Optical axis motion in a ring resonator is investigated as a function of resonator mirror misalignment by constructing an equivalent paraxial model and applying the ray matrix formalism. Analytical expressions are derived for the optical axis motion. The paraxial model of the ring is shown to imply a linear resonator as a specific case, and the ring resonator expressions collapse to the familiar Krupke-Sooy results for that case. Using this method, new misalignment expressions are determined for more complex linear resonators. Uncorrectable misalignment conditions caused by toroidal mirror parameter errors are studied analytically and with a geometric optics code, and resulting phase front errors are given for a special case. These results are also examined as a basis for toroidal mirror quality specifications.