  We numerically study a triangulated surface model in R^2 by taking into
account a viewpoint of string model. The models are defined by a mapping X from
a two-dimensional surface M to R^2, where the mapping X and the metric g of M
are the dynamical variables. The sum over g in the partition function is
simulated by the sum over bond lengths and deficit angles by using the Regge
calculus technique, and the sum over g is defined to be performed independently
of the sum over X. We find that the model undergoes a first-order transition of
surface fluctuations, which accompanies a collapsing transition, and that the
transitions are reflected in the internal geometry of surface. Fluid surface
models are also studied on dynamically triangulated surfaces, and the
transitions are found to be of second order. The order of the transition
remains unchanged from that of the conventional model defined only by the
variable X both in the fixed-connectivity and the fluid models.
