A concave mirror and a cat's eye reflector acting as a resonator mirror coustitute a cat's eye cavity. The parallelism of the incident and the reflected beams achieved by using three kinds of cat's eye reflectors (one ideal cat's eye reflector and two reflectors with errors), and the parameters of Gaussian beam in resonators which respectively use the three kinds of cat's eye reflectors are accurately calculated. We arive at the conclusion that for the Gaussian beam the cat's eye reflector is ideal and gives best parallelism of the incident and the reflected beam when the focal length of the convex lens, the radius of curvature of the concave mirror and the distance between them are equal. In this case the cat's eye cavity has the highest stability along with the largest fundamental mode volume and the least far-field divergence angle of fundamental mode. In the cavity design, one should strive to decrease the focal length of the convex lens in the cat's eye reflector and increase the radius of curvature of the output mirror. The influence on resonator performance when there are errors in the cat's eye reflector is also discussed. This paper gives a theoretical basis for the design of cat's eye cavity lasers.