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A deformation of Penner's simplicial coordinate

We produce a one-parameter family of coordinates $\{Ψ_h\}_{h\in\mathbb{R}}$ of the decorated Teichmüller 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 Teichmüller space in the $Ψ_h$ coordinate becomes an explicit convex polytope $P(T)$ independent of $h$, and if $h<0$, the decorated Teichmüller 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 Teichmüller space is reproduced.