Shape transformation of a growing string in a plane

We propose a model of a string which grows at a constant rate and study the characteristic properties of shape transformations. Numerical calculations show that the string grown from a straight line comes to assume a complex shape confined in a small region in which we can observe a characteristic length. From a statistical point of view, we can say that the string with the complex shape is an assembly of many arcs whose mean radius is approximately equal to the characteristic length. The radius varies, depending on several parameters, but the curvature histogram can be modified to a unique form by rescaling the length scale. This means that all solutions in our model of a growing string are statistically equivalent. The scaling properties and several other quantities which characterize the system are also studied.