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Using the canonical Monte Carlo simulation technique, we study a Reggecalculus model on triangulated spherical surfaces. The discrete model isstatistical mechanically defined with the variables $X$, $g$ and $\rho$, whichdenote the surface position in ${\bf R}^3$, the metric on a two-dimensionalsurface $M$ and the surface density of $M$, respectively. The metric $g$ isdefined only by using the deficit angle of the triangles in {$M$}. This is insharp contrast to the conventional Regge calculus model, where {$g$} dependsonly on the edge length of the triangles. We find that the discrete model inthis paper undergoes a phase transition between the smooth spherical phase at$b to infty$ and the crumpled phase at $b to 0$, where $b$ is the bendingrigidity. The transition is of first-order and identified with the one observedin the conventional model without the variables $g$ and $\rho$. This impliesthat the shape transformation transition is not influenced by the metricdegrees of freedom. It is also found that the model undergoes a continuoustransition of in-plane deformation. This continuous transition is reflected inalmost discontinuous changes of the surface area of $M$ and that of $X(M)$,where the surface area of $M$ is conjugate to the density variable $\rho$.

In-plane deformation of a triangulated surface model with metric degrees of freedom