  In \cite{rigidity}, Luo introduced a $\psi_{\lambda}$ edge invariant which
turns out to be a coordinate of the Teichm\"uller space of a surface with
boundary. And he proved that for $\lambda \geq 0$, the image of the
Teichm\"uller space under $\psi_{\lambda}$ edge invariant coordinate is an open
cell. In this paper we verify his conjecture that for $\lambda<0$, the image of
the Teichm\"uller space is a bounded convex polytope.
