Nash twist and Gaussian noise measure on isometric $C^1$ maps

Starting with a short map $f_0:I\to \mathbb R^3$ on the unit interval $I$, we construct random isometric map $f_n:I\to \mathbb R^3$ (with respect to some fixed Riemannian metrics) for each positive integer $n$, such that the difference $(f_n - f_0)$ goes to zero in the $C^0$ norm. The construction of $f_n$ uses the Nash twist. We show that the distribution of $n^{1/2} (f_n - f_0)$ converges (weakly) to a Gaussian noise measure.