  The scaling behavior of the closed trajectories of a moving particle
generated by randomly placed rotators or mirrors on a square or triangular
lattice in the critical region are investigated. We study numerically two
scaling functions: $f(x)$ related to the trajectory length distribution $n_S$
and $h(x)$ related to the trajectory size $R_S$ (gyration radius) as introduced
by Stauffer for the percolation problem, where $S$ is the length of a closed
trajectory. The scaling function $f(x)$ is in most cases found to be symmetric
double Gaussians with the same characteristic size exponent $\sigma=0.43\approx
3/7$ as was found at criticality. In contrast to previous assumptions of an
exponential dependence of $n_S$ on $S$, the Gaussian functions lead to a
stretched exponential dependence of $n_S$ on $S$, $n_S\sim e^{-S^{6/7}}$.
However, for the rotator model on the partially occupied square lattice, an
alternative scaling function near criticality is found, leading to a new
exponent $\sigma '=1.6\pm0.3$ and a super exponential dependence of $n_S$ on
$S$. The appearance of the same exponent $\sigma=3/7$ describing the behavior
at and near the critical point is discussed. Our numerical simulations show
that $h(x)$ is essentially a constant, which depends on the type of lattice and
on the concentration of the scatterers.
