Relationships between the curvatures of tooth surfaces in three-dimensional gear systems

A three-dimensional gear system between crossing and intersecting axes is considered under the assumption that the first derivative of the transmission ratio is zero in the vicinity of the point of contact. The following are obtained; (1) an equation that relates the normal curvatures of the tooth surfaces in the section that passes through the vector of relative velocity; (2) a relation between the principal curvatures and principal directions of the two tooth surfaces; and (3) new formulas for determining the reduced curvatures of the surfaces.