【正文】 In 3D xyz Cartesian space , a 1parameter family of surfaces can be given in parametric form as (1)where ζ is the parameter of the family, and u1, u 2, are the parameters for a particular surface of the family. Then, the envelope for the family described in Equation 1 satisfies equation 1 and the following Equation: (2)where the righthand side is a constant zero7. Litvin showed the proving process of the theorem in detail. If we can solve Equation2 and substitute into equation1to eliminate one of the three parameters u1, u 2, and ζ , we may obtain the envelope in parametric form. However, one important thing should be pointed out here. Equations 1 and 2 can also be satisfied by the singular points of surfaces described below I the family, even if they do not belong to the envelope. Points which are regular points of surfaces of the family and satisfy Equation 2 lie on the envelope. The condition for the singular points of a surface is discussed here.. a parametric representation of a surface is (3)where u1 and u 2 are the parameters of the surface. A point of the surface that corresponds to in a given parameterization is called a singular point of the parameterization. A point of a surface is called singular if it is singular for every parameterization of the surface7. A point that is singular in one parameterization of a surface may not be singular in other parameterizations. For a fixed value of ζ, equations 1 and 2 represent, in general, a curve on the surface which corresponds to this value of the parameter. If this is not a line of singular points, the curve slso lies on the envelope. The surface and the envelope are tangent to each other along this curve. Such curves are called characteristic lines of the family7. they can be used to find the contact lines between the surfaces of the cylindrical cam and the follower.THEORY OF ENVELOPES FOR DETERMINATION OF CYLINDRICALCAM PROFILESOn the basis of the theory of envelopes, the profile of a cylindrical cam can be regarded as the envelope of the family of follower surfaces in relative positions between the cylindrical cam and the follower while the motion of cam proceeds. In such a condition, the input parameters of the cylindrical cam serve as the family parameters. Because the cylindrical or conical follower surface can be expressed in parametric form without difficulty, the theory of envelopes for a 1parameter of surfaces represented in parametric form (see equations 1 and 2) is used in determining the analytical equations of cylindricalcam profiles. As stated in the last section, a check for singular points on the follower surface is always needed. Figure 1a shows a cylindricalcam mechanism with a translating conical follower. The axis which the follower translates along is parallel to the axis of rotation of the cylindrical cam. a is the offset, that is, the normal distance between the longitudinal axis of the cam and that of the follower. R and L are the radius and the axial length of the cam, respectively.The rotation angle of the cylindrical cam is Ф2 about its axis. The distance traveled by the follower is s1 , which is a function of parameter Ф2 ,as follows: (4)The displacement relationship (see equation 4 ) for the translating follower is assumed to be given.In figure 1b, the relative position of the follower when the follower moves is shown. The follower is in the form of a frustum of a cone. The semicone angle is α, and the smallest radius is r. δ1 is the height, and μ is the normal distance from the xz plane to the base of the cone. The fixed coordinate system Oxyz is located in such a way that the z axis is along the rotation axis of the cam, and the y axis is parallel to the longitudinal axis of the conical follower. the unit vectors of the x axis, y axis and z axis are i , j and k, respectively. By the use of the envelope technique to generate the cylindricalcam profile, the cam is assumed to be stationary. The follower rotates about the dam axis in the