The technical note proposes the use of the generalized cyclic pursuit control strategy for distributed control of a set of agents to render global formation patterns that are characterized in terms of epicyclic motions. By investigating the root distribution of the generalized cyclic pursuit control system, a unique configuration with multiple imaginary-axis eigenvalues is identified. Design criteria to achieve such a configuration are delineated. The design leads to a class of epicyclic trajectories for agents and, as a result, a class of patterns can be established to serve the purpose of cooperative tracking, area surveillance, and exploration.