Urgency of the research. The grinding surface effectiveness of the rotation surfaces with increasing productivity, improvement quality and energy conservation is a problem of relevance to modern mechanical engineering, which can be solved on CNC machine tools using face finishes with a refilled cup circle.New promising methods make it possible to control the geometry of contact and the loading of local areas within the contact due to a change in the distance between the rotation axes of the detail and tool.Target setting. In the advanced studies sanctified to the similar subjects, specified, that an angular management the orientation of instrument is able considerably to promote the productivity and quality of treatment; it is possible to process the various forms of surface the instrument of geometrical simply-shaped at simple relative motions. The actual selection parameters are debugging of software grinding machines for the effective implementation of new methods.Actual scientific researches and issues analysis. The problem of effectively forming surfaces, including rotation, while driving a mutual orientation, usually angular, engaged in a number of local scientists, including Rodyn P.R., Ravska N.S., Kalchenko V.I., and other. In addition, the specialists of UNKER (Germany) created a way and efficient equipment for the processing of stepped shafts.Uninvestigated parts of general matters defining. However on efficiency his form and sharpening kinematics influence largely. These questions in modern literature have only a partial ground and need creation new equipments.The research objective. The purpose of this work is to determine the processing parameters when managing the contact by changing the axial distance. On this basis, theoretically explore the new highly effective methods of grinding surfaces of rotation by the end of the abrasive tool of the cup shape.The statement of basic materials. For precise shaping of the surface of the part, it is necessary at the contact point to maintain the equality of the inclination of the tangent, the conformity of the diametric dimensions and to impose conditions on the radii of curvature of the component in the axial plane and the tool. For processing the fallen section of the profile of the part in the direction of the axial feed, such conditions are lower values of the radius of curvature of the tool, approaching the required value of the corresponding section of the workpiece. These requirements can be ensured by rational choice of diametric sizes, by changing the values of the angle, which characterizes the cone of the tight end, relative longitudinal and transverse feeds. Using matrices of coordinate transformations, mathematical 3D models of the circle surface, machining and forming on the machine tool are constructed taking into account the values of longitudinal and transverse feeds.Conclusions. The resulting system of equations and work programs, compiled in the MathCAD program package, allows us to determine the mode parameters and the thickness of the sections in a convenient for practical use, taking into account all kinematic components, in order to load the tool to the marginal level determined by the cutting power. The materials of theoretical studies, calculations and their comparison prove the possibility of a significant increase in the efficiency of the treatment by grinding the surfaces of the rotation, filled to the cone by the end of the circle. The results of the work can be implemented on existing equipment after a minor upgrade, and simulation - to assess the thermal tension of these methods.