An invariant approach for curvature analysis of conjugate surfaces

Curvature analysis of conjugate surfaces is important and difficult in the theory of gearing. By introducing a rotation tensor for rigid-body motion and a surface curvature tensor for the curvatures of surface, tensor expressions of curvature relations between conjugate surfaces of line contact and point contact are directly obtained based on tensor analysis. Especially the result of point contact shows a clear explanation for Litvinʹs local synthesis. When involving practical computation, convenient reference coordinate systems can be chosen and the methods and tools for matrix computation can be employed. Application to generation, local synthesis and tooth contact analysis of face-milled spiral bevel gears is presented. The proposed invariant approach and the obtained results can be applied to design, manufacture and simulation of other complex surfaces.