  We produce a one-parameter family of coordinates
$\{\Psi_h\}_{h\in\mathbb{R}}$ of the decorated Teichm\"{u}ller space of an
ideally triangulated punctured surface $(S,T)$ with negative Euler
characteristic, which is a deformation of Penner's simplicial coordinate
\cite{P1}. If $h\geqslant0$, the decorated Teichm\"{u}ller space in the
$\Psi_h$ coordinate becomes an explicit convex polytope $P(T)$ independent of
$h$, and if $h<0$, the decorated Teichm\"{u}ller space becomes an explicit
bounded convex polytope $P_h(T)$ so that $P_h(T)\subset P_{h'}(T)$ if $h<h'$.
As a consequence, Bowditch-Epstein and Penner's cell decomposition of the
decorated Teichm\"{u}ller space is reproduced.

Geometric Topology;Differential Geometry