The paper presents a general and versatile model describing the kinematic and geometric properties relating the centrodes, the conjugate curves, and the path of contact in a transmission through direct contact between any bodies. From a prescribed motion transmission and a desired path of contact, the characteristics of the conjugate tooth profiles, such as sliding ratio, tooth curvature, induced tooth curvature, conjugate boundary, and interference can be explicitly expressed even before the tooth profiles are obtained. The model is made possible by expressing the path of contact with the Frenet frame on the centrodes. The paper establishes a general theory suitable for optimal tooth profile design. The model is applicable in situations with constant or various transmission ratio as well as constant or various center distance.

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