  Harmonic generalized barycentric coordinates (GBC) functions have been used
for cartoon animation since an early work in 2006\cite{JMDGS06}. A
computational procedure was further developed in \cite{SH15} 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.
