Formation of Surfaces Under Kinetic Displaying

This work is the development of previously published ones in the journal "Geometry and Graphics" as follows: "Kinematic Correspondence of Rotating Spaces" (№ 1, 2013) and "Formation of Cyclic Surfaces in Kinetic Geometry" (№ 4, 2017). Many of mechanisms make rotational movement, wherein rotating parts of one mechanism "invade" into the zone of rotation for another rotating mechanism’s parts. At the same time, in addition to rotation, they can make other movements, both translational and rotational nature. The theory of kinetic geometry, of which this work is an integral part, is developed in order to avoid collisions of two or more parts of different mechanisms with each other. This is a rather complicated problem in mechanical engineering, in the mining industry, in metallurgy, and in space navigation, where there are no objects that are at rest. Therefore, the kinetic theory of matching for rotating spaces R1 3 and R23 when they are independent from each other movement is quite relevant. In this work have been considered cases for mapping of geometric figures of one space to another one when these figures are moving inside their space R13 . A theory which is presented has been called kinetic geometry, as it relates to engineering problems associated with gearings. These problems were addressed for the first time and drew-up as inventions. A monograph entitled "Introduction to Kinetic Geometry" is currently being prepared for publication.