$[\xi_{1},\ldots,\xi_{N}]$. Actually, we shall only be interested in the cases
of dimensions $N=1,2,3$.

Given a subset $\Omega$ of $\mathbb{R}^{N}$, we write ${\cal C}(\Omega)$ for
the family of all continuous functions with domain $\Omega$. If the interior
$\Omega^{\circ}$ of $\Omega$ is dense in $\Omega$, we define ${\cal
C}^{1}(\Omega)$ as the set consisting of all functions $f\in{\cal C}(\Omega)$
with continuous partial derivatives $D_{1}f,\ldots,D_{N}f$ on $\Omega^{\circ}$
admitting a continuous extension to $\Omega$ with the value denoted also by
$D_{k}f({\bf x})$ at the points ${\bf x}\in\Omega\setminus\Omega^{\circ}$. As
a folklore consequence of Whitney’s extension theorem [7], if $\Omega$ is
closed in $\mathbb{R}^{N}$ then every function $f\in{\cal C}^{1}(\Omega)$ can
be regarded as the restriction of a continuously differentiable function
defined on $\mathbb{R}^{N}$.

For any function $f\in{\cal C}^{1}(\Omega)$, we write $f^{\prime}$ for its
Fréchet derivative defined at any point ${\bf x}\in\Omega$ as the linear
functional

$f^{\prime}({\bf x}){\bf u}=\sum_{k=1}^{N}D_{k}f({\bf
x})\upsilon_{k}\qquad\big{(}{\bf
u}=[\upsilon_{1},\ldots,\upsilon_{N}]\big{)}.$

In particular $f^{\prime}({\bf x}){\bf u}=\lim\limits_{t\to
0}t^{-1}\big{[}f({\bf x}+t{\bf u})-f{\bf u}\big{]}$ is the familiar
directional derivative at the points ${\bf x}\in\Omega^{\circ}$. If $\Omega$
is a closed set, we define ${\cal C}_{0}^{k}(\Omega)=\big{\\{}f\in{\cal
C}^{k}(\Omega):f({\bf x})=0\ \big{(}{\bf
x}\in\Omega\setminus\Omega^{o}\big{)}\big{\\}}$ $(k=0,1)$.

Given any subset ${\bf P}\subset\mathbb{R}^{2}$, we write ${\rm Conv}({\bf
P})$ for its convex hull. If ${\bf P}=\\{{\bf p}_{1},{\bf p}_{2},{\bf
p}_{3}\\}$ and the triangle ${\bf T}={\rm Conv}({\bf P})$ is non-degenerate,
the barycentric weights (weight functions associated to the vertices) [8] are
the functions

$\lambda_{i}=\lambda_{{\bf p}_{i}}^{{\bf T}}:{\bf x}\mapsto{{\rm
det}\big{[}{\bf x}-{\bf p}_{j},{\bf x}-{\bf p}_{k}\big{]}\over{\rm
det}\big{[}{\bf p}_{i}-{\bf p}_{j},{\bf p}_{i}-{\bf
p}_{k}\big{]}}\quad\hbox{with}\ (i,j,k)\in S_{3}$

where

$S_{3}=\big{\\{}(1,2,3),(2,1,3),(2,3,1),(3,2,1),(3,1,2),(1,3,2)\big{\\}}$

denotes the set of all pemutations of the indices $1,2,3$. For the sake of
brevity, in the sequel we omit the background parameters in most formulas
(like writing $\lambda_{i}\\!=\\!\lambda_{{\bf p}_{i}}^{{\bf T}}$ above)
without danger of confusion. It is well-known that the mapping ${\bf
x}\mapsto[\lambda_{1}({\bf x}),\lambda_{2}({\bf x}),\lambda_{3}({\bf x})]$ is
a homeomorphism between ${\bf T}$ and the 3D-unit simplex

$\Delta_{3}:=\big{\\{}[t_{1},t_{2},t_{3}]\in\mathbb{R}_{+}^{3}:t_{1}+t_{2}+t_{3}=1\big{\\}},$

moreover $[\lambda_{1}({\bf x}),\lambda_{2}({\bf x}),\lambda_{3}({\bf x})]$ is
the unique triple of non-negative numbers such that ${\bf
x}\\!=\\!\sum_{k}\lambda_{k}({\bf x}){\bf p}_{k}$ and
$\sum_{k}\lambda_{k}({\bf x})\\!=\\!1$. Furthermore
$\lambda_{1},\lambda_{2},\lambda_{3}:\mathbb{R}^{2}\\!\to\\!\mathbb{R}$ are
affine (linear+constant) functions with the necessarily constant Fréchet
derivatives

$G_{i}{\bf u}:=\lambda_{i}^{\prime}({\bf x})={d\over
dt}\Big{|}_{t=0}\lambda_{i}({\bf x}+t{\bf u})\quad\hbox{independently of }{\bf
x}.$

Given any pair ${\bm{\Psi}}=[\Psi_{0},\Psi_{1}]$ with
$\Psi_{0},\Psi_{1}\in{\cal C}^{1}\big{(}[0,1])$ such that

$\Psi_{0}(0)=\Psi_{0}^{\prime}(0)=\Psi_{1}(0)=\Psi_{1}^{\prime}(0)=\Psi_{0}^{\prime}(1)=0,\quad\Psi_{0}(1)=\Psi_{1}(1)=1$
$None$

we introduce the basic triangular interpolation of first order with the shape
functions $\Psi_{0},\Psi_{1}$ as the operator

$\hbox{\Goth F}_{0}^{\bm{\Psi}}:\big{\\{}({\bf
p}_{k},f_{k},A_{k})\big{\\}}_{k=1}^{3}\mapsto\sum_{k=1}^{3}\Big{\\{}\Psi_{0}\big{(}\lambda_{k}\big{)}f_{k}\\!+\\!\Psi_{1}\big{(}\lambda_{k}\big{)}A_{k}\big{(}{\bf
x}\\!-\\!{\bf p}_{k}\big{)}\Big{\\}}$ $None$

defined for all first order function germs $\big{\\{}({\bf
p}_{k},f_{k},A_{k}):k\\!=\\!1,2,3\big{\\}}$ with ${\bf T}={\rm Conv}\\{{\bf
p}_{1},{\bf p}_{2},{\bf p}_{1}\\}\subset\mathbb{R}^{2}$ being a non-degenerate
triangle, $f_{k}\in\mathbb{R}$ and $A_{k}\in{\cal
L}(\mathbb{R}^{2},\mathbb{R})$ in terms of the weights
$\lambda_{k}=\lambda_{k}^{\bf T}$. By definition, the domain of the function
$F_{0}=\hbox{\Goth F}_{0}^{\bm{\Psi}}\big{\\{}({\bf
p}_{k},f_{k},A_{k})\big{\\}}_{k=1}^{3}$ is only the triangle ${\bf T}$ and
$F_{0}\in{\cal C}^{1}({\bf T})$. By straightforward calculation, for its
Fréchet derivative we have

$F_{0}^{\prime}=\sum_{k=1}^{3}\Big{\\{}\big{[}\Psi_{0}^{\prime}\big{(}\lambda_{k}\big{)}f_{k}+\big{[}\Psi_{1}^{\prime}\big{(}\lambda_{k}\big{)}A_{k}\big{(}{\bf
x}-{\bf
p}_{k}\big{)}\big{]}G_{k}+\Psi_{1}\big{(}\lambda_{k}\big{)}A_{k}\Big{\\}}.$
$None$

2.4. Remark. (a) Conditions (2.1) do not restrict the value of
$\Psi_{1}^{\prime}(1)$.

(b) In view of $(2.3)$, it is not hard to see that $(2.1)$ is suffiecient and
necessary for $F_{0}$ to satisfy the relations

$F_{0}\big{(}{\bf p}_{i}\big{)}=f_{i},\quad F_{0}^{\prime}\big{(}{\bf
p}_{i}\big{)}=A_{i}\qquad(i=1,2,3)$ $None$

under any choice of the first order function germs $\big{\\{}{\bf
p}_{k},f_{k},A_{k})\big{\\}}_{k=1}^{3}$.

(c) We can express the terms $A_{k}({\bf x}-{\bf p}_{k})$ appearing in
$(1.3)$, $(2.2)$ resp. $(2.5)$ in the form of linear combination of the
weights due to the identity ${\bf x}=\sum_{i=1}^{3}\lambda_{i}({\bf x}){\bf
p}_{i}$ as

$A_{k}({\bf x}-{\bf p}_{k})=\sum_{i=1}^{3}A_{k}({\bf p}_{i}-{\bf
p}_{k})\lambda_{i}.$

2.6. Definition. Henceforth we say that ${\bm{\Psi}}=[\Psi_{0},\Psi_{1}]$ is a
pair of admissible shape functions if $\Psi_{0},\Psi_{1}\in{\cal
C}^{1}([0,1])$ and $(2.1)$ holds. We also say for short that $\big{\\{}({\bf
p}_{k},f_{k},A_{k})\big{\\}}_{k=1}^{3}$ is an admissible function germ if
${\bf p}_{1},{\bf p}_{2},{\bf p}_{3}\in\mathbb{R}^{2}$ form a non-degenete
triangle, $f_{k}\in\mathbb{R}$ and $A_{k}\in{\cal
L}(\mathbb{R}^{2},\mathbb{R})$.

As an immediate consequence of $(2.3)$, since

$\displaystyle\lambda_{i}\big{(}{\bf p}_{j}\big{)}=\delta_{ij},\quad
G_{i}\big{(}{\bf p}_{j}-{\bf p}_{k}\big{)}=\delta_{ij}-\delta_{ik},$ $None$
$\displaystyle\lambda_{i}({\bf x}_{t}^{k})=t,\quad\lambda_{j}({\bf
x}_{t}^{k})=1-t,\quad\lambda_{k}({\bf x}_{t}^{k})=0,$ $\displaystyle{\bf
x}^{k}_{t}-{\bf p}_{i}=(1-t)({\bf p}_{j}-{\bf p}_{i}),\quad{\bf
x}^{k}_{t}-{\bf p}_{j}=t({\bf p}_{i}-{\bf p}_{j}),$

we obtain the following observation.

2.8. Lemma. Given an admissible pair ${\bm{\Psi}}=[\Psi_{0},\Psi_{1}]$ of
shape functions along with an admissible function germ $\hbox{\Goth
g}=\big{\\{}({\bf p}_{k},f_{k},A_{k})\big{\\}}_{k=1}^{3}$, if
$i,j,k\in\\{1,2,3\\}$ are three different indices then, at the generic point

${\bf x}_{t}^{k}:=t{\bf p}_{i}+(1-t){\bf p}_{j}\qquad(0\leq t\leq 1)$ $None$

of the edge $[{\bf p}_{i},{\bf p}_{j}]$ in the triangle ${\bf T}={\rm
Conv}\\{{\bf p}_{1},{\bf p}_{2},{\bf p}_{3}\\}$, for the function
$F_{0}=\hbox{\Goth F}^{\bm{\Psi}}_{0}\hbox{\Goth g}$ we have

$\displaystyle F_{0}\big{(}{\bf x}_{t}^{k}\big{)}$
$\displaystyle=\Psi_{0}(t)f_{i}+\Psi_{1}(t)(1-t)A_{i}({\bf p}_{j}-{\bf
p}_{i})+$ $None$ $\displaystyle\hskip
14.22636pt+\Psi_{0}(1-t)f_{j}+\Psi_{1}(1-t)tA_{j}({\bf p}_{i}-{\bf p}_{j});$
$\displaystyle F_{0}^{\prime}\big{(}{\bf x}_{t}^{k}\big{)}$
$\displaystyle=\big{[}\Psi_{0}^{\prime}(t)f_{i}+\Psi_{1}^{\prime}(t)(1-t)A_{i}({\bf
p}_{j}-{\bf p}_{i})\big{]}G_{i}+\Psi_{1}(t)A_{i}+$
$\displaystyle\hskip-19.91692pt+\big{[}\Psi_{0}^{\prime}(1-t)f_{j}+\Psi_{1}^{\prime}(1-t)tA_{j}({\bf
p}_{i}-{\bf p}_{j})\big{]}G_{j}+\Psi_{1}(1-t)A_{j}.$

3\. Generic algorithm of reduced side derivatives (RSD)

In this section we are looking for ${\cal C}^{1}$-spline interpolation
procedures analogous to those described in [1] but with more general shape
functions $\Psi_{0},\Psi_{1}\in{\cal C}^{1}[0,1]$ instead of $\Phi,\Theta$
there.

Henceforth let ${\bf p}_{1},{\bf p}_{2},{\bf p}_{3}\in\mathbb{R}^{2}$ be the
vertices of some (arbitrarily fixed) non-degenerate triangle ${\bf T}$ with
respective weight functions $\lambda_{m}:\mathbb{R}^{2}\\!\to\\!\mathbb{R}$
and derivative weights $G_{m}\\!:=\lambda_{m}^{\prime}\\!\in\\!{\cal
L}(\mathbb{R}^{2},\mathbb{R})$ $(m\\!=\\!1,2,3)$ and let ${\bf u}_{1},{\bf
u}_{2},{\bf u}_{3}\in\mathbb{R}^{2}$ be given vectors such that ${\bf
u}_{k}\not\parallel{\bf p}_{i}-{\bf p}_{j}$ whenever $(i,j,k)\in S_{3}$.

Furthermore ${\bm{\Psi}}\\!=\\![\Psi_{0},\Psi_{1}]$ resp. $\hbox{\Goth
g}\\!=\\!\big{\\{}({\bf p}_{k},\\!f_{k},\\!A_{k})\big{\\}}_{k=1}^{3}$ denote a
fixed admissible pair of shape functions and a function germ. As earlier, we
write

$F_{0}=\hbox{\Goth F}_{0}^{\bm{\Psi}}\hbox{\Goth g}.$

Our starting point is the following immediate consequence of $(2.10)$:

3.1. Remark. Let $H\in{\cal C}^{1}_{0}({\bf T})$. Then the function
$F:=F_{0}-H$ coincides with $F_{0}$ along the edges of ${\bf T}$ and hence, at
the vertices, it has also the properties $F({\bf p}_{i})=f_{i}$, $F({\bf
p}_{i})=A_{i}$ $(i=1,2,3)$ analogous to $(2.5)$.

3.2. Definition. We say that the function

$F=F_{0}-H,\quad H\in{\cal C}^{1}_{0}({\bf T})$ $None$

is an RSD modification of $F_{0}$ (with respect to the directions ${\bf
u}_{1},{\bf u}_{2},{\bf u}_{3}$ along the edges $\big{[}{\bf p}_{2},{\bf
p}_{3}\big{]},\big{[}{\bf p}_{3},{\bf p}_{1}\big{]},\big{[}{\bf p}_{1},{\bf
p}_{2}\big{]}$ respectively) if for any index pemutation $(i,j,k)\in S_{3}$,

$F^{\prime}\big{(}{\bf x}\big{)}{\bf
u}_{k}\\!\\!=\\!\Psi_{1}\big{(}\lambda_{i}({\bf x})\big{)}A_{i}{\bf
u}_{k}+\Psi_{1}\big{(}\lambda_{j}({\bf x})\big{)}A_{j}{\bf u}_{k}\
\hbox{whenever}\ {\bf x}\\!\in\\!\\!\big{[}{\bf p}_{i},{\bf p}_{j}\big{]}.$

Conveniently, in this case we refer to $H$ as an RSD modifier of $F_{0}$.
According to Lemma $2.8$, we have the following.

3.4. Corollary. A function $H\in{\cal C}^{1}_{0}({\bf T})$ is an RSD modifier
of $F_{0}$ with respect to the directions ${\bf u}_{k}$ along the edges
$\big{[}{\bf p}_{i},{\bf p}_{j}\big{]}$ if and only if

$\displaystyle H^{\prime}\big{(}{\bf x}_{t}^{k}\big{)}{\bf u}_{k}$
$\displaystyle=\big{[}\Psi_{0}^{\prime}(t)f_{i}+\Psi_{1}^{\prime}(t)(1-t)A_{i}({\bf
p}_{j}-{\bf p}_{i})\big{]}[G_{i}{\bf u}_{k}]+$
$\displaystyle+\big{[}\Psi_{0}^{\prime}(1-t)f_{j}+\Psi_{1}^{\prime}(1-t)tA_{j}({\bf
p}_{i}-{\bf p}_{j})\big{]}[G_{j}{\bf u}_{k}]$

whenever ${\bf x}_{t}^{k}=t{\bf p}_{i}+(1-t){\bf p_{j}}$ with $(i,j,k)\in
S_{3}$ and $0\leq t\leq 1$.

3.5. Theorem. Let $\Omega\subset\mathbb{R}^{3}$ be a set whose interior
contains $\Delta_{3}\cap(0,1)^{3}$. Assume $\chi_{0},\chi_{1}\in{\cal
C}^{1}(\Omega)$ are functions vanishing along the edges
$\Delta_{3,k}\\!=\\!\big{\\{}(t_{1},t_{2},t_{3})\\!\in\\!\Delta_{3}:t_{k}\\!=\\!0\big{\\}}$
$(k=1,2,3)$ of $\Delta_{3}$ such that

$D_{3}\chi_{0}(t,1-t,0)=\Psi_{0}^{\prime}(t),\quad
D_{3}\chi_{1}(t,1-t,0)=\Psi_{1}^{\prime}(t)\cdot(1-t)$ $None$

with the following marginal conditions on the derivatives

$\chi_{r}^{\prime}({\bf t})=0\ \ \big{(}{\bf
t}\in\Delta_{3,1}\cup\Delta_{3,2}\big{)};\quad D_{m}\chi_{r}({\bf t})=0\ \
\big{(}{\bf t}\in\Delta_{3,3},\ m=1,2\big{)}.$ $None$

Then the function $H$ in $(1.3)$ is an RSD modifier for $F_{0}$ under any
choice of the vectors ${\bf u}_{1},{\bf u}_{2},{\bf u}_{3}$ with ${\bf
u}_{i}\not\parallel{\bf p}_{j}-{\bf p}_{k}$ $\big{(}(i,j,k)\in S_{3}\big{)}$.

Proof. Consider any permutation $(i,j,k)\in S_{3}$. According to Corollary
$3.4$, it suffices to see that, at the points ${\bf x}_{t}=t{\bf
p}_{i}+(1-t){\bf p}_{j}$ we have

$\displaystyle H({\bf x}_{t})$ $\displaystyle=0,$ $None$ $\displaystyle
H^{\prime}({\bf x}_{t}){\bf u}_{k}$
$\displaystyle=\big{[}\Psi_{0}^{\prime}(t)f_{i}+\Psi_{1}^{\prime}(t)(1-t)A_{i}({\bf
p}_{j}-{\bf p}_{i})\big{]}G_{i}{\bf u}_{k}+$ $\displaystyle\hskip
17.07164pt+\big{[}\Psi_{0}^{\prime}(1-t)f_{j}+\Psi_{1}^{\prime}(1-t)tA_{j}({\bf
p}_{i}-{\bf p}_{j})\big{]}G_{j}{\bf u}_{k}.$

Notice that, in terms of the triples

${\bf t}_{\ell,m,n}\\!=\\!\big{(}\lambda_{\ell}({\bf x}_{t}),\lambda_{m}({\bf
x}_{t}),\lambda_{n}({\bf x}_{t})\big{)}\quad\big{(}(\ell,m,n)\in
S_{3}\big{)},$

and by setting
$(\ell,m,n)_{1}\\!=\\!\ell,\,(\ell,m,n)_{2}\\!=\\!m,\,(\ell,m,n)_{3}\\!=\\!n$,
we can write

$\displaystyle H({\bf x}_{t})=\sum_{(\ell,m,n)\in S_{3}}{G_{\ell}{\bf
u}_{n}\over G_{n}{\bf u}_{n}}\bigg{[}f_{\ell}\chi_{0}({\bf
t}_{\ell,m,n})+A_{\ell}({\bf p}_{m}-{\bf p}_{\ell})\chi_{1}({\bf
t}_{\ell,m,n})\bigg{]},$ $\displaystyle H^{\prime}({\bf x}_{t}){\bf
u}_{k}=\hskip-5.69054pt\sum_{(\ell,m,n)\in S_{3}}\hskip-5.69054pt{G_{\ell}{\bf
u}_{n}\over G_{n}{\bf
u}_{n}}\bigg{[}f_{\ell}\Big{[}\sum_{q=1}^{3}D_{q}\chi_{0}({\bf
t}_{\ell,m,n})G_{(\ell,m,n)_{q}}{\bf u_{k}}\Big{]}+$ $\displaystyle\hskip
91.04881pt+A_{\ell}({\bf p}_{m}-{\bf
p}_{\ell})\Big{[}\sum_{q=1}^{3}D_{q}\chi_{1}({\bf t}_{\ell
mn})G_{(\ell,m,n)_{q}}{\bf u_{k}}\Big{]}\bigg{\\}}.$

Observe that we have ${\bf t}_{ijk}\\!=\\!(t,1-t,0)\\!\in\\!\Delta_{3,3}$,
${\bf t}_{jik}\\!=\\!(1-t,t,0)\\!\in\\!\Delta_{3,3}$, ${\bf
t}_{ikj}\\!=\\!(t,0,1-t)\\!\in\\!\Delta_{3,2}$, ${\bf
t}_{kij}\\!=\\!(0,t,1-t)\\!\in\\!\Delta_{3,2}$, ${\bf
t}_{jki}\\!=\\!(1-t,0,t)\\!\in\\!\Delta_{3,1}$, ${\bf
t}_{kji}\\!=\\!(0,1-t,t)\\!\in\\!\Delta_{3,1}$. Hence the relation $H({\bf
x}_{t})=0$ is immediate. According to $(3,7)$, all the terms
$D_{q}\chi_{r}({\bf t}_{\ell,m,n})$ vanish except for the cases with
$\lambda_{n}({\bf x}_{t})=0$ and $q=3$ that is for
$(q,r,\ell,m,n)\in\big{\\{}(3,r,i,j,k),(3,r,j,i,k):r=0,1\big{\\}}$. In view of
$(3.6)$ we have $D_{3}\chi_{0}({\bf t}_{ijk})=\Psi_{0}^{\prime}(t)$,
$D_{3}\chi_{0}({\bf t}_{jik})=\Psi_{0}^{\prime}(1-t)$, $D_{3}\chi_{1}({\bf
t}_{ijk})=\Psi_{1}^{\prime}(t)(1-t)$, $D_{3}\chi_{1}({\bf
t}_{jik})=\Psi_{1}^{\prime}(1-t)t$ which completes the proof of $(3.8)$ and
hence the theorem.

3.9. Definition. We shall say that $[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ is
an RSD tuple if $[\Psi_{0},\Psi_{1}]$ is an admissible pair of shape functions
and $\chi_{0},\chi_{1}$ are functions satisfying the requirements of Theorem
$3.5$.

3.10. Lemma. Given any function $h\\!\in\\!{\cal C}_{0}(\mathbb{R}^{2}_{+})$,
there exists $\chi\\!\in\\!{\cal C}^{1}_{0}(\mathbb{R}_{+}^{3})$ such that for
all $t_{1},t_{2},t_{3}\geq 0$ we have

$\displaystyle
0=D_{m}\chi(0,t_{2},t_{3})=D_{m}(t_{1},0,t_{3})=D_{m}(t_{1},t_{2},0)\quad(m=1,2),$
$\displaystyle h(t_{1},t_{2})=D_{3}\chi(t_{2},t_{2},0),\ \
0=D_{3}\chi(0,t_{2},t_{3})=D_{3}\chi(t_{1},0,t_{3}).$

Proof. For ${\bf t}=(t_{1},t_{2},t_{3})\in\mathbb{R}^{3}_{+}$, define the
smoothing

$\widehat{h}({\bf t})\\!=\\!{1\over
t_{3}^{2}}\\!\int_{s_{1}=t_{1}}^{t_{1}+t_{3}}\\!\int_{s_{2}=t_{2}}^{t_{2}+t_{3}}\\!\\!h(s_{1},s_{2})\,ds_{2}ds_{1}\
\ (t_{3}\\!>\\!0),\quad\widehat{h}({\bf t})\\!=\\!h(t_{1},t_{2})\ \
(t_{3}\\!=\\!0)$

of $h(t_{1},t_{2},0)$ and let us fix a function $\phi\in{\cal
C}^{1}\big{(}[0,1]\big{)}$ (e.g. $\phi(t)=3t^{2}-2t^{3}$) with bounded
derivative and being such that $\phi(0)=\phi^{\prime}(0)=\phi^{\prime}(1)=0$
and $\phi(1)=1$. We show that the function

$\chi(t_{1},t_{2},t_{3})=\phi\Big{(}{t_{1}\over
t_{1}+t_{3}}\Big{)}\phi\Big{(}{t_{2}\over
t_{2}+t_{3}}\Big{)}t_{3}\widehat{h}(t_{1},t_{2},t_{3})\qquad(t_{1},t_{2}\geq
0,\ t_{3}>0)$

extended with $\chi(t_{1},t_{2},0)=0$ suits the requirements of the lemma.

It is folklore that, by the Newton-Leibniz theorem, the function $\widehat{h}$
is continuous, moreover its restriction to
$\mathbb{R}^{2}_{+}\times\mathbb{R}_{++}\big{(}=\\{(t_{1},t_{2},t_{3}):t_{1},t_{2}\geq
0,\ t_{3}>0\\}\big{)}$ is ${\cal C}^{1}$ smooth. In particular, for any point
${\bf t}=(t_{1},t_{2},t_{3})$ with $t_{3}>0$ and $t_{1}=0$ or $t_{2}=0$ we
have $\chi({\bf t})=0$ and $\chi^{\prime}({\bf t})=0$. Furthermore, for any
point ${\bf t}\in\mathbb{R}_{+}\times\mathbb{R}_{++}$, with the indices
$m=1,2$ resp. 3 we can write

$\displaystyle D_{m}\chi({\bf t})=\\!\Phi\Big{(}{t_{3-m}\over
t_{3-m}\\!+\\!t_{3}}\Big{)}\bigg{[}\Phi^{\prime}\Big{(}{t_{m}\over
t_{m}\\!+\\!t_{3}}\Big{)}{t_{3}^{2}\over(t_{m}\\!+\\!t_{3})^{2}}\widehat{h}({\bf
t})\\!+\\!\Phi\Big{(}{t_{m}\over
t_{m}\\!+\\!t_{3}}\Big{)}{\partial[t_{3}\widehat{h}]\over\partial
t_{1}}\bigg{]},$ $\displaystyle D_{3}\chi({\bf
t})=-\Phi^{\prime}\Big{(}{t_{1}\over t_{1}+t_{3}}\Big{)}\Phi\Big{(}{t_{2}\over
t_{2}+t_{3}}\Big{)}{t_{1}t_{3}\over(t_{1}+t_{3})^{2}}\widehat{h}({\bf t})\ -$
$\displaystyle-\Phi\Big{(}{t_{1}\over
t_{1}+t_{3}}\Big{)}\Phi^{\prime}\Big{(}{t_{2}\over
t_{2}+t_{3}}\Big{)}{t_{2}t_{3}\over(t_{2}+t_{3})^{2}}\widehat{h}({\bf
t})+\Phi\Big{(}{t_{1}\over t_{1}+t_{3}}\Big{)}\Phi\Big{(}{t_{2}\over
t_{2}+t_{3}}\Big{)}{\partial[t_{3}\widehat{h}]\over\partial t_{3}}.$

Therefore it only remains to prove that given any point ${\bf
t}^{*}=(t_{1}^{*},t_{2}^{*},0)\in\mathbb{R}_{+}^{3}$ we have

$\displaystyle D_{m}\chi({\bf t})\\!\\!\to\\!0\ (m\\!=\\!\\!1,2),\quad
D_{3}\chi({\bf t})\\!\\!\to\\!h(t_{1}^{*},t_{2}^{*})$ $None$
$\displaystyle\hbox{whenever}\quad\mathbb{R}^{2}_{+}\\!\times\mathbb{R}_{++}\\!\ni\\!{\bf
t}\\!\to\\!{\bf t}^{*}.$

Actually, these relation follow from the mean value expressions

$\displaystyle{\partial[t_{3}\widehat{h}]\over\partial t_{m}}$
$\displaystyle={1\over
t_{3}}\int_{s_{2}=t_{3-m}}^{t_{3-m}+t_{3}}\big{[}h(t_{m}+t_{3},s_{2})-h(t_{m},s_{2})\big{]}\,ds_{2}=$
$\displaystyle=h\big{(}t_{m}+t_{3},r_{3-m}({\bf
t})\big{)}-h\big{(}t_{m},q_{3-m}({\bf t})\big{)},$
$\displaystyle{\partial[t_{3}\widehat{h}]\over\partial t_{3}}$
$\displaystyle=-{1\over
t_{3}^{2}}\int_{s_{1}=t_{1}}^{t_{1}+t_{3}}\int_{s_{2}=t_{2}}^{t_{2}+t_{3}}h(s_{1},s_{2})\
ds_{2}\ ds_{1}\ +$ $\displaystyle\hskip 14.22636pt+{1\over
t_{3}}\int_{s_{2}=t_{2}}^{t_{2}+t_{3}}h(t_{1}+t_{3},s_{2})\ ds_{2}\ +{1\over
t_{3}}\int_{\\!s_{1}=t_{1}}^{t_{1}+t_{3}}h(s_{1},t_{2}+t_{3})\ ds_{1}=$
$\displaystyle=-h\big{(}p_{1}({\bf t}),p_{2}({\bf
t})\big{)}+h\big{(}t_{1}+t_{3},r_{2}({\bf t})\big{)}+h\big{(}r_{1}({\bf
t}),t_{2}+t_{3}\big{)}$

with suitable

$p_{1}({\bf t}),q_{1}({\bf t}),r_{1}({\bf t})\in[t_{1},t_{1}+t_{3}],\quad
p_{2}({\bf t}),q_{2}({\bf t}),r_{2}({\bf t})\in[t_{2},t_{2}+t_{3}].$

Indeed, let $\mathbb{R}_{+}^{2}\times\mathbb{R}_{++}\ni{\bf t}\to{\bf
t}^{*}=(0,t_{2}^{*},t_{3}^{*})$. Then ${\bf p}({\bf
t})\to(t_{1}^{*},t_{2}^{*})$, ${\bf q}_{m}({\bf t}),{\bf r}_{m}({\bf t})\to
t_{m}^{*}$ $(m=1,2)$, whence $\partial[t_{3}\widehat{h}]/\partial t_{m}\to 0$
$(m=1,2)$ and $\partial[t_{3}\widehat{h}]/\partial t_{3}\to
h(t_{1}^{*},t_{2}^{*})$. Since the function $(s,t)\\!\mapsto\\!s/(s\\!+\\!t)$
is analytic on the open half plane
$\\{(s,t)\\!\in\\!\mathbb{R}^{2}:s\\!+\\!t\\!>\\!0\\}$ containing the rays
$R_{1}\\!=\\!\mathbb{R}_{++}\\!\times\\!\\{0\\}$ resp.
$R_{2}\\!=\\!\\{0\\}\\!\times\\!\mathbb{R}_{++}$, $(3.11)$ is immedite in the
cases ${\bf t}^{*}\in R_{m}$ $(m=1,2)$. If
$\mathbb{R}^{2}_{+}\times\mathbb{R}_{++}\ni{\bf t}\to(0,0,0)$, we can deduce
$(3.11)$ from the facts that, for $\ell=1,2,3$, the functions
$t_{\ell}t_{3}/(t_{\ell}+t_{3})^{2},t_{3}^{2}/(t_{\ell}+t_{3})^{2}$ resp.
$\Phi\big{(}t_{\ell}/(t_{\ell}+t_{3})\big{)},\Phi^{\prime}\big{(}t_{\ell}/t_{\ell}+t_{3})\big{)}$
are bounded, furthermore $\partial[t_{3}\widehat{h}]/\partial t_{\ell}\to
h(0,0)=0$.

3.12. Corollary. By Theorem $3.5$ and Lemma $3.10$, for any admissible pair
$[\Psi_{0},\Psi_{1}]$, there exist $\chi_{0},\chi_{1}\\!\in\\!{\cal
C}^{1}_{0}\big{(}\mathbb{R}_{+}^{3}\setminus\\{(0,0,0\\}\big{)}$ such that
$[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ is an RSD-tuple.

3.13. Remark. Unfortunately, our convolution type generic construction
provided by the proof of Lemma $3.10$ is far from being optimal in most cases
from numerical view points.

In contrast, in view of Theorem $3.5$, in many cases we may apply the
following satisfactory construction:

3.14. Proposition. If $[\Psi_{0},\Psi_{1}]$ is an admissible pair of shape
functions and $h_{0},h_{1}\\!\in\\!{\cal C}^{1}_{0}(\mathbb{R}^{2}_{+})$ are
functions such that $h(t,1\\!-\\!t)\\!=\\!\Psi_{0}^{\prime}(t)$ and
$h(t,1\\!-\\!t)\\!=\\!\Psi_{1}^{\prime}(t)(1\\!-\\!t)$
$(0\\!\leq\\!t\\!\leq\\!1)$ then $[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ is an
RSD tuple with

$\chi_{r}(t_{1},t_{2},t_{3})=h_{r}(t_{1},t_{2})t_{3}\qquad(r=0,1).$

3.15. Examples. Cases with a factorization

$\Psi_{0}^{\prime}(t)=w_{01}(t)w_{02}(1-t),\quad\Psi_{1}^{\prime}(t)(1-t)=w_{11}(t)w_{12}(1-t)$

for suitable functions $w_{rk}\in{\cal C}^{1}_{0}(\mathbb{R}_{+})$ such that
$w_{rk}(0)=w_{rk}^{\prime}(0)=0$.

(a) The procedure $(1.1)$ with the shape functions $(1.2)$ corresponds to the
case $[\Psi_{0},\Psi_{1}]=[\Phi,\Theta]$,
$\chi_{0}(t_{1},t_{2},t_{3})=30t_{1}^{2}t_{2}^{2}t_{3}$,
$\chi_{1}(t_{1},t_{2},t_{3})\\!=\\!12t_{1}^{2}t_{2}^{2}t_{3}$ with
$w_{01}(t)\\!=\\!30t^{2}$, $w_{11}(t)\\!=\\!12t^{2}$,
$w_{02}(t)\\!=\\!w_{12}(t)\\!=\\!t^{2}$ since
$\Phi^{\prime}(t)\\!=\\!30t^{2}(1\\!-\\!t)^{2}$ and
$\Theta^{\prime}(t)\\!=\\!12t^{2}(1\\!-\\!t)$.

(b)
$\big{[}\Phi,\Phi,30t_{1}^{2}t_{2}^{2}t_{3},30t_{1}^{2}t_{2}^{3}t_{3}\big{]}$
is an RSD tuple by Proposition $3.14$, corresponding to the case
$w_{01}(t)=w_{11}(t)=30t^{2}$, $w_{02}(t)=t^{2}$, $w_{12}(t)=t^{3}$.

3.16. Remark. It is an easy consequence of Whitney’s extension theorem [7]
that for any function $\psi\in{\cal C}([0,1])$ with
$\psi(0)=\psi^{\prime}(0)=\psi^{\prime}(1)=0$ there exists a function
$h\in{\cal C}^{1}_{0}(\mathbb{R}^{2}_{+})$ such that
$\psi^{\prime}(t)=h(t,0-t)$. Hence every admissible pair $[\Psi_{0},\Psi_{1}]$
with the property $\Psi_{0}^{\prime}(1)=0$ admits functions
$h_{0},h_{1}\in{\cal C}(\mathbb{R}^{2}_{+})$ such that
$[\Psi_{,}\Psi_{1},h_{0}(t_{1},t_{2})t_{3},h_{0}(t_{1},t_{2})t_{3}]$ is an RSD
tuple.

We complete this section with a brief description of the local approximation
operator corresponding to Theorem $3.5$ and its use in constructing ${\cal
C}^{1}$-splines over a 2D triangular mesh analogously as done in [1,
Algorithm].

3.17. Definition. Let ${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ be an
RSD tuple and write ${\bm{\Psi}}=[\Psi_{0},\Psi_{1}]$ resp.
${\bm{X}}=[\chi_{0},\chi_{1}]$. We introduce the SDR modification of
$\hbox{\Goth F}^{\bm{\Psi}}$ by means of the complentary shape functions
${\bm{X}}$ as the operator

$\hbox{\Goth F}^{\bm{\Pi}}\\!\\!:\\!\big{\\{}({\bf
p}_{k},\\!f_{k},\\!A_{k},\\!{\bf u}_{k})\big{\\}}_{k=1}^{3}\mapsto\hbox{\Goth
F}^{\bm{\Psi}}\big{\\{}({\bf
p}_{k},\\!f_{k},\\!A_{k})\big{\\}}_{k=1}^{3}-\hbox{\Goth
H}^{\bm{X}}\big{\\{}({\bf p}_{k},\\!f_{k},\\!A_{k},\\!{\bf
u}_{k})\big{\\}}_{k=1}^{3}$

defined for all tuples $\big{\\{}({\bf p}_{k},\\!f_{k},\\!A_{k},\\!{\bf
u}_{k})\big{\\}}_{k=1}^{3}$ where $\big{\\{}({\bf
p}_{k},\\!f_{k},\\!A_{k})\big{\\}}_{k=1}^{3}$ is a non-degenerate function
germ and ${\bf u}_{k}\not\parallel{\bf p}_{i}\\!-{\bf p}_{j}$
$\big{(}(i,j,k)\in S_{3}\big{)}$ and

$\hbox{\Goth H}^{\bm{X}}\big{\\{}({\bf p}_{k},\\!f_{k},\\!A_{k},\\!{\bf
u}_{k})\big{\\}}_{k=1}^{3}=\Big{[}\hbox{ $H$ in Theorem $3.5$}\Big{]}.$

Recall [1] that, given a set ${\bf V}=\big{\\{}{\bf v}_{1},\ldots,{\bf
v}_{R}\big{\\}}$ of points in $\mathbb{R}^{2}$, by a triangular mesh over
${\bf V}$ we mean a family $\big{\\{}{\bf T}_{1},\ldots,{\bf
T}_{M}\big{\\}}\subset\mathbb{R}^{2}$ of non-degenerate triangles of the form

${\bf T}_{m}={\rm Conv}\big{\\{}{\bf v}_{i(m,1)},{\bf v}_{i(m,2)},{\bf
v}_{i(m,3)}\big{\\}},\quad i(m,1)<i(m,2)<i(m,3)$

with pairwise disjoint interiors whose intersections are either empty or a
common vertice or a common edge, furthermore ${\bf
V}\subset\bigcup_{m=1}^{M}{\bf T}_{m}$. A crutial ingreadient of the
construction, we enumerate the edges of the mesh triangles ${\bf T}_{m}$ in
the form ${\bf E}_{1},\ldots,{\bf E}_{S}$ and associate a vector ${\bf
u}_{s}\not\parallel{\bf E}_{s}$ with each edge ${\bf E}_{s}$, furthermore let
$s(m,k)$ denote the index of the opposite edge to the vertex ${\bf
v}_{m(i,k)}$ in the triangle ${\bf T}_{m}$. By replacing $\Phi,\Theta$ with
$\Psi_{0},\Psi_{1}$, respectively the functions
$30\lambda_{i}^{2}\lambda_{j}^{2}\lambda_{k}$,
$12\lambda_{i}^{2}\lambda_{j}^{2}\lambda_{k}$ with $\chi_{0},\chi_{1}$ in a
straightforward manner in [1, Proof of Thm. 2], we can conclude the folowing:
Given any germ $\big{\\{}({\bf v}_{i},f_{i},A_{i})\big{\\}}_{i=1}^{R}$ of
first order function data over ${\bf V}$, the union of the functions

$F_{m}\\!=\hbox{\Goth F}^{\bm{\Pi}}\Big{\\{}\big{(}{\bf
v}_{i(m,k)},f_{i(m,k)},A_{i(m,k)},{\bf
u}_{s(m,k)}\big{)}\Big{\\}}_{\\!k=1}^{\\!3}\\!\in{\cal C}^{1}({\bf
T}_{m})\quad(m\\!=\\!1,\ldots,M)$

is continuously differentiable on the mesh domain ${\bf
D}=\bigcup_{m=1}^{M}{\bf T}_{m}$.

4\. Range shift property

One of the most frequent applications of splines over triangular meshes is
reconstucting approximately a smooth surface with a function graph passing
through a family of points. Such procedures correspond to a model of the
following pattern: We are given a smooth surface ${\bf
S}\subset\mathbb{R}^{3}$ which can be represented in the form ${\bf S}={\rm
graph}(f)=\big{\\{}\big{[}{\bf p},f({\bf p})\big{]}:{\bf
p}\in\mathbb{R}^{2}\big{\\}}$ with some (a priori unknown) function $f\in{\cal
C}^{1}(\mathbb{R}^{2})$. Accurate data (obtained with scanner equipments) are
available for a finite family of points ${\bf v}_{1},\ldots,{\bf
v}_{N}\in\mathbb{R}^{2}$ concerning the first order data $f^{[1]}({\bf
p})=\big{[}{\bf p},f({\bf p}),f^{\prime}({\bf p})\big{]}$ and we approximate a
piece of ${\bf S}$ with the graph of a ${\cal C}^{1}$-spline function
$F\in{\cal C}^{1}({\bf D})$ on the domain ${\bf D}$ of a triangular mesh with
vertices ${\bf v}_{n}$. The procedures used in constructing $F$ are usually
investigated from the view points of estimating various biases between ${\rm
graph}(F)$ and ${\bf S}$, respectively from exhibiting features of algebraic-
geometric character. In this note we only discuss two kinds of the last
category in the context of RSD methods: if the the spline graps are invariant
with respect to tranlations of first order data and the property that plane
surfaces remain invariant.

4.1. Definition. Throughout this section $[{\bf P,U}]$ denotes an admissible
pair of triples ${\bf P}=\big{\\{}{\bf p}_{i}\big{\\}}_{i=1}^{3},{\bf
U}=\big{\\{}{\bf u}_{i}\big{\\}}_{i=1}^{3}$ with ${\bf p}_{i},{\bf
u}_{i}\in\mathbb{R}^{2}$. We write ${\bf T}={\rm Conv}({\bf P})$ and
$\lambda_{1},\lambda_{2},\lambda_{3}$ resp. $G_{1},G_{2},G_{3}$ for the weight
functions resp. their derivatives associated with the vertices of the triangle
${\bf T}$. Given any function $f\in{\cal C}^{1}(\mathbb{R}^{2})$, we shall use
the shorthand notations

$f^{[1]}_{\bf P,U}=\big{\\{}\big{[}{\bf p}_{i},f({\bf p}_{i}),f^{\prime}({\bf
p}_{i}),{\bf u}_{i}\big{]}\big{\\}}_{i=1}^{3},\quad\hbox{\Goth
F}^{\bm{\Pi}}_{\bf P\\!,U}f=\hbox{\Goth F}^{\bm{\Pi}}f^{[1]}_{\bf P,U}.$

An RSD-tuple ${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ has the range
shift property if the procedure $\hbox{\Goth F}^{\bm{\Pi}}$ leaves the
constant functions invariant, that is if for all admissible pairs ${\bf P,U}$
and the constant unit function ${\bf 1}:\mathbb{R}^{2}\ni{\bf x}\mapsto 1$ we
have

$\hbox{\Goth F}^{\bm{\Pi}}_{\bf P,U}{\bf 1}={\bf 1}|{\rm Conv}({\bf P})$

4.2. Remark. Since ${\bf 1}({\bf p}_{i})\\!=\\!1,\ {\bf 1}^{\prime}({\bf
p_{i}}){\bf u}_{i}\\!=\\!0$, we can write

$\displaystyle\hbox{\Goth F}^{{\bm{\Pi}}}_{\bf P\\!,U}{\bf 1}$
$\displaystyle=\sum_{i=1}^{3}\Psi_{0}(\lambda_{i})-\hskip-11.38109pt\sum_{(\ell,m,n)\in{\rm
S}_{3}}\hskip-2.84526pt{G_{\ell}{\bf u}_{n}\over G_{n}{\bf
u}_{n}}\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})=$
$\displaystyle\hskip-22.76219pt=\sum_{n=1}^{3}\left\\{\Psi_{0}(\lambda_{n})\\!-\\!\left[{G_{\ell_{n}}{\bf
u}_{n}\over G_{n}{\bf
u}_{n}}\chi_{0}(\lambda_{\ell_{n}},\lambda_{m_{n}},\lambda_{n})\\!+\\!{G_{m}{\bf
u}_{n}\over G_{n}{\bf
u}_{n}}\chi_{0}(\lambda_{m_{n}},\lambda_{\ell_{n}},\lambda_{n})\right]\right\\}$

with $(\ell_{1},m_{1})=(2,3)$, $(\ell_{2},m_{2})=(1,3)$,
$(\ell_{3},m_{3})=(1,2)$.

4.3. Lemma. For any $\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}$ there
exist ${\bf u}_{1},{\bf u}_{1},{\bf u}_{1}\in\mathbb{R}^{2}$ such that

$\left[{G_{i}{\bf u}_{j}\over G_{j}{\bf
u}_{j}}\right]_{i,j=1}^{3}=\left[\begin{matrix}1&-1-\alpha_{2}&\alpha_{3}\cr\alpha_{1}&1&-1-\alpha_{3}\cr-1-\alpha_{1}&\alpha_{2}&1\end{matrix}\right].$

Proof. A suitable choce is ${\bf u}_{1}={\bf p}_{1}-{1\over 2}[{\bf
p}_{2}+{\bf p}_{3}]+({1\over 2}+\alpha_{1})[{\bf p}_{2}-{\bf p}_{3}]$, ${\bf
u}_{2}={\bf p}_{2}-{1\over 2}[{\bf p}_{3}+{\bf p}_{1}]+({1\over
2}+\alpha_{2})[{\bf p}_{3}-{\bf p}_{1}]$, ${\bf u}_{3}={\bf p}_{3}-{1\over
2}[{\bf p}_{1}+{\bf p}_{2}]+({1\over 2}+\alpha_{3})[{\bf p}_{1}-{\bf p}_{2}]$.

4.4. Proposition. An RSD tuple
${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ has range shift property if
and only if

$\displaystyle\Psi_{0}(t)+\Psi_{0}(1-t)=1,\ \
\Psi_{0}^{\prime}(t)=\Psi_{0}^{\prime}(1-t)\hskip 28.45274pt(0\leq t\leq 1),$
$None$
$\displaystyle\chi_{0}(t_{1},t_{2},t_{3})=\chi_{0}(t_{2},t_{1},t_{3})\quad\hskip
65.44133pt\big{(}(t_{1},t_{2},t_{3})\in\Delta_{3}\big{)}$

and ${\bf
1}=\sum\limits_{i=1}^{3}\Psi_{0}(\lambda_{i})+\hskip-25.60747pt\sum\limits_{\hskip
19.91692pt(\ell,m,n)\in{\rm
S}_{3}^{+}}\\!\\!\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})$.

Proof. Consider the points ${\bf x}_{t}=t{\bf p}_{1}+(1-t){\bf p}_{2}$ $(0\leq
t\leq 1)$. Since ${\bf 1}^{\prime}\equiv 0$, ${\bf 1}({\bf x}_{t})=1$,
$\lambda_{1}({\bf x}_{t})=t$, $\lambda_{2}({\bf x}_{t})=1-t$, and
$\lambda_{3}({\bf x}_{t})=0$, by assumption

$1={\bf 1}({\bf x}_{t})=\hbox{\Goth F}^{\bm{\Pi}}_{\bf P,U}{\bf 1}\,({\bf
x}_{t})=\sum_{i=1}^{3}\Psi_{0}\big{(}\lambda_{i}({\bf
x}_{t})\big{)}=\Psi_{0}(t)+\Psi_{0}(1-t)+\Psi_{0}(0)$

where $\Psi_{0}(0)=0$ due to $(2.1)$. Thus in view of Remark $4.2$ and Lemma
$4.3$, the expression of $\hbox{\Goth F}^{{\bm{\Pi}}}_{\bf P\\!,U}{\bf 1}$ has
the form

$\hbox{\Goth F}^{{\bm{\Pi}}}_{\bf P\\!,U}{\bf
1}=\sum_{n=1}^{3}\left\\{\Psi_{0}(\lambda_{n})\\!-\\!\big{[}\alpha_{n}\chi_{0}(\lambda_{\ell_{n}},\lambda_{m_{n}},\lambda_{n})\\!+\\!(-1-\alpha_{n})\chi_{0}(\lambda_{m_{n}},\lambda_{\ell_{n}},\lambda_{n})\big{]}\right\\}$

where the coefficients $\alpha_{1},\alpha_{2},\alpha_{3}$ may assume any real
value. This is possible only if $\hbox{\Goth F}^{{\bm{\Pi}}}_{\bf P\\!,U}{\bf
1}=\sum\limits_{i=1}^{3}\Psi_{0}(\lambda_{i})+\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})+\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})+\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})$
and

$\chi_{0}(\lambda_{\ell_{n}},\lambda_{m_{n}},\lambda_{n})=\chi_{0}(\lambda_{m_{n}},\lambda_{\ell_{n}},\lambda_{n})\qquad(n=1,2,3).$

In particular, for $n\\!=\\!3$ and a generic point ${\bf
x}\\!=\\!\sum\limits_{k=1}^{3}t_{k}{\bf p}_{k}$ $(\sum_{k}t_{k}\\!=\\!1,\
t_{k}\\!\geq\\!0)$ of the triangle ${\rm Conv}({\bf P})$, we get
$\chi_{0}(t_{1},t_{2},t_{3})\\!\\!=\\!\chi_{0}\big{(}\lambda_{1}({\bf
x}),\lambda_{2}({\bf x}),\lambda_{3}({\bf
x})\\!\big{)}\\!\\!=\\!\chi_{0}\big{(}\lambda_{2}({\bf x}),\lambda_{1}({\bf
x}),\lambda_{3}({\bf x})\\!\big{)}\\!\\!=\\!\\!\chi_{0}(t_{2},t_{1},t_{3})$
which completes the proof.

4.6. Lemma. Given any smooth function $\phi$ defined on a domain
$\Omega\\!\subset\\!\mathbb{R}^{3}$ containing $\Delta_{3}$ and given any
point ${\bf x}\\!\in\\!{\bf T}$, for the function
$f\\!=\\!\phi(\lambda_{1},\lambda_{2},\lambda_{3})$ we have $f^{\prime}({\bf
x})=0$ if and only if

$D_{1}\phi({\bf t})=D_{2}\phi({\bf t})=D_{3}\phi({\bf t})\quad\hbox{\rm
where}\quad{\bf t}=\big{(}\lambda_{1}({\bf x}),\lambda_{2}({\bf
x}),\lambda_{3}({\bf x})\big{)}.$

Proof. For any vector ${\bf u}\in\mathbb{R}^{2}$ we have $f^{\prime}({\bf
x}){\bf u}=\sum_{k=1}^{3}D_{k}\varphi({\bf t})G_{k}{\bf u}$. Hence the
statement is immediate from the relation $\big{\\{}(G_{1}{\bf u},G_{2}{\bf
u},G_{3}{\bf u}):{\bf
v}\in\mathbb{R}^{2}\big{\\}}=\big{\\{}(\alpha_{1},\alpha_{2},\alpha_{3}):\sum_{k}\alpha_{k}=0\big{\\}}$.
$\big{(}$Actually, given any triple $(i,j,k)\in S_{3}$ of indices, by taking
${\bf u}={\bf p}_{i}-{\bf p}_{j}$ we have $(G_{i}{\bf u}=1,(G_{j}{\bf
u}=-1,(G_{k}{\bf u}=0$ implying $D_{i}\varphi({\bf t}-D_{j}\varphi({\bf t}$ if
$f^{\prime}({\bf x}){\bf u}=0\big{)}$.

4.7. Corollary. The RSD tuple
${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ has range shift property if
and only if $(4.5)$ holds and

${\partial\Sigma\over\partial
t_{1}}(t_{1},t_{2},t_{3})\\!=\\!{\partial\Sigma\over\partial
t_{2}}(t_{1},t_{2},t_{3})\\!=\\!{\partial\Sigma\over\partial
t_{3}}(t_{1},t_{2},t_{3})\quad\big{(}(t_{1},t_{2},t_{3})\\!\in\\!\Delta_{3}\big{)},$
$None$

$\Sigma(t_{1},t_{2},t_{3})\\!=\\!\sum\limits_{i=1}^{3}\\!\Psi_{0}(t_{i})+\hskip-14.22636pt\sum\limits_{(\ell,m,n)\in
S_{3}^{+}}\hskip-14.22636pt\chi_{0}(t_{\ell},t_{m},t_{n})$,
$S_{3}^{+}\\!=\\!\\{(1,\\!2,\\!3),(2,\\!3,\\!1),(3,\\!1,\\!2)\\}$.

Proof. We established that ${\bm{\Pi}}$ has range shift property if and only
if

$\Sigma\big{(}\lambda_{1}({\bf x}),\lambda_{2}({\bf x}),\lambda_{3}({\bf
x})\big{)}=1\qquad\big{(}{\bf x}\\!\in\\!{\rm Conv}({\bf P})\big{)}.$ $None$

Since ${\bm{\Pi}}$ is an RSD tuple, in any case we have $\hbox{\Goth
F}^{\bm{\Pi}}_{\bf P,U}({\bf p}_{i})=1$ $(i=1,2,3)$. Hence the relation
$(4.9)$ holds if and only if the Fréchet derivative of the function
$\hbox{\Goth F}^{\bm{\Pi}}_{\bf P,U}{\bf 1}$ vanishes, that is if

$\big{[}\Sigma(\lambda_{1},\lambda_{2},\lambda_{3})\big{]}^{\prime}({\bf
x})=0\qquad\big{(}{\bf x}\\!\in\\!{\rm Conv}({\bf P})\big{)}.$ $None$

An application of Lemma $4.6$ with $\phi=\Sigma$ completes the proof.

4.10. Example. The RSD procedures corresponding to tuples of the form
$\big{[}t^{3}(10-15t+6t^{2}),*,30t_{1}^{2}t_{2}^{2}t_{3},*\big{]}$ as in
Examples $3.15$ have range shift property. This highly non-trivial fact can
easily be established by verifying $(4.8)$ as follows. In such cases, we have

$\Sigma(t_{1},t_{2},t_{3})\\!=\\!\sum\limits_{i=1}^{3}t_{i}^{3}(10-15t_{i}+6t_{i}^{2})+30\big{(}t_{1}^{2}t_{2}^{2}t_{3}+t_{2}^{2}t_{3}^{2}t_{1}+t_{3}^{2}t_{1}^{2}t_{2})$

and it suffices to show that the polynomial ${\partial\Sigma\over\partial
t_{1}}$ is symmetric in $t_{1},t_{2},t_{3}$ when replacing
$t_{1}=1-t_{2}-t_{3}$. Since
$\big{[}t^{3}(10-15t+6t^{2})\big{]}^{\prime}=30t^{2}(1-t)^{2}$, if
$(t_{1},t_{2},t_{3})\in\Delta_{3}$ then
$\big{[}t_{1}^{3}(10-15t_{1}+6t_{1}^{2})\big{]}^{\prime}=30t_{1}^{2}(t_{2}+t_{3})^{2}$
and

$\displaystyle{1\over 30}{\partial\Sigma\over\partial
t_{1}}=t_{1}^{2}(t_{2}+t_{3})^{2}+2t_{1}t_{2}^{2}t_{3}+t_{2}^{2}t_{3}^{2}+2t_{3}^{2}t_{1}t_{2}=$
$\displaystyle=t_{1}^{2}t_{2}^{2}+t_{1}^{2}t_{3}^{2}+t_{2}^{2}t_{3}^{2}+2t_{1}t_{2}t_{3}(t_{1}+t_{2}+t_{3})=t_{1}^{2}t_{2}^{2}+t_{1}^{2}t_{3}^{2}+t_{2}^{2}t_{3}^{2}+2t_{1}t_{2}t_{3}.$

4.11. Definition. We say that a function $\Psi\in{\cal C}^{1}([0,1])$ is
d-symmetric if

$\Psi^{\prime}(t)=\Psi^{\prime}(1-t)\quad(0\leq t\leq 1).$

4.12. Corollary. If ${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ is an
RSD tuple with d-symmetric shape function $\Psi_{0}$ then
${\bm{\Pi}}^{[s]}=[\Psi_{0},\Psi_{1},\chi_{0}^{[s]},\chi_{1}]$ with

$\chi_{0}^{[s]}(t_{1},t_{2},t_{3})={1\over
2}\chi_{0}(t_{1},t_{2},t_{3})+{1\over 2}\chi_{0}(t_{2},t_{1},t_{3})$ $None$

is also an RSD tuple such that

$\hbox{\Goth F}^{{\bm{\Pi}}^{[s]}}_{\bf P,U}{\bf 1}=\sum_{(\ell,m,n)\in
S_{3}}\Big{\\{}{1\over
2}\Psi_{0}(\lambda_{\ell})+\chi^{[s]}(\lambda_{\ell},\lambda_{m},\lambda_{n})\Big{\\}}.$
$None$

Proof. The marginal conditions $(3.7)$ for $\chi_{0}^{[s]}$ are immediate. As
a consequence of the d-symmetry of $\Psi_{0}$, we have

$\displaystyle D_{3}\chi_{0}^{[s]}(t,1-t,0)$ $\displaystyle={1\over
2}D_{3}\chi_{0}(t,1-t,0)+{1\over 2}D_{3}\chi_{0}(1-t,t,0)=$
$\displaystyle={1\over 2}\Psi_{0}(t)+{1\over 2}\Psi_{0}(1-t)=\Psi_{0}(t)$

establishing that ${\bm{\Pi}}^{[s]}$ is an RSD tuple. Due to the symmetry of
$\chi_{0}^{[s]}$ in the variables $t_{1},t_{2}$ in Remark $4.2$ applied to
$\chi_{0}^{[s]}$ we can write

$\hbox{\Goth F}^{{\bm{\Pi}}}_{\bf P\\!,U}{\bf
1}=\sum_{n=1}^{3}\left\\{\Psi_{0}(\lambda_{n})\\!-\\!\left[\Big{[}{G_{\ell_{n}}{\bf
u}_{n}\over G_{n}{\bf u}_{n}}\\!+\\!{G_{m}{\bf u}_{n}\over G_{n}{\bf
u}_{n}}\Big{]}\chi_{0}(\lambda_{\ell_{n}},\lambda_{m_{n}},\lambda_{n})\right]\right\\}.$

Taking Lemma $4.3$ into account, we obtain $(4.14)$.

4.15. Theorem. Let $[\Psi_{0},\Psi_{1}]$ be an admissible pair of shape
functions, with $\Psi_{0}$ being d-symmetric. Then we can find
$\chi_{0}^{*},\chi_{1}^{*}\in{\cal C}_{0}(\mathbb{R}^{3}_{+})$ such that
${\bm{\Pi}}^{*}=[\Psi_{0},\Psi_{1},\chi_{0}^{*},\chi_{1}^{*}]$ is an RSD-tuple
with range shift property where $\chi_{0}^{*}$ is symmetric in its first two
variables.

Proof. According to Theorem $3.5$ there exist an RSD tuple of the form
${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ with
$\chi_{0},\chi_{1}\in{\cal C}_{0}^{1}(\mathbb{R}_{0}^{3})$. By Corollary
$(4.12)$, ${\bf\Pi}^{[s]}=[\Psi_{0},\Psi_{1},\chi_{0}^{[s]},\chi_{1}]$ is also
an RSD tuple where the complementary shape function $\chi_{0}^{[s]}$ is
symmetric in its first two variables and such that $(4.14)$ holds. Then,
independently of the choice of the vector triple ${\bf U}=[{\bf u}_{1},{\bf
u}_{2},{\bf u}_{3}]$, we can write the difference function of ${\bf 1}$ and
$\hbox{\Goth F}^{{\bm{\Pi}}^{[s]}}_{\bf P\\!,U}{\bf 1}$ in the form

$\displaystyle d={\bf 1}-\hbox{\Goth F}^{{\bm{\Pi}}^{[s]}}_{\bf P\\!,U}{\bf
1}=\delta(\lambda_{1},\lambda_{2},\lambda_{3})\quad\hbox{with}$ $None$
$\displaystyle\delta({\bf t})=d\big{(}\hbox{$\sum_{k=1}^{3}t_{k}{\bf
p}_{k}\big{)}$}\\!=\hskip-11.38109pt\sum_{(\ell,m,n)\in{\rm
S}_{3}}\\!\\!\\!\left\\{{1\over 6}-{1\over
2}\Psi_{0}\big{(}t_{\ell}\big{)}-\chi_{0}^{[s]}\big{(}t_{\ell},t_{m},t_{n}\big{)}\right\\}.$

Observe that

$d({\bf x})=0,\quad d^{\prime}({\bf x})=0\qquad({\bf x}\in\partial{\bf T}).$
$None$

Indeed, at the generic point $(2.9)$ of the edge opposite to the vertex ${\bf
p}_{k}$, by $(2.2)$ we have $d({\bf
x}^{k}_{t})\\!=\\!1\\!-\\![\Psi_{0}(t)\\!+\\!\Psi_{0}(1\\!-\\!t)]\\!=\\!0$. In
particular the functions ${\bf 1}$ and $\hbox{\Goth F}^{\Pi^{[s]}}_{\bf P,U}$
coincide along the edge $[{\bf p}_{i},{\bf p}_{j}]$, thus $d^{\prime}({\bf
x}_{t}^{k})({\bf p}_{j}\\!-\\!{\bf p}_{i})\\!=\\!0$ with $d({\bf x})\\!=\\!0$
for ${\bf x}\\!\in\\!\partial{\bf T}$. On the other hand, since trivially
${\bf 1}^{\prime}({\bf p})\\!=\\!0$ everywhere, the RSD construction ensures
that $d^{\prime}({\bf x}^{k}_{t}){\bf u}_{k}\\!=\\!{\bf 1}^{\prime}({\bf
x}_{t}^{k}){\bf u}_{k}\\!-\\!\big{[}\Psi_{0}(t)1^{\prime}({\bf p}_{k})){\bf
u}_{k}\\!+\\!\Psi_{0}(1\\!-\\!t)1^{\prime}({\bf p}_{k})){\bf
u}_{k}\big{]}\\!=\\!0$. Thus $\delta^{\prime}({\bf x}_{t}^{k})$ vanishes in
two linearly independent directions implying that $d^{\prime}({\bf
x}_{t}^{k}){\bf u}=0$ $({\bf u}\in\mathbb{R}^{2})$ i.e. $d^{\prime}({\bf
x}^{k}_{t})=0$.

Notice that ${\bf x}\in\partial{\bf T}$ if and only if
$\big{(}\lambda_{1}({\bf x}),\lambda_{2}({\bf x}),\lambda_{3}({\bf
x})\big{)}\in\bigcup_{k=1}^{3}\Delta_{3,k}$. Hence, due to $(4.17)$, we can
apply Lemma $4.6$ with $f=d$ and $\phi=\delta$ to conclude that

$\delta({\bf t})=0,\quad D_{1}\delta({\bf t})=D_{2}\delta({\bf
t})=D_{3}\delta({\bf t})\quad\hbox{whenever}\ \ {\bf
t}\in\hbox{$\bigcup\limits_{k=1}^{3}\Delta_{3,k}$}.$ $None$

On the domain $\Omega:=\mathbb{R}_{+}^{3}\setminus\\{0\\}$, define

$\displaystyle\chi_{0}^{*}=\chi_{0}^{[s]}+{1\over
6}\widehat{\delta}\quad\hbox{where}$ $\displaystyle\widehat{\delta}({\bf
t})=\delta\Big{(}{t_{1}\over t_{1}\\!+\\!t_{2}\\!+\\!t_{3}},{t_{2}\over
t_{1}\\!+\\!t_{2}\\!+\\!t_{3}},{t_{3}\over
t_{1}\\!+\\!t_{2}\\!+\\!t_{3}}\Big{)}.$

By writing $s({\bf t})=t_{1}+t_{2}+t_{3}$ for short, for $k=1,2,3$ we have

$\displaystyle D_{k}$ $\displaystyle\widehat{\delta}({\bf
t})={\partial\over\partial t_{k}}\delta\Big{(}{t_{1}\over s({\bf
t})},{t_{2}\over s({\bf t})},{t_{3}\over s({\bf t})}\Big{)}=$
$\displaystyle=D_{1}\delta\Big{(}{{\bf t}\over s({\bf
t})}\Big{)}{\partial\over\partial t_{k}}\,{t_{1}\over s({\bf
t})}+\\!D_{2}\delta\Big{(}{{\bf t}\over s({\bf
t})}\Big{)}{\partial\over\partial t_{k}}\,{t_{2}\over s({\bf
t})}+\\!D_{3}\delta\Big{(}{{\bf t}\over s({\bf
t})}\Big{)}{\partial\over\partial t_{k}}\,{t_{3}\over s({\bf t})}.$

In particular, if $0\neq{\bf t}\in\mathbb{R}^{3}_{+}$ then $s({\bf
t})^{-1}{\bf t}\in\Delta_{3}$, furthemore $s({\bf t})^{-1}{\bf
t}\in\bigcup_{k=1}^{3}\Delta_{3,k}$ whenever $0\neq{\bf
t}\in\partial\mathbb{R}^{3}_{+}$. Thus, as a consequence of $(4.17)$, we get

$D_{k}\widehat{\delta}({\bf t})\\!=\\!\sum_{\ell=1}^{3}D_{1}\delta\Big{(}{{\bf
t}\over s({\bf t})}\Big{)}{\partial\over\partial t_{k}}{t_{\ell}\over s({\bf
t})}\\!=\\!D_{1}\delta\Big{(}{{\bf t}\over s({\bf
t})}\Big{)}{\partial\over\partial t_{k}}{s({\bf t})\over s({\bf
t})}\\!=\\!0\quad\big{(}0\\!\neq\\!{\bf
t}\\!\in\\!\partial\mathbb{R}^{3}_{+}\big{)}$

It follows $D_{k}\chi_{0}^{*}({\bf t})=D_{k}D_{k}\chi_{0}({\bf t})$ for
$0\neq{\bf
t}\in\partial\mathbb{R}^{3}_{0}=\big{\\{}(t_{1},t_{2},t_{3})\in\mathbb{R}_{+}^{3}:t_{1}t_{2}t_{3}=0\big{\\}}$
and $k=1,2,3$ whence, in view of Theorem $3.5$ we concluce that
${\bm{\Pi}}^{*}$ with $\chi_{1}^{*}=\chi_{1}$ is indeed an RSD tuple.

5\. Affinity invariance

5.1. Definition. An RSD tuple
${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ (and the procedure
$\hbox{\Goth F}^{\bm{\Pi}}$) is affinity invariant if

$\hbox{\Goth F}^{\bm{\Pi}}_{\bf P\\!,U}f=f|{\rm Conv}({\bf P})$

for all admissible pairs $[{\bf P,\\!U}]$ $\big{(}$i.e. ${\rm Conv}({\bf P})$
is a non-degenerate triangle, ${\bf p}_{\ell}\\!-\\!{\bf
p}_{m}\\!\not\parallel\\!{\bf p}_{n}$ $\big{(}(\ell,m,n)\in
S_{3}\big{)}\big{)}$ and affine (linear+costant) functions
$f\\!\\!:\\!\mathbb{R}^{2}\\!\\!\to\\!\mathbb{R}$.

5.2. Remark. (a) Affinity invariance implies range shift property.

(b) Given any non-degenerate triangle ${\bf T}\\!=\\!{\rm Conv}\\{{\bf
p}_{1},{\bf p}_{2},{\bf p}_{3}\\}\\!\subset\\!\mathbb{R}^{2}$, any affine
function $\mathbb{R}^{2}\to\mathbb{R}$ is a linear combination of the weights
$\lambda_{1},\lambda_{2},\lambda_{3}$. Hence to verify the affinity invariance
of ${\bf\Pi}$, it suffices to prove the relation $\hbox{\Goth
F}^{\bm{\Pi}}_{\bf P\\!,U}\lambda_{1}({\b{x}})=\lambda_{1}({\bf x})\ \
\big{(}{\bf x}\in{\rm Conv}({\bf P})\big{)}$ for all admissible pairs $[{\bf
P,\\!U}]$.

(c) In view of Remark $2.4$(c) and the identities $G_{1}({\bf x}-{\bf
p}_{k})=\lambda({\bf x})-\delta_{1k}$ resp. $G_{i}({\bf p}_{j}-{\bf
p}_{k})=\lambda_{i}({\bf p}_{j})-\lambda_{i}({\bf
p}_{k})=\delta_{ij}-\delta_{ik}$, we can write

$\displaystyle{\hbox{\Goth F}}^{\bf\Pi}_{\bf
P,\\!U}\lambda_{1}=\sum_{k=1}^{3}\Big{[}\lambda_{1}({\bf
p}_{k})\Psi_{0}(\lambda_{k})+G_{1}({\bf x}-{\bf
p}_{k})\Psi_{1}(\lambda_{k})\Big{]}-$ $\displaystyle\hskip
8.53581pt-\hskip-8.53581pt\sum_{(\ell,m,n)\in
S_{3}}\hskip-2.84526pt{G_{\ell}{\bf u}_{n}\over G_{n}{\bf
u}_{n}}\Big{[}\lambda_{1}({\bf
p}_{\ell})\chi_{0}(\lambda_{\ell},\lambda_{m},\lambda_{n})+G_{1}({\bf
p}_{m}\\!-\\!{\bf
p}_{\ell})\chi_{1}(\lambda_{\ell},\lambda_{m},\lambda_{n})\Big{]}=$
$\displaystyle\hskip
14.22636pt=\Psi_{0}(\lambda_{1})+\Psi_{1}(\lambda_{1})(\lambda_{1}-1)+\Psi_{1}(\lambda_{2})\lambda_{1}+\Psi(\lambda_{3})\lambda_{1}-$
$\displaystyle\hskip 19.91692pt-\\!{G_{1}{\bf u}_{3}\over G_{3}{\bf
u}_{3}}\Big{\\{}\chi_{0}(\lambda_{1},\lambda_{2},\lambda_{3})\\!-\\!\chi_{1}(\lambda_{1},\lambda_{2},\lambda_{3})\Big{\\}}\\!-\\!{G_{2}{\bf
u}_{3}\over G_{3}{\bf u}_{3}}\chi_{1}(\lambda_{2},\lambda_{1},\lambda_{3})-$
$\displaystyle\hskip 19.91692pt-\\!{G_{1}{\bf u}_{2}\over G_{2}{\bf
u}_{2}}\Big{\\{}\chi_{0}(\lambda_{1},\lambda_{3},\lambda_{2})\\!-\\!\chi_{1}(\lambda_{1},\lambda_{3},\lambda_{2})\Big{\\}}\\!-\\!{G_{3}{\bf
u}_{2}\over G_{2}{\bf
u}_{2}}\chi_{1}(\lambda_{3},\lambda_{1},\lambda_{2})\\!-\\!0\\!-\\!0.$

with the ordering
$(1,\\!2,\\!3),(2,\\!1,\\!3),(1,\\!3,\\!2),(3,\\!1,\\!2),(2,\\!3,\\!1),(3,\\!2,\\!1)$
for ${\rm S}_{3}$.

5.3. Lemma. If the RSD tuple
${\bm{\Pi}}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ is affinity invariant then
necessarily

$\displaystyle\Psi_{1}(t)=\Psi_{0}(t)\ \ \hbox{\rm and}\ \
\Psi_{0}(t)+\Psi_{0}(1-t)=1\quad(0\leq t\leq 1),$ $None$
$\displaystyle\chi_{0}(t_{1},t_{2},t_{3})=\chi_{1}(t_{1},t_{2},t_{3})+\chi_{1}(t_{2},t_{1},t_{3})\hskip
19.91692pt\big{(}(t_{1},t_{2},t_{3})\in\Delta_{3}\big{)}.$

Proof. Since ${\bm{\Pi}}$ is an RSD tuple, by Definition $3.2$, the
directional derivatives $F^{\prime}({\bf x}^{3}_{t}){\bf u}_{3}$ along the
side $[{\bf p}_{2},{\bf p}_{3}]$ with generic point ${\bf x}^{1}_{t}=t{\bf
p}_{2}+(1-t){\bf p}_{3}$ of the function $F=\hbox{\Goth F}^{\bm{\Pi}}_{\bf
P,\\!U}\lambda_{1}$ are the linear combinations

$\displaystyle F^{\prime}({\bf x}_{t}^{1}){\bf u}_{1}$
$\displaystyle=\Psi_{1}(t)F^{\prime}({\bf p}_{2}){\bf
u}_{1}+\Psi_{1}(1-t)F^{\prime}({\bf p}_{3}){\bf u}_{1}=$
$\displaystyle=\Psi_{1}(t)G_{1}{\bf u}_{1}+\Psi_{1}(1-t)G_{1}{\bf
u}_{1}=\big{[}\Psi_{1}(t)+\Psi_{1}(1-t)\big{]}G_{1}{\bf u}_{1}.$

Due to the admissibility of $[{\bf P,\\!U}]$, $G_{1}{\bf u}_{1}\neq 0$. In
case of the affinity invariance of ${\bm{\Pi}}$, $F=\lambda_{1}$ with
$F^{\prime}({\bf x}_{t}^{1}){\bf u}_{1}=G_{1}{\bf u}_{1}$ implying
$\Psi_{1}(t)+\Psi_{1}(1-t)=1$. Also if $F=\hbox{\Goth
F}^{\bm{\Pi}}\lambda_{1}=\lambda_{1}$ then, as a consequence of the relations
$(2.7)$, at the points ${\bf x}^{3}_{t}=t{\bf p}_{1}+(1-t){\bf p}_{2}$ we have

$\displaystyle t$ $\displaystyle=\lambda_{1}({\bf x}^{3}_{t})=F({\bf
x}^{3}_{1})=\Psi_{0}(t)-\Psi_{1}(t)(1-t)+\Psi_{1}(1-t)t,$
$\displaystyle=\big{[}\Psi_{0}(t)-\Psi_{1}(t)\big{]}+t\big{[}\Psi_{1}(t)+\Psi_{1}(1-t)\big{]}=\big{[}\Psi_{0}(t)-\Psi_{1}(t)\big{]}+t$

whence the first statement in $(5.4)$ is immediate.

The proof of the second statement relies upon the fact that the formula for
$\hbox{\Goth F}^{\bm{\Pi}}_{\bf P,U}$ in Remark $5.2$(c) must be independent
of the choice of the vectors in ${\bf U}$. Using the argument of the proof of
Proposition $4.4$, this means that the expression

$\displaystyle\Psi_{0}(\lambda_{1})+\Psi_{1}(\lambda_{1})(\lambda_{1}-1)+\Psi_{1}(\lambda_{2})\lambda_{1}+\Psi(\lambda_{3})\lambda_{1}-$
$\displaystyle\hskip
19.91692pt-\\!\alpha_{3}\Big{\\{}\chi_{0}(\lambda_{1},\lambda_{2},\lambda_{3})\\!-\\!\chi_{1}(\lambda_{1},\lambda_{2},\lambda_{3})\Big{\\}}\\!-\\!(-1-\alpha_{3})\chi_{1}(\lambda_{2},\lambda_{1},\lambda_{3})-$
$\displaystyle\hskip
19.91692pt-\\!\alpha_{2}\Big{\\{}\chi_{0}(\lambda_{1},\lambda_{3},\lambda_{2})\\!-\\!\chi_{1}(\lambda_{1},\lambda_{3},\lambda_{2})\Big{\\}}\\!-\\!(-1-\alpha_{2})\chi_{1}(\lambda_{3},\lambda_{1},\lambda_{2})$

must be independent of the scalars
$\alpha_{1},\alpha_{2},\alpha_{3}\in\mathbb{R}$. In particular it follows
$\chi_{0}(\lambda_{1},\lambda_{2},\lambda_{3})\\!-\\!\chi_{1}(\lambda_{1},\lambda_{2},\lambda_{3})\\!-\\!\chi_{1}(\lambda_{3},\lambda_{1},\lambda_{3})=0$
which completes the proof.

5.5. Remark. (a) According to $(5.4)$, affinity invariant tuples are of the
form

${\bm{\Pi}}=\big{[}\Psi,\Psi,\chi_{0},\chi_{1})\big{]},\qquad\chi_{0}\big{|}\Delta_{3}=2\chi_{1}^{[s]}\big{|}\Delta_{3}$
$None$

with a d-symmetric shape function $\Psi$ and the symmetrization $(4.13)$ of
$\chi_{1}$.

(b) From the final argument in the proof of Lemma $5.3$ it readily follows the
below converse statement.

5.7. Corollary. Assume ${\bm{\Pi}}=[\Psi,\Psi,\chi_{0},\chi_{1}]$ is an RSD-
tuple with $(5.6)$. Then, independently of the choice of the vectors ${\bf
u}_{1},{\bf u}_{2},{\bf u}_{3}$ we get

$\hbox{\Goth F}^{\bm{\Pi}}_{\bf
P\\!,U}\lambda_{1}=\sum_{i=1}^{3}\Psi(\lambda_{i})\lambda_{1}+\chi_{1}(\lambda_{2},\lambda_{1},\lambda_{3})+\chi_{1}(\lambda_{3},\lambda_{1},\lambda_{2}).$
$None$

5.9. Proposition. An RSD tuple ${\bm{\Pi}}$ is affinity invariant if and only
if it is of the form ${\bf\Pi}=[\Psi,\Psi,\chi_{0},\chi_{1}]$ with range shift
property, $(5.6)$ and

$\chi_{1}(t_{2},t_{1},t_{3})\\!+\\!\chi_{1}(t_{3},t_{1},t_{2})\\!=t_{1}\hskip-19.91692pt\sum_{\hskip
14.22636pt(\ell,m,n)\in
S_{3}}\hskip-19.91692pt\chi_{1}(t_{\ell},t_{m},t_{n})\quad\big{(}(t_{1},t_{2},t_{3})\\!\in\\!\Delta_{3}\big{)}.$
$None$

Proof. Necessity: Assume the affinity invariance of ${\bf\Pi}$. According to
$5.5$ and Corollary $5.7$, we can write
${\bm{\Pi}}=[\Psi,\Psi,\chi_{0},\chi_{1}]$ with

$\lambda_{1}=\sum_{i=1}^{3}\Psi(\lambda_{i})\lambda_{1}+\chi_{1}(\lambda_{2},\lambda_{1},\lambda_{3})+\chi_{1}(\lambda_{3},\lambda_{1},\lambda_{2}).$
$None$

On the other hand (Remark $5.2$(a), Lemma $5.3)$ ${\bm{\Pi}}$ has range shift
property with $(5.6)$ and
$1=\sum\limits_{i=1}^{3}\Psi(\lambda_{i})\\!+\\!\chi_{0}(\lambda_{1},\lambda_{2},\lambda_{3})\\!+\\!\chi_{0}(\lambda_{2},\lambda_{3},\lambda_{1})\\!+\\!\chi_{0}(\lambda_{3},\lambda_{1},\lambda_{2})$
by $(4.9)$. Hence, in view of $(5.6)$ we conclude that

$1=\sum_{i=1}^{3}\Psi(\lambda_{i})+\hskip-8.53581pt\sum_{(\ell,m,n)\in
S_{3}}\hskip-5.69054pt\chi_{1}(\lambda_{\ell},\lambda_{m},\lambda_{n}).$
$None$

We obtain $(5.10)$ by subtracting $(5.11)$ from $\lambda_{1}\cdot(5.12)$.

Sufficiency: We obtain $(5.11)$ meaning the affinity invariance of
${\bm{\Pi}}$ by adding the equations
$0=\chi_{1}(\lambda_{2},\lambda_{1},\lambda_{3})\\!+\\!\chi_{1}(\lambda_{3},\lambda_{1},\lambda_{2})-\\!\lambda_{1}\hskip-19.91692pt\sum\limits_{\hskip
14.22636pt(\ell,m,n)\in
S_{3}}\hskip-22.76219pt\chi_{1}(\lambda_{\ell},\lambda_{m},\lambda_{n})$

and $\lambda_{1}\cdot(5.12)$.

5.13. Corollary. A tuple ${\bf\Pi}=[\Psi_{0},\Psi_{1},\chi_{0},\chi_{1}]$ with
range shift property and such that $\chi_{1}=\chi_{0}\varphi$ for some
$\varphi\in{\cal C}^{1}\big{(}{\rm dom}(\chi_{0})\big{)}$ is affinity
invariant if for every $(t_{1},t_{2},t_{3})\in\Delta_{3}$ we have

$\displaystyle 1=\varphi(t_{1},t_{2},t_{3})+\varphi(t_{2},t_{1},t_{3}),$
$\displaystyle\chi_{0}(t_{3},t_{1},t_{2})\varphi(t_{3},t_{1},t_{2})+\chi_{0}(t_{2},t_{1},t_{3})\varphi(t_{2},t_{1},t_{3})=$
$\displaystyle\hskip
19.91692pt=t_{1}\Big{[}\chi_{0}(t_{1},t_{2},t_{3})+\chi_{0}(t_{2},t_{3},t_{1})+\chi_{0}(t_{2},t_{3},t_{1})\Big{]}.{5.14}$

5.15. Example. The tuple
${\bm{\Pi}}\big{[}\Phi,\Phi,\chi_{0},\chi_{0}\varphi\big{]}$ is affinity
invariant with

$\Phi(t)=t^{3}(10-15t+6t^{2}),\ \
\chi_{0}(t_{1},t_{2},t_{3})=30t_{1}^{2}t_{2}^{2}t_{3},\ \
\varphi(t_{1},t_{2},t_{3})=t_{2}+{1\over 2}t_{3}.$

Proof. The range shift property of ${\bm{\Pi}}$ is established in Example
$4.10$. The identities $(5.14)$ follow with straightforward calculation.

References

  * [1] Stachó, L.L.; A Simple Affine-Invariant Spline Interpolation over Triangular Meshes Mathematics 2022, 10(5), 776;

  * [2] Stachó, L.L.; Locally Generated Polynomial C1-Splines over Triangular Meshes. Miskolc Mathematical Notes 2022, 23(2), 897–911.

  * [3] Zlámal, M.; Zěnišek, A. Mathematical aspects of the FEM. In Technical Physical and Mathematical Priciples of the FEM; Kolaž, V., Kratoci-Ivil, F., Zlamal, M., Zenisek, A., Eds.; Publising House: Praha, Czech Republic, 1971; pp. 15–39.

  * [4] I.V. Sergienko, O.M. Lytvyn, O.O. Lytvyn and O.I. Denisova; Explicit formulas for interpolating splines of degree 5 on the triangle, Cybernetics and Systems Analysis 2014, 50(5), 670–678.

  * [5] Hahmann, S.; Bonneau, G.-P. Triangular $G^{1}$ interpolation by 4-splitting domain triangles. Comput. Aided Geom. Des. 2000, 17, 731–757.

  * [6] Cao, J.; Chen, Z.; Wei, X.; Zhang, Y.J. A finite element framework based on bivariate simplex splines on triagle configurations. Comput. Methods Appl. Mech. Eng. 2019, 357, 112598.

  * [7] Whitney, Hassler; Analytic extensions of differentiable functions, Transactions of the Amer. Math. Soc. 1934, 36(1), 63–89.

  * [8] Berger, Marcel; Geometry I; Springer, New York, 2009.