# The Harmonic GBC Function Map is a Bijection if the Target Domain is Convex

Chongyang Deng111School of Science, Hangzhou Dianzi University, Xiasha,
Hangzhou 310018, China. This author is supported by the National Science
Foundation of China under grant # 61872121 and Zhejiang Provincial Science and
Technology Program in China under Grant # 2021C01108. Tsung-wei
Hu222Department of Mathematics, University of Georgia, Athens, GA 30602. Ming-
Jun Lai<EMAIL_ADDRESS>Department of Mathematics, University of Georgia,
Athens, GA 30602. This author is supported by the Simons Foundation
collaboration grant #864439.

###### Abstract

Harmonic generalized barycentric coordinates (GBC) functions have been used
for cartoon animation since an early work in 2006[15]. A computational
procedure was further developed in [20] for deformation between any two
polygons. The bijectivity of the map based on harmonic GBC functions is still
murky in the literature. In this paper we present an elementary proof of the
bijection of the harmonic GBC map transforming from one arbitrary polygonal
domain $V$ to a convex polygonal domain $W$. This result is further extended
to a more general harmonic map from one simply connected domain $V$ to a
convex domain $W$ if the harmonic map preserves the orientation of the
boundary of the domain $V$. In addition, we shall point out that the harmonic
GBC map is also a diffeomorphism over the interior of $V$ to the interior of
$W$. Finally, we remark on how to construct a harmonic GBC map from $V$ to $W$
when the number of vertices of $V$ is different from the number of vertices of
$W$ and how to construct harmonic GBC functions over a polygonal domain with a
hole or holes. We also point out that it is possible to use the harmonic GBC
map to deform a nonconvex polygon $V$ to another nonconvex polygon $W$ by a
good arrangement of the boundary map between $\partial V$ and $\partial W$.
Several numerical deformations based on images are presented to show the
effectiveness of the map based on bivariate spline approximation of the
harmonic GBC functions.

Key Words: GBC functions, harmonic GBC map, harmonic map, bijection

Mathematics Subject Classification: 31A05, 35J25, 30C60, 53A10

##  1 Introduction

We are interested in the bijective property of the map based on harmonic GBC
functions which transforms from one arbitrary polygonal domain
$V\subset\mathbb{R}^{2}$ to a convex polygonal domain
$W\subset\mathbb{R}^{2}$. Let us first recall the GBC functions. Given a
polygon $V\in\mathbb{R}^{2}$ of $n$-sides with vertices ${\bf
v}_{i},i=1,\cdots,n$, the functions $\phi_{i},i=1,\cdots,n$ satisfying the
following conditions:

$\begin{cases}\sum_{i=1}^{n}\phi_{i}({\bf x})&=1\cr\sum_{i=1}^{n}\phi_{i}({\bf
x}){\bf v}_{i}&={\bf x}\cr\phi_{i}({\bf x})&\geq 0,\quad
i=1,\cdots,n,\end{cases}$ (1)

for all ${\bf x}=(x,y)\in V$ and being linear on edges of $V$ with
$\phi_{i}({\bf v}_{j})=1$ if $i=j$ and $=0$ if $i\not=j$. Such functions are
called generalized barycentric coordinates (GBC) or GBC functions over $V$.

The study of generalized barycentric coordinates (GBC) started from a seminal
work in [22]. Since then, there are many GBCs which have been constructed. For
example, harmonic GBC functions in the 3D setting was explained in [15]
together with many interesting and useful properties. We refer to a recent
survey in [5] and a book [12] edited by leading experts K. Hormann and N.
Sukumar. Recently, a general minimization method is proposed to construct GBC
functions (cf. [4]). More precisely, consider the following minimization
problem:

$\begin{array}[]{ll}\min\\{F(\phi_{1},\cdots,\phi_{n})\mathrel{\mathop{\ordinarycolon}}\hbox{subject
to }&\sum_{i=1}^{n}\phi_{i}({\bf x})=1,\sum_{i=1}^{n}\phi_{i}({\bf x}){\bf
v}_{i}={\bf x},\cr&{\bf x}\in V,\phi_{i}\geq 0,i=1,\cdots,n\\},\end{array}$
(2)

where $F$ is a convex function and $V$ has $n$ boundary vertices. For example,
one can choose

$\displaystyle
F(\phi_{1},\cdots,\phi_{n})=\sum_{i=1}^{n}\int_{V}|\nabla\phi_{i}({\bf
x})|^{2}d{\bf x}$

which leads to a new way to compute harmonic GBCs based on bivariate spline
functions (cf. [3] and [17]). We refer to [4] for a detail explanation. The
GBCs have found their applications in geometric design. See, e.g. [13], [24],
[16], and etc.. In addition, they found their applications in numerical
solution of partial differential equations (e.g. [18], [19], [7], and etc.).

One interesting property of the GBCs functions is that it can form a transform
or map from one polygon to another polygon easily. That is, letting $V$ be a
polygon with boundary vertices ${\bf v}_{i},i=1,\cdots,n$ arranged in the
counter-clockwise direction and $W$ be another polygon with boundary vertices
${\bf w}_{i},i=1,\cdots,n$ in the counter-clockwise direction, if
$\phi_{i},i=1,\cdots,n$ are GBC functions over $V$ satisfying (1), we can
define

${\bf F}(x,y)=\sum_{i=1}^{n}\phi_{i}(x,y){\bf w}_{i},\quad(x,y)\in V.$ (3)

Then ${\bf F}$ is a map from $V$ to $W$ when $W$ is a convex polygon. Clearly,
the image $Im({\bf F})\subset W$ due to the properties of (1). One can show
that the map is onto which will be proved in the next section. The injectivity
of the Wachspress from a convex polygon $V$ to a convex polygon $W$ was
established in [FK10]. However, the mean value coordinates may not be
injective when a convex polygon $V$ is a pentagon as shown in [FK10]. It is
interesting to know if the harmonic GBC map ${\bf F}$ is injective or not.
This paper is also motivated by the work in [20], where the researchers
proposed an approach to construct a map ${\bf F}$ from a polygonal domain $V$
to another polygonal domain $W$ by introducing an intermediate convex
polygonal domain $\Theta$ such that

${\bf F}=\Phi_{1}^{-1}\circ\Phi_{0},$

where $\Phi_{0},\Phi_{1}$ are two harmonic GBC maps from $V\mapsto\Theta$ and
$W\mapsto\Theta$, respectively. The researchers in [20] simply used the Radó-
Kneser-Choquet theorem to explain that $\Phi_{0}$ is bijective from
$V\mapsto\Theta$ and so is $\Phi_{1}$. Note that the classic Radó-Kneser-
Choquet theorem only explains the bijectivity of the unit ball to a convex
domain. See the wiki page of the Radó-Kneser-Choquet theorem on-line. See [1]
for a recent revisit of the Radó-Kneser-Choquet theorem. If one combines it
with a Riemann map from a polygonal domain $V$ to the unit ball $B$, one can
establish the bijection between an arbitrary polygon $V$ to a convex function
$W$. Unfortunately, the Radó-Kneser-Choquet theorem does not hold for a higher
dimensional setting other than $\mathbb{R}^{2}$. One can not use the Radó-
Kneser-Choquet theorem for the bijectivity of 3D harmonic GBC maps. Although
the researchers in [20] have many nice examples in the $\mathbb{R}^{3}$
setting, it is necessary to study the bijectivity of harmonic mappings in a
general dimensional space. These motivate us to study the bijectivity of the
harmonic map based on GBC functions from polygon $V$ to a convex polygon $W$.
For simplicity, we consider only the situation in the Euclidean space
$\mathbb{R}^{2}$ in this paper.

The purpose of this paper is to give a proof of the bijective property of
$\Phi_{i}$, $i=0,1$ in the $\mathbb{R}^{2}$ setting. See Theorem 2.1. In
addition, we extend the study to the bijectivty of a general harmonic map
under some additional boundary condition. See Theorem 2.2 in the next section.
We leave the study of the nontrivial extension of the results in this paper to
the multidimensional setting in a future publication. Finally, we point out
that polygons $V$ and $W$ of interest may not need to have the same number of
vertices. One may add inactive vertices to the boundary of $V$ or $W$ so that
the number of vertices of $V$ and $W$ will be the same and the computation of
the bijection ${\bf F}$ can be carried out based on the method in [20]. See
three numerical examples in Figure 1– Figure 3 which are computed by a similar
method to the one in [20] with a difference that we employed bivariate spline
approximation of harmonic GBC functions discussed in [4].

Figure 1: An Example of Image Deformation Based on Bijections from Polygons to
Rectangle. Clearly the polygon on the right has more than 4 vertices. We added
inactive vertices on the left domain to match the number of the vertices of
the domain on the right.  Figure 2: An Example of Image Deformation Based on
Bijections from Polygons to Rectangle. Note that the polygon on the right has
more than 4 vertices.  Figure 3: An Example of Image Deformation from an
L-shaped Polygon to the H-shaped polygon via a square domain based on the
approach in [20]

From these images we can see how the points on the right-hand sides in Figure
1 and Figure 2 are mapped to the points on the left-hand sides by using the
harmonic GBC map.

In addition to the proof of Theorems 2.1 and 2.2, we also make a few comments
in the end of this paper. One example will be shown that a harmonic GBC map
from one nonconvex domain to another nonconvex domain can be bijective.
Harmonic GBC functions can be also constructed over a polygonal domain with
hole or holes. Does the harmonic GBC map based on these GBC functions have a
bijective property? Besides the bijectivity, we will mention that the harmonic
GBC map is also a diffeomorphism when $W$ is convex. Although the harmonic GBC
map does not have the property like a conforming map, the harmonic GBC map can
be easily computed since harmonic GBC functions over all boundary vertices of
$V$ can be done in parallel and once these functions are found, they can be
used to form a bijection to any convex set $W$. A similar study of the
bijectivity of harmonic GBCs in the 3D setting will be commented and
difficulties will be pointed out. It seems the extension to the 3D setting is
nontrivial.

##  2 Main Results and Their Proofs

In this section, let us start with harmonic functions over simply connected
domains in $\mathbb{R}^{2}$. Let $V$ and $W$ be two simply connected domains.
Typically, $V$ and $W$ are polygons. Let
$u\mathrel{\mathop{\ordinarycolon}}V\rightarrow\mathbb{R}$ is a harmonic
function, that is, $\Delta u=0$ over $\mathring{V}$, where
$\Delta=D_{x}^{2}+D_{y}^{2}$ is the standard Laplace operator. For two
harmonic functions $u,v$ on $V$, let ${\bf F}=(u(x,y),v(x,y))$ which will be
called a harmonic map on $V$ which maps $V$ to $\hbox{Im}({\bf F})$, the image
of ${\bf F}$ over $V$.

The harmonic functions have been studied for more than 100 years and many
properties are known. Let us mention a few properties without proofs which
will be used in the following study. The harmonic function $u$ will not
achieve its maximum nor minimum inside $V$ except for a constant function $u$.
When $u$ is a harmonic, so are its derivatives. If a harmonic function $u$ is
a constant over a disk $D$ (open set) inside $V$, then $u$ is a constant over
$V$. Indeed, $\displaystyle\frac{\partial u}{\partial x}$ over $V$ will be
zero over $D$ and hence $\displaystyle\frac{\partial u}{\partial x}$ is zero
over $V$ as a harmonic function is like an analytic function.

Note that when $V,W\subset\mathbb{R}^{2}$ be two separate polygons, $\partial
V$ and $\partial W$ are piecewise linear boundaries, respectively. By the
properties of GBC functions (1) and the definition of ${\bf F}$ (3), one can
easily check that ${\bf F}|_{\partial V}$ is a homeomorphism from $\partial V$
to $\partial W$. We can also see that ${\bf F}|_{\partial V}$ preserve the
orientation of the boundary vertices of $W$ as both of them are in the
counter-clockwise direction.

Our first main result in this paper is to establish the following

######  Theorem 2.1.

Let $V,W\subset\mathbb{R}^{2}$ be two polygons. Let ${\bf
F}=(u(x,y),v(x,y))\mathrel{\mathop{\ordinarycolon}}V\mapsto W$ be a harmonic
GBC function which maps from $V$ to $W$ according to (3). If $W$ is convex,
then ${\bf F}$ is a bijection from $V$ to $W$.

We remark that the above result can be generalized to the general harmonic map
${\bf F}=(u,v)\mathrel{\mathop{\ordinarycolon}}V\mapsto W$ under the
assumptions that

  * •

(1) ${\bf F}|_{\partial V}$ is an orientation preserving homeomorphism from
the boundary $\partial V$ of $V$ to the boundary $\partial W$ of $W$.

  * •

(2) ${\bf F}\in C^{0}(V)\cap C^{1}(\mathring{V})$.

In this general setting, we have

######  Theorem 2.2.

Let $V,W\subset\mathbb{R}^{2}$ be two simply connected domains in
$\mathbb{R}^{2}$. Suppose $u,v$ are harmonic functions over $V$. Let ${\bf
F}=(u(x,y),v(x,y))\mathrel{\mathop{\ordinarycolon}}V\mapsto W$ be a harmonic
map from $V$ to $W$. Suppose that ${\bf F}$ satisfies (1) and (2) above. Then
if $W$ is convex, ${\bf F}$ is a bijection from $V$ to $W$.

Note that the map ${\bf F}$ is a bijection if ${\bf F}$ is surjective and also
injective in the sense that ${\bf F}$ maps $V$ onto $W$ and the mapping is
one-to-one. Before proving Theorem 2.1, we first establish the surjectivity.

######  Lemma 2.3.

As the setting in Theorem 2.2, ${\bf F}$ maps $V$ onto $W$ if $W$ is convex.

###### Proof.

We will show that the image $\hbox{Im}({\bf F})$ of ${\bf F}$ over $V$ is $W$,
i.e. $\hbox{Im}({\bf F})=W$.

First of all, if ${\bf F}$ is a harmonic GBC, it is easy to see
$\hbox{Im}({\bf F})\subset W$ by the GBC properties (1). In general, we claim
$\hbox{Im}({\bf F})\subset W$ for a general harmonic map ${\bf F}$. Otherwise,
there exists a point ${\bf p}=(u_{0},v_{0})\in\hbox{Im}({\bf F})\setminus W$.
Then there is a line separate $W$ and ${\bf p}$ because $W$ is convex. That
is, there exist real numbers $\alpha$ and $\beta$ such that $\alpha\cdot
u_{0}+\beta\cdot v_{0}>0$ , and $\alpha\cdot u^{\prime}+\beta\cdot
u^{\prime}<0$ for all $(u^{\prime},v^{\prime})\in W$. For ${\bf x}=(x,y)\in
V$, let $\Phi({\bf x})=\alpha\cdot u(x,y)+\beta\cdot v(x,y)$ be an harmonic
function over $V$ and ${\bf F}^{-1}({\bf p})\in\mathring{V}$. Then we have
$\Phi({\bf F}^{-1}({\bf p}))>0$ and $\Phi({\bf x})<0$ for all ${\bf
x}\in\partial V$, this leads to a contradiction because it violate the maximum
principle of the harmonic function $\Phi$. Hence, the claim follows.

On the other hand, we want to show that $W\subset\hbox{Im}({\bf F})$. To do
so, we introduce the concept called the contractible curve. If a Jordan curve
$\gamma$ inside $V$ can be continuously shrunk within $V$ to a point in $V$,
then $\gamma$ is called a contractible curve in $V$. We note that $V$ is
simply connected, $\partial V$ is a contractible curve in $V$. Since ${\bf
F}|_{\partial V}$ is a homeomorphism (continuously bijection) to $\partial W$,
$\partial W$ must also be contractible in $\hbox{Im}({\bf F})$.

Now suppose $W\not\subset\hbox{Im}({\bf F})$. Then there exists a point ${\bf
w}\in W\setminus\hbox{Im}({\bf F})$. Thus, $\partial W$ is contractible in
$\hbox{Im}({\bf F})\subset(W\setminus{\bf w})$. As $W\setminus{\bf w}$ is
larger than $\hbox{Im}{\bf F}$, $\partial W$ would be contractible in
$(W\setminus{\bf w})$ which is obviously not possible as $W\setminus{\bf w}$
is like a punctured disk. Hence, $W\subset\hbox{Im}({\bf F})$. ∎

Next we shall show that ${\bf F}$ is an injection. Recall that ${\bf
F}=(u,v)^{\top}$ is homeomorphism at $(x_{0},y_{0})\in V$ if its Jacobian
matrix $J_{0}$ is not singular at the point $(x_{0},y_{0})\in\mathring{V}$.
Suppose that ${\bf F}(x,y)$ has a nonzero Jacobian determinant at ${\bf
x_{0}}=(x_{0},y_{0})$, that is,

${\rm det}(J_{0})={\rm det}([\nabla u(x_{0},y_{0});\nabla
v(x_{0},y_{0})])={\rm det}\begin{bmatrix}\frac{\partial}{\partial
x}u(x_{0},y_{0})&\frac{\partial}{\partial
y}u(x_{0},y_{0})\cr\frac{\partial}{\partial
x}v(x_{0},y_{0})&\frac{\partial}{\partial y}v(x_{0},y_{0})\end{bmatrix}\neq
0.$ (4)

Since ${\rm det}(J_{0})\neq 0$, $\left\|J_{0}\cdot{\bf x}\right\|\geq
c\left\|{\bf x}\right\|$ for some positive constant $c$ for all ${\bf x}$ in a
neighborhood of $(x_{0},y_{0})$. The Taylor expansion at ${\bf
x_{0}}=(x_{0},y_{0})$ gives us

$\displaystyle\left\|{\bf F}({\bf x})-{\bf F}({\bf x_{0}})\right\|$
$\displaystyle=$ $\displaystyle\left\|J_{0}\cdot({\bf x}-{\bf
x_{0}})\right\|+o(\left\|{\bf x}-{\bf x_{0}}\right\|)$ (5) $\displaystyle\geq$
$\displaystyle c\left\|{\bf x}-{\bf x_{0}}\right\|+o(\left\|{\bf x}-{\bf
x_{0}}\right\|).$ (6)

Hence there exist a $r_{0}>0$ such that ${\bf F}({\bf x})\neq{\bf F}({\bf
x_{0}})$ for all ${\bf x}\in B_{r_{0}}({\bf x_{0}})$. Now consider the closed
ball $\overline{B}_{r_{0}}({\bf x_{0}})\subset\mathring{V}$if $r_{0}>0$ small
enough. Apply the same trick to each point on $\overline{B}_{r_{0}}({\bf
x_{0}})$. We let

$r=\min\left\\{r_{k}\,|\,{\bf x_{k}}\in\overline{B}_{r_{0}},{B}_{r_{k}}({\bf
x}_{k})\subset\mathring{V},{\bf F}({\bf x})\neq{\bf F}({\bf
x_{k}})\,for\,all\,{\bf x}\in B_{r_{k}}({\bf x_{k}})\right\\}.$

The compactness of $\overline{B_{r_{0}}({\bf x}_{0})}$ implies that $r>0$.
Then it is clear that ${\bf F}$ restricted on $B_{r/2}({\bf x_{0}})$ is a
homeomorphism from $B_{r/2}({\bf x_{0}})$ to its image $\hbox{Im}({\bf F})$
over the ball.

The discussion above leads to the following

######  Lemma 2.4.

Suppose that the Jacobian determinant of ${\bf F}$ is not zero at every point
of $V$. Then ${\bf F}$ is a local homeomorphism on $\mathring{V}$, i.e..
continuous, one-to-one and onto map over a neighborhood $N(x,y)\subset V$ for
$(x,y)\in\mathring{V}$.

Unfortunately, the nonzero Jacobian determinant of ${\bf F}$ over $V$ does not
guarantee the global injectivity as shown in the following example:

######  Example 2.5.

Let ${\bf F}=(u,v)$ with $u(x,y)=\exp(x/2)\cos(y\exp(-x))$ and
$v(x,y)=\exp(x/2)\sin(y\exp(-x))$. Then the Jacobian determinant is equals to
$1/2$ for all $(x,y)\in\mathbb{R}^{2}$. However, ${\bf F}(0,0)={\bf
F}(0,2\pi)$. i.e. ${\bf F}$ is not injective globally.

On the other hand, due to the GBC property, we can see that when $W$ is a
triangle, the map ${\bf F}$ is an injection. Indeed, suppose that there are
two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ in $V$ which is also a triangle
such that ${\bf F}(x_{1},y_{1})={\bf F}(x_{2},y_{2})$. Then we have

$\sum_{i=1}^{3}(\phi_{i}(x_{1},y_{1})-\phi_{i}(x_{2},y_{2})){\bf w}_{i}=0.$

Without loss of generality, we may assume that ${\bf w}_{1}=(0,0)$. As $W$ is
a triangle with nonzero area, it follows that
$(\phi_{i}(x_{1},y_{1})-\phi_{i}(x_{2},y_{2}))=0,i=2,3$. That is,
$\phi_{i}(x_{1},y_{1})=\phi_{i}(x_{2},y_{2}),i=2,3$. By the GBC property,
$1=\phi_{1}(x_{2},y_{2})+\phi_{2}(x_{2},y_{2})+\phi_{3}(x_{2},y_{2})$ and
hence, $\phi_{1}(x_{1},y_{1})=\phi_{1}(x_{2},y_{2})$. Now

$(x_{1},y_{1})=\sum_{i=1}^{3}\phi_{i}(x_{1},y_{1}){\bf
v}_{i}=\sum_{i=1}^{3}\phi_{i}(x_{2},y_{2}){\bf v}_{i}=(x_{2},y_{2}).$

We see that $x_{1}=x_{2},y_{1}=y_{2}$ and hence, ${\bf F}$ is injective. It is
also known that when $W$ is not a convex polygon, the GBC map may not be an
injection. See a counter example in [14].

In the following, let us prove that the determinant of the Jacobian matrix of
${\bf F}$ is nonzero if ${\bf F}$ is harmonic and $W$ is convex. To do so, we
need more properties of harmonic functions. Let us begin with the set of
critical points of harmonic function.

######  Definition 2.6.

Suppose that $u$ is harmonic on $\Omega\subset\mathbb{R}^{2}$. The critical
set $C(u)$ of $u$ is the subset of $\Omega$ where the $\nabla u$ vanish. More
precisely,

$C(u)\coloneqq\\{{\bf x}\in\Omega|\quad\nabla u({\bf x})=0\\}.$

######  Lemma 2.7.

If $u$ is non-constant harmonic function, then $C(u)$ is an empty set or a
collection of finitely many isolated points.

###### Proof.

Let ${\bf i}=\sqrt{-1}$ and write ${\bf z}=x+{\bf i}y$. Denote by
$u_{x},u_{y}$ the partial derivatives of $u$. Note that $f({\bf z})=u_{x}-{\bf
i}u_{y}$ is a holomorphic complex function as one can easily check that $f$
satisfied Cauchy-Riemann equations. A fundamental theory of complex analysis
states that the set of zeros of an holomorphic function could only be empty or
finitely many isolated point(s). Since $\nabla u(x,y)=0$ if and only if
$f({\bf z})=0$, we have $C(u)=\\{{\bf x}\in\Omega|f({\bf z})=0\\}$ is empty or
contains finitely many isolated point(s). ∎

######  Remark.

There are few different ways to prove Lemma 2.7. For example, a good way to
see is that the critical set $C(u)$ of a harmonic function
$u\mathrel{\mathop{\ordinarycolon}}\mathbb{R}^{n}\rightarrow\mathbb{R}$ have
Hausdorff dimension at most $n-2$ as discussed in [9]. So when $n=2$, $C(u)$
is empty or isolated point(s) which are only cases when a set is of Hausdorff
dimension $0$.

Next we discuss the level curve(s) of harmonic function $u$. Fix ${\bf
a}\in\Omega$. Let $\Gamma_{u}({\bf
a})\mathrel{\mathop{\ordinarycolon}}=\\{{\bf x}\in\Omega|u({\bf x})=u({\bf
a})\\}$ be the level curve set of $u$ at $u({\bf a})$. The set will have the
following properties.

######  Lemma 2.8.

Let $u\mathrel{\mathop{\ordinarycolon}}\Omega\rightarrow\mathbb{R}$ be a non-
constant harmonic function. If ${\bf a}\in C(u)$, then $\Gamma_{u}(\bf a)$ are
curves intersect at ${\bf a}$. Moreover, the angle between two neighboring
curves is $\pi/n$, where $n$ is the number of curves in $\Gamma_{u}({\bf a})$.
Also, any of these curves is not a loop.

###### Proof.

WLOG, let ${\bf a}=0$. Consider a function $f({\bf z})=u_{x}-{\bf i}u_{y}$.
Then $f(0)=0$ and $f$ is holomorphic and hence, $f$ has a finite multiplicity,
say $n$ at $0$. Write $f=nz^{n-1}+O(z^{n})$ be the Taylor expansion of $f$ at
$0$. So $u=Re(z^{n})+O(z^{n+1})$. Note that $\Gamma_{z^{n}}(0)$ is $n$ lines
intersects at $0$ with angle $\frac{2\pi}{n}$, so $\Gamma_{u}(0)$ are curves
as described above.

Next if there is a level curve which has a loop, say it encloses a nonempty
open set $D\subset V$. Because $u$ is harmonic on $V$, $u$ is harmonic on $D$
and hence, $u$ must be a constant on $D$. That is, the critical point set of
$u$ contains a nonempty open set $D$. which contradicts to Lemma 2.7. $u$ can
have only isolated critical points. Hence, any curve in the level set is not a
loop. ∎

Next we need the set of extremes.

######  Definition 2.9.

Let $Max_{u}\mathrel{\mathop{\ordinarycolon}}=\\{{{\bf
x}}\in\partial\Omega|u({\bf x})\geq u({\bf y})\forall{\bf y}$ in a
neighborhood of ${\bf x}\\}$ be the set of all local maximal locations.
Similarly, let $Min_{u}\mathrel{\mathop{\ordinarycolon}}=\\{{\bf
x}\in\partial\Omega|u({\bf x})\leq u({\bf y})\forall{\bf y}$ in a neighborhood
of ${\bf x}\\}$ be the set of local minimal locations. We denote by
$\\#Max_{u}$ the number of connected components in the set $Max_{u}$.
Similarly, $\\#Min_{u}$ is the number of connected components in the set
$Min_{u}$.

######  Theorem 2.10.

Suppose that
$u\mathrel{\mathop{\ordinarycolon}}\Omega\rightarrow\mathbb{R}^{2}$ is a
harmonic function. If $\\#Max_{u}=\\#Min_{u}=1$, then $C(u)=\emptyset$.

###### Proof.

If there exist any ${\bf a}\in C(u)$, by Lemma 2.8, $\Gamma_{u}(\bf a)$ are
curves intersect at $\bf a$. Since ${\bf a}\in\Omega$, these curves will
intersect with $\partial\Omega$ $2n$ times, where $n$ is number of level
curves in $\Gamma_{u}(\bf a)$ and $n\geq 2$. That is, the level curves cut the
boundary $\partial\Omega$ into 2n pieces with $n\geq 2$. There must have some
components of $Max_{u}$ or $Min_{u}$ between two intersecting points. So
$\\#Max_{u}\geq 2$ or $\\#Min_{u}\geq 2$ which is a contradiction. ∎

Furthermore, we need more definitions and one more useful lemma. Define a
continuous function $f_{\alpha}=\cos(\alpha)u+\sin(\alpha)v$ over $V$ for each
$\alpha\in[0,\pi)$. Let

$S_{\alpha}\mathrel{\mathop{\ordinarycolon}}=\\{(u,v)\in
W|\cos(\alpha)u+\sin(\alpha)v=\max_{(u^{\prime},v^{\prime})\in
W}\cos(\alpha)u^{\prime}+\sin(\alpha)v^{\prime}\\}$

and

$s_{\alpha}\mathrel{\mathop{\ordinarycolon}}=\\{(u,v)\in
W|\cos(\alpha)u+\sin(\alpha)v=\min_{(u^{\prime},v^{\prime})\in
W}\cos(\alpha)u^{\prime}+\sin(\alpha)v^{\prime}\\}.$

######  Lemma 2.11.

Let ${\bf F}$ , $V$ and $W$ be the same in Theorem 2.2. Then
$\\#Max_{u}=\\#Min_{u}=1$ for any $\alpha\in[0,\pi)$.

###### Proof.

We denote by $\\#S$ the number of connected components in the set $S$ in the
proof. Fixed an $\alpha$ in $[0,\pi]$. Note that $f_{\alpha}$ is the inner
product $(\cos\alpha,\sin\alpha)\cdot{\bf F}$. See an illustration of
$f_{\alpha}$ in Figure 4.

$W$$\bullet$$\bullet$$s_{\alpha}$$\bullet$$S_{\alpha}$$\alpha$$f_{\alpha}$$f_{\alpha}$$W$$\bullet$$s_{\alpha}$$S_{\alpha}$
Figure 4: An illustration of the sets ${S_{\alpha}}$ and $s_{\alpha}$ which
have multiple components when $W$ is a nonconvex (left) and the sets
$S_{\alpha}$ and $s_{\alpha}$ when $W$ is a convex domain (right)

Since $W$ is convex, both $S_{\alpha}$ and $s_{\alpha}$ are a set consisting
of one connected component. See Figure 4 for the cases when $W$ is convex and
is not convex. When $W$ is convex, $S_{\alpha}$ contains only one component of
the image ${\bf F}$. When $W$ is not convex, $S_{\alpha}$ may have two or more
separate components as in the left of Figure 4 where $\alpha=\pi/4$. Because
$f_{\alpha}$ is harmonic, $f_{\alpha}$ can only achieve its maximum on the
boundary. So $S_{\alpha}\subset\partial W$. Because ${\bf F}$ is a continuous
bijection on the boundary $\partial V$ to $\partial W$, ${\bf
F}^{-1}(S_{\alpha})$ is a set of one component and hence $\\#Max_{u}=\\#{\bf
F}^{-1}(S_{\alpha})=1$. Similarly, $\\#Min_{u}=\\#{\bf F}^{-1}(s_{\alpha})=1$.
∎

Note that even though the non-zero determinant of Jacobian implies local
injection, we cannot extend it to $\partial V$ because ${\bf F}$ is not
necessary be differentiable on $\partial V$ as $V$ has corners. However, we
can prove the following

######  Lemma 2.12.

Suppose that ${\bf F}\mathrel{\mathop{\ordinarycolon}}K\rightarrow T$, where
$K\subset V$ is a compact set and $T\subset W$ is connected. Also, ${\bf F}\in
C^{0}(K)$ is a local homeomorphism. The pre-image set ${\bf F}^{-1}(u,v)$ is
finite for all $(u,v)\in\hbox{Im}({\bf F})$.

###### Proof.

Suppose that there is a point $(u,v)\in T$ such that the set ${\bf
F}^{-1}(u,v)\subset K$ has an infinitely many points. Then due to the
compactness of $K$, there is a subset of points which converge to a cluster
point $(x,y)\in K$. Then ${\bf F}$ is no longer a local homeomorphism at
$(x,y)$ which contradicts to the result in Lemma 2.4. ∎

Due to the fact that the pre-image set ${\bf F}^{-1}(u,v)$ consists of
finitely many isolated points, we can further show

######  Lemma 2.13.

Under the same assumptions in Lemma 2.12, the number of all isolated points in
the pre-image set ${\bf F}^{-1}(u,v)$ is a constant overall
$(u,v)\in\mathring{W}$.

###### Proof.

Suppose $T\subset\mathring{W}$ is a connected compact set. Since ${\bf F}$ is
continuous, $K\mathrel{\mathop{\ordinarycolon}}={\bf F}^{-1}(T)$ is a compact
set. ${\bf F}|_{K}$ is a local homeomorphism because $K\subset\mathring{V}$.
We first show that $T_{i}\mathrel{\mathop{\ordinarycolon}}=\\{(u,v)\in
T|\\#{\bf F}^{-1}(u,v)=i\\}$ is an open set for any fixed integer $i\geq 1$,
where $\\#{\bf F}^{-1}(u,v)$ stands for the cardinality of the pre-image set
$\\#{\bf F}^{-1}(u,v)$ of $(u,v)\in T$.

Let $(u,v)\in T_{i}$. Consider ${\bf F}^{-1}(u,v)=\\{{\bf x}_{1},{\bf
x}_{2},\cdots,{\bf x}_{i}\\}$, where ${\bf x}_{j}$ are isolated points in
$\mathring{V}$. Since ${\bf F}$ is a local homeomorphism, we can choose
disjoint neighborhood $N_{j}$ of ${\bf x}_{j}$, $j=1,2,\cdots,i$, where ${\bf
F}$ is a homeomorphism from $N_{j}$ to the image set ${\bf F}(N_{j})$. Define
$R=V\setminus(\cup_{j=1}^{i}N_{j})$. Then $R$ is a close set as $V$ is compact
and hence, ${\bf F}(R)$ is a close set. Now we define an open set
$M=(\cap_{j=1}^{i}{\bf F}(N_{j}))\setminus{\bf F}(R)$ on $T$. Obviously, $M$
is a neighborhood of $(u,v)$, and ${\bf F}^{-1}(M)$ is a union of disjoint
open sets belongs to $\cup_{j=1}^{i}N_{j}$. By the definition of $N_{j}$,
${\bf F}^{-1}(M)\cap N_{j}$ is a homemorphism which means the cardinality
$\\#{\bf F}^{-1}((u^{\prime},v^{\prime}))=i$ for all
$(u^{\prime},v^{\prime})\in M$, so $M\subset T_{i}$ and hence $T_{i}$ is an
open set.

Now since $T_{i}$ is open for any fixed $i$, it follows that
$T={\bigcup}_{i}T_{i}$. As $T$ is connected, it cannot be a disjoint union of
more than $2$ open set. This implies that $T=T_{i}$ for some $i\geq 1$.
Finally, because this is true for all connected set $T\subset\mathring{W}$, we
can conclude that the number of the pre-image set ${\bf F}^{-1}(u,v)$ is a
constant overall $(u,v)\in\mathring{W}$. ∎

We now show that $i=1$. Suppose that $i\geq 2$ on $\mathring{W}$. We pick 4
points $A$, $B$, $C$, and $D$ in the counterclockwise order as shown on the
right of Figure 5. Then we pick a curve called $\Gamma_{1}$ connected $A$ and
$C$, and a curve $\Gamma_{2}$ connected $B$ and $D$ over $W$, where 2 curves
intersect at $x=(u,v)$ on the right of Figure 5. Note that ${\bf F}$ is a
homeomorphism on boundary, but not on the interior. We can see that ${\bf
F}^{-1}(\Gamma_{1})$ has $i$ curves jointing ${\bf F}^{-1}(A)$ and ${\bf
F}^{-1}(C)$ and do not intersects over $\mathring{V}$. Similarly, ${\bf
F}^{-1}(\Gamma_{2})$ has $i$ curves jointing ${\bf F}^{-1}(B)$ and ${\bf
F}^{-1}(D)$ and do not intersects over $\mathring{V}$. By the fact that the
intersections of ${\bf F}^{-1}(\Gamma_{1})$ and ${\bf F}^{-1}(\Gamma_{2})$ is
${\bf F}^{-1}(u,v)$. We conclude that the cardinality $\\#{\bf
F}^{-1}(u,v)=i^{2}>i$, which is a violation of Lemma 2.13 unless $i=1$. Figure
5 shows the case when $i=2$.

Figure 5: An example when the cardinality of the pre-image set is 2

Let us summarize the discussion above in the following

######  Theorem 2.14.

Suppose $V$, $W$ are two polygons in $\mathbb{R}^{2}$. Let ${\bf
F}\mathrel{\mathop{\ordinarycolon}}V\rightarrow W$ be a harmonic GBC map. Then
${\bf F}$ is an injection from $V$ to $W$.

###### Proof.

As we have seen that $i=1$, i.e. the number of isolated points in the pre-
image ${\bf F}^{-1}(u,v)$ is one for each $(u,v)\in\mathring{W}$, the map
${\bf F}$ is injective. ∎

Combine with Lemma 2.4 we have the following

######  Theorem 2.15.

Suppose $V$, $W$ are two simply connected domains in $\mathbb{R}^{2}$. Let
${\bf F}\mathrel{\mathop{\ordinarycolon}}V\rightarrow W$ be a harmonic map
satisfying ${\bf F}\in C^{1}(\mathring{V})\cap C^{0}(V)$. Suppose that ${\bf
F}|_{\partial V}=\partial W$ is an orientation preserved homeomorphism and
determinant of Jacobian is non-vanish on $\mathring{V}$. Then ${\bf F}$ is an
injection from $V$ to $W$.

Now we ready to prove Theorem 2.2.

###### Proof of Theorem 2.2.

We write $(u,v)={\bf F}(x,y)$ and use $f_{\alpha}$ for $\alpha\in[0,\pi)$. By
Lemma 2.11, and Theorem 2.10, we have $C(f_{\alpha})=\emptyset$. This implies
the vector

$\nabla f_{\alpha}=\nabla{\bf
F}(x,y)\begin{bmatrix}\cos(\alpha)\cr\sin(\alpha)\end{bmatrix}\not=0$

for all $(x,y)\in\mathring{V}$ and for any $\alpha\in[0,\pi)$. So $\nabla u$
and $\nabla v$ are linearly independent in $V$. Hence, ${\rm det}(\nabla{\bf
F})={\rm det}([\nabla u;\nabla v])\neq 0$ for all $(x,y)\in\mathring{V}$,
which implies that ${\bf F}$ is a local homeomorphism on $\mathring{V}$ by
Lemma 2.4.

Hence, ${\bf F}$ is an injection by Theorem 2.14 if ${\bf F}$ is a harmonic
GBC map and by Theorem 2.15 if ${\bf F}$ is a general harmonic map. Together
with Lemma 2.3, we complete the proof. ∎

##  3 Remarks

We have the following remarks in order.

  * •

1\. Note that the convexity condition in Theorem 2.1 is a sufficient
condition. It is interesting to see how the map ${\bf F}$ from $V$ to $W$
behaviors when $W$ may not be a convex domain. We use our bivariate spline
harmonic map ${\bf F}$ (cf. [4]) to explore the possibility if the map ${\bf
F}$ is a bijection. It is easy to find many examples that ${\bf F}$ does not
map into $W$ if $W$ is not a convex polygon. However we found a few examples
of polygons which can be mapped into an L-shaped domain bijectively using our
${\bf F}$ in the numerical sense. See Figure 6 for such an example. We show a
J-shaped domain with labeled vertices in $J$ and in $L$. We compute the
determinants of Jacobi matrix numerically at more than 25,000 points and only
found 3 points on the boundary of the $L$-shaped domain where the determinant
of the Jacobi matrix is less than $1e-5$. These three points locate on the
boundary of the L-shaped domain as shown in the red points in Figure 6. The
contour lines on the right of Figure 6 show that the mapping is surjective.
However we are not able to show such a map is indeed a bijection
theoretically.

Figure 6: Contour lines of ${\bf F}$ from a J-shaped domain to an L-shaped
domain

In fact, the mapping between the boundary of $V$ and the boundary $W$ is
important. Many different arrangements of the boundary map between $V$ and $W$
lead to the harmonic GBCC map ${\bf F}$ which is not bijective in our
numerical experiment. We leave the question how to find a good arrangement of
the boundaries of $V$ and $W$ so that the map is bijective as an open problem
for further study.

  * •

2\. As harmonic GBC functions are infinitely many differentiable, we can
conclude that ${\bf F}$ defined in (3) is not only a bijection, it also a
diffeomorphism from $\mathring{V}$ to $\mathring{W}$. It is interesting to
compare our results, e.g. Theorem 2.1 to the main result, i.e. Theorem 1.3 in
[1]. For convenience, let us start the main result below. Let
$B\mathrel{\mathop{\ordinarycolon}}=\\{(x,y)\in\mathbb{R}^{2}\mathrel{\mathop{\ordinarycolon}}x^{2}+y^{2}<1\\}$
denote the unit disk. Recall the following theorem from [1]:

######  Theorem 3.1 (G. Alexssandrini and V. Nesi, 2008[1]).

Let $\Phi\mathrel{\mathop{\ordinarycolon}}\partial
B\mapsto\gamma\subset\mathbb{R}^{2}$ be an orientation preserving
diffeomorphism of class $C^{1}$ onto a simple closed curve $\gamma$. Let $D$
be the bounded domain such that $\partial D=\gamma$. Let ${\bf U}\in
C^{2}(\mathring{B};\mathbb{R}^{2})\cap C^{1}(\overline{B};\mathbb{R}^{2})$ be
the solution to the Dirichlet boundary problem of Laplace equation:

$\displaystyle\Delta{\bf U}(x,y)$ $\displaystyle=$ $\displaystyle
0,\quad(x,y)\in B$ $\displaystyle{\bf U}(x,y)$ $\displaystyle=$
$\displaystyle\Phi,\quad(x,y)\in\partial B.$

Then the mapping ${\bf U}$ is a diffeomorphism of $B$ onto $D$ if and only if

${\rm det}(\nabla{\bf U})>0\hbox{ everywhere on }\partial B.$ (7)

Our result uses a convex domain $W$ instead of the unit $B$. Also, our ${\bf
F}$ maps from a polygon $V$ to $W$ instead of ${\bf U}$ from $B$ to $D$ in
[1]. Our ${\bf F}$ is an orientation preserving homeomorphism over the
piecewise linear boundary $\partial V$ while the theorem above requires the
diffeomorphism $\Phi$. We do not have a condition similar to (7). One
significance of the harmonic GBC map ${\bf F}$ is that the diffeomorphism from
$V$ to $W$ can be easily constructed using the GBC functions.

  * •

3\. It is also interesting to compare the harmonic GBC map with the Riemann
map from any polygon to the disk. The computation of such a Riemann map can be
found in book [8]. Both maps are diffeomorphism from $V$ to $W$ when $W$ is a
disk. It is well-known that the Riemann map is a conforming map which
preserves the angle and relative locations of the domain $V$ as shown in the
images inside [8]. However, a harmonic GBC map gives a distorted image called
image warping as shown in Figure 3 which may be used for creative artworks
such as cartoon images, e.g. in [15]. One advantage of the harmonic GBC map is
the convenience of the construction: once GBC functions over a polygon $V$ are
computed (which can be done in parallel, e.g. GPU), they can be used to map
bijectively to any convex polygon $W$. In addition, the map between the
boundary of $V$ and the boundary of $W$ can be chosen by the user which is a
convenience as the Riemann map does not have this freedom.

  * •

4\. Although the harmonic GBC functions are defined over a polygon, they are
possible to be defined over a polygonal domain with a hole or holes. We refer
to [10] for mean value coordinates over domains with holes. Indeed, suppose a
polygonal domain $P$ with a hole has $n$ outer boundary vertices ${\bf
v}_{i},i=1,\cdots,n$ and $m$ vertices ${\bf h}_{j},j=1,\cdots,m$ on the inner
boundary of the hole. One simply defines $\phi_{1},\cdots,\phi_{n}$ are
nonnegative functions which satisfy the standard GBC boundary conditions with
additional property: $\phi_{i}$’s are zero at the boundary of the hole. Also,
$\phi_{n+i},\cdots,\phi_{n+m}$ are nonnegative functions which are zero on the
outer boundary of $P$ and satisfy the standard GBC boundary conditions over
the boundary of the hole. All these functions $\phi_{1},\cdots,\phi_{n}$ and
$\phi_{n+i},\cdots,\phi_{n+m}$ satisfy the standard GBC properties inside $P$:

$\sum_{i=1}^{n+m}\phi_{i}=1\hbox{ and }\sum_{i=1}^{n}\phi_{i}{\bf
v}_{i}+\sum_{j=1}^{m}\phi_{n+j}{\bf h}_{j}=(x,y),$

where ${\bf v}_{i},i=1,\cdots,n$ are the vertices of the outer boundary of $P$
and ${\bf h}_{j},i=1,\cdots,m$ are the vertices of the inner boundary of $P$.
See Figs. 7 and 8 for contour plots of some harmonic GBC functions
($\phi_{17}$ is now shown) computed based on the bivariate spline solution to
the corresponding Laplace equation over the polygonal domain $P$ with a hole
with an appropriate boundary condition (cf. [3]). It is interesting to know
the bijectivity of the harmonic map based on these GBC functions. We leave the
question to an open problem.

Figure 7: Contour plots of GBC functions over a domain with a hole

Figure 8: Contour plots of more GBC functions over a domain with a hole

  * •

5\. Although we can use 3D GBC functions to construct a deforming mapping from
a 3D polyhedron to a convex polyhedron, say solid ball as in [20], it is easy
to see that any extension of the proof of Theorem 2.1 to the 3D setting may
not be easy. This is because that the proof uses the zero property of
holomorphic functions which has no simple extension in the 3D setting.

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