An approximate analytical theory is constructed for the optical anisotropy of a circular weakly bending filament. It is argued that such circles exhibit very nearly dynamical mean local cylindrical symmetry, despite their inherent curvature. This theory and the corresponding approximate analytical theory for weakly bending rods are tested by fitting each theory to the results of Brownian dynamics simulations, in which all of the relevant forces are included. A rigorous derivation is presented for the force arising from the torsion potential, which couples torsional strain to writhing and crankshaft motions, and a new more convenient expression is obtained. Simulations of equilibrium trajectories performed with and without this force show that it has no significant effect on the optical anisotropy of either circular or linear filaments with parameters appropriate for DNA. However, when large net torsional strains are introduced into planar circles, this force enormously enhances the rate at which twist is converted into writhe during the subsequent nonequilibrium trajectories (Chirico, G.; Langowski, J. Biopolymers 1994, 34, 415-433). The analytical theories give good fits to the simulated anisotropies, and the ratios of best-fit torsion constants to the input values are relatively close to 1.0. These ratios are tabulated as correction factors to be applied to best-fit torsion constants obtained by fitting experimental anisotropy data.