AbstractThe paper represents the first effort that extends the free-form geometric modeling technique to conjugation design between surfaces and tooth profile synthesis in gearing. Traditionally, conjugate profiles in gearing are limited to specific curves or surfaces such as involutes that are known satisfying the conjugation condition. The difficulty of a conjugation design lies on the coupling of two conjugate surfaces. This paper introduces spline conjugation and thus offers unlimited freedom and flexibility to the design of conjugate tooth profiles that allows one to pursue profile optimization for the best performance. A general model is introduced to describe the relationship between conjugate curves via contact path and sufficient conditions are obtained to justify non-undercutting. The use of splines in contact path offers the versatility of free-form surface modeling to conjugation design while maintains the traditional involutes as a special case. The paper also demonstrates how control points ...