Cycloid mechanism, with its cycloidal path of the output link, is renowned for the exceptional fluid dynamics it provides. Achieving this cycloidal motion through controlling the mechanism’s motion is crucial. In this study, firstly a kinematic model of the cycloid mechanism is constructed to elucidate the uniqueness of the control point; then a cycloid mechanism based on the constraints of the control point is proposed to explore the influence of the control point phase on the motion law; finally, the motion law of the cycloid mechanism under different control points is regulated and analyzed, and the motion law is classified based on the radial offset and axial deviation angle of the control point. These findings provide a comprehensive foundation for the motion control of cycloid mechanisms.