Formation of Cyclic Surfaces in Kinetic Geometry

This paper is an evolution of the "Kinematic Compliance of Rotating Spaces" paper, previously published in the "Geometry and Graphics" journal №1, 2013. A great many of mechanisms are making rotational movement, wherein rotating parts of one mechanism are "invading" into a rotation zone belonging to parts of another rotating mechanism. The challenge is to prevent the collision of rotating parts belonging to two or more details with each other. This problem is particularly sensitive for machine engineering. In space navigation, where, in principle, there are no objects that are at rest, the problem of satellites collision with astronomical bodies rotating around their axes is also the urgent one. Therefore, the theory of kinematic matching for rotating spaces R31 and R32 when they are moving independently from each other is urgent too. Each of two considered spaces may have a uniform or non-uniform movement in a given direction, a curved movement or a rotational movement around the axis specified for each space. In this paper has been considered the formation of cyclic surfaces obtained by rotation of one space relative to another one and different orientations of the generating line relative to the axes. Has been considered one of the options for rotating spaces, when their axes are parallel. In such a case the generating line is located in the following positions: it is straight and parallel to the axis; it is straight and intersects the axis; the rectilinear generator is in a plane that is parallel to the plane of the axes; the generating line is a straight line of general position; the generating line is a space curve. Has been demonstrated application of the rotating spaces theory in mining, chemical and machine tool industries, made in the form of inventions, confirmed by copyright certificates of the USSR.