Geometrical simulation of the driving gear detail surface as an envelope

The mathematical model of the hipoid driving gear on the basis of ec-gearing is under consideration. Equations are obtained for the in-coming detail surface motion that generates a family of surfaces for which one can find an exact equation of the envelope, i.e., of the out-coming detail surface. It is proved that this way of obtaining an exact equation of the out-coming detail surface can be applied for any in-coming detail surface under the condition that the family belongs to the $C^2$ class.