The Frénet trihedron, known in differential geometry, is accompanying for a spatial and, as a special case, for a flat curve. Its three mutually perpendicular unit orts are defined uniquely for any point on the curve except for some special ones. Unlike the Frénet trihedron, the Darboux trihedron relates to the surface. Two of its unit orts are located in a plane tangent to the surface, and the third is directed normally to the surface. It can also be accompanying for the curve, which is located on the surface. To this end, one of the orts in the plane tangent to the surface must be tangent to the curve. Trihedra are movable and, with respect to a fixed coordinate system, change their position due to movement and rotation. The object of research is the process of formation of curves and surfaces, as a result of the geometric sum of the bulk motion of the Darboux trihedron and the relative motion of the point in its system under given conditions. In the study of the geometric characteristics of curves and surfaces, it is necessary to have formulas for the transition from the position of the elements of these objects in the system of a moving trihedron to the position in a fixed Cartesian coordinate system. This is exactly what needs to be solved. The results obtained are parametric equations of curves and surfaces that are tied to the initial surface. Nine guide cosines were found, three for each ort. A distinctive feature of this approach in comparison with the traditional one is the use of two systems: fixed and mobile, which is the Darboux trihedron. This approach allows us to consider in a new way the problem of the construction of curves and surfaces. The scope of practical application can be the construction of geometric shapes on a given surface. An example of such a construction is the laying of a pipeline along a given line on the surface. In addition, the sum of the relative motion of a point in a trihedron and the bulk motion of the trihedron itself over the surface gives an absolute trajectory of motion. Its sequential differentiation produces absolute speed and absolute acceleration without finding individual components, including the Coriolis acceleration. This could be used in point dynamics problems

Bevel gears are used to transmit torque between intersecting axes. They demonstrate high reliability and durability of work, as well as a constant gear ratio. The disadvantage of such a transmission is the mutual sliding of the surfaces of the teeth of the gears, which leads to the emergence of friction forces and wear of their working surfaces. In this regard, there is a task to design such bevel gears that would have no slip. Non-circular wheels are understood as a pair of closed curves that rotate around fixed centers and at the same time roll over each other without sliding. They can serve as centroids for the design of cylindrical gears between parallel axes. If the axes of rotation of the wheels intersect, then the gears are called conical. An analog of gears between parallel axes, in which centroids are flat closed curves, for gears with intersecting axes are spherical closed curves. For a bevel gear with a constant gear ratio, such spherical curves are circles on the surface of the sphere, and with a variable gear ratio, spatial spherical curves. This paper considers the construction of closed spherical curves that roll around each other without sliding when they rotate around the axes intersecting in the center of the spheres. These curves are formed from symmetrical arcs of the loxodrome, a curve that crosses all the meridians of the ball at a constant angle. This angle should be 45°, which ensures the intersection of the loxodrome at right angles. Analytical dependences have been derived underlying the calculations of profiles of spherical non-circular wheels and their visualization by means of computer graphics. The results could be used to design non-circular wheels for textile machines, hydraulic machine pumps, pump dispensers, etc.