  The scaling behavior of the closed trajectories of a moving particle
generated by randomly placed rotators or mirrors on a square or triangular
lattice is studied numerically. For most concentrations of the scatterers the
trajectories close exponentially fast. For special critical concentrations
infinitely extended trajectories can occur which exhibit a scaling behavior
similar to that of the perimeters of percolation clusters. In addition to the
two critical exponents $\tau=15/7$ and $d_{f}=7/4$ found before, the critical
exponent $\sigma=3/7$, which is associated with the scaling function for
trajectory size away from criticality, also appears. This exponent determines
structural scaling properties of closed trajectories of finite size when they
approach infinity, at criticality. New scaling behavior was found for the
square lattice partially occupied by rotators, indicating a different
universality class from that of percolation clusters. An argument for the
scaling behavior found along the critical lines is presented.
