  We first extend the construction of generalized barycentric coordinates (GBC)
based on the vertices on the boundary of a polygon $\Omega$ to a new kind of
GBCs based on vertices inside the $\Omega$ of interest. For clarity, the
standard GBCs are called boundary GBCs while the new GBCs are called interior
GBCs. Then we present an analysis on these two kinds of harmonic GBCs to show
that each GBC function whose value is $1$ at a vertex (boundary or interior
vertex of $\Omega$) decays to zero away from its supporting vertex
exponentially fast except for a trivial example. Based on the exponential decay
property, we explain how to approximate the harmonic GBC functions locally.
That is, due to the locality of these two kinds of GBCs, one can approximate
each of
  these GBC functions by its local versions which is supported over a
sub-domain of $\Omega$. The local version of these GBC function will help
reduce the computational time for shape deformation in graphical design. Next,
with these two kinds of GBC functions at hand, we can use them to approximate
the solution of the Dirichlet problem of the Poisson equation. This may provide
a more efficient way to solve the Poisson equation by using a computer which
has graphical processing unit(GPU) with thousands or more processes than the
standard methods using a computer with one or few CPU kernels.
