Loci of points (LOP) equally spaced from two given geometrical figures are considered. Has been proposed a method, giving the possibility to systematize the loci, and the key to their study. The following options have been considered. A locus equidistant from N point and l straight line. N belongs to l. We have a plane that is perpendicular to l and passing through N. N does not belong to l – parabolic cylinder. A locus equidistant from F point and a plane. In the general case, we have a paraboloid of revolution. The F point belongs to the given plane. We get a straight line perpendicular to the plane and passing through the F point. A locus equidistant from a point and a sphere. The point coincides with the sphere center. We get the sphere with a radius of 0.5 R. The point lies on the sphere. We get the straight line passing through the sphere center and the point. The point does not coincide with the sphere center, but is inside the sphere. We get the ellipsoid. The point is outside the sphere. We have parted hyperboloid of rotation. A locus equidistant from a point and a cylindrical surface. The point lies on the cylindrical surface’s axis. We get the surface of revolution which generatix is a parabola. The point lies on the generatrix of the cylindrical surface of rotation. We get a straight line, perpendicular to that generatrix and passing through the cylinder axis. The point does not lie on the axis, but is located inside the cylindrical surface. We get the surface with a horizontal sketch line – the ellipse, and a front sketch lines – two different parabolas. The point is outside the cylindrical surface. A locus consists of two surfaces. The one with the positive Gaussian curvature, and the other – with the negative one.

In technical inventions related to innovative developments, the role of one of the main components belongs to geometry. A follow hypothesis has been adopted: in technical inventions the geometrical component is the determining one. This hypothesis applied to technical inventions can be confirmed by any copyright certificate, any patent both in Russia and abroad. In proposed paper this statement is proved by examples developed based on geometry of following inventions. 1. Screen feeder for sticky masses. Screen feeder’s grates are made in pairs, and between grate pairs there are gaps for screening of material’s size-defined fractions. In the screen-feeder has been proposed such geometry of grates that grates of each pair could clean each other, thereby preventing sticking on the surface and destroying the gaps between the pairs, which transforms the usual screen-feeder with cylindrical grates at an ordinary feeder. 2. Double-screw mixer for paste-like masses. The mixer consists of two contiguous worms. Their surfaces are the helical ones, in cross-section consisting of two quarters of circles stacked at the ends. Such cross-section allows homogenize the mixed paste-like material in the best way, and also deliver it under higher pressure in an extrusion head, that improves a final product. 3. Machine for processing of multi-faceted surfaces. This invention serves for manufacture of worms with a cross-section composed of two, three, etc. pieces of circles of the same radius and angle. Worms, made with this machine, are designed for the above mentioned two-screw mixer. 4. Method of mechanical processing. This method is also intended to manufacture of worms for two-screw mixer.

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.

Dupin cyclide belongs to channel surfaces. These surfaces are the single known ones whose focal surfaces, i.e. surfaces consisting of point sets of centers of curvatures, have been degenerated into two confocal second order curves. In the works devoted to Dupin cyclide and published in the "Geometry and Graphics" journal, are presented various cyclides’ properties and demonstrated application of these surfaces in various industries, mostly in construction. Based on Dupin cyclides’ properties have been developed several inventions relating to drawing devices and having the opportunity to apply in various geometric constructions with the use of computer technologies. It is possible because the Dupin cyclides’ geometric properties suppose not only to create devices recognized as inventions, but also provide an opportunity to apply these properties to write programs for drawing v arious kinds of curves on a display screen: the second order curves, their equidistant in the direction of normals or tangents, as well as to perform other constructions. It should be said that in inventions belonging to technical areas, which include the drawing devices, the geometric component is always decisive. This position with the express aim of technical inventions can justify any copyright certificate of the USSR, any patent of Russia and foreign countries. Unfortunately, currently in schools geometry is not studied as a component of pupil’s intellectual horizons, that broadens his area of interests and teaches to analytical understanding the world, but as an inevitable, almost unnecessary appendage in preparation for the Unified State Examination.

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.