Determining regularities in the construction of curves and surfaces using the Darboux trihedron

Abstract

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

Construction of mathematical model of particle movement by an inclined screw rotating in a fixed casing

“…One such approach is to move a single particle without taking into account its interaction with other particles. At the same time, the Frénet trihedron [9] or the Darboux trihedron [10] can be used to study the patterns of movement of particles on surfaces. With the help of Frénet trihedron, the complex motion of a point on a plane with a specified displacement of the latter was studied in [9].…”

“…With the help of Frénet trihedron, the complex motion of a point on a plane with a specified displacement of the latter was studied in [9]. Article [10] describes a procedure for determining the regularities of the formation of curves and surfaces using the Darboux trihedron.…”

“…The vector equation ma F = must be written in projections on the axis of the OXYZ coordinate system, taking into account that the friction force F is directed in the opposite direction of the vector V r (10) of the relative velocity of the particle, and the friction force F с is directed in the opposite direction of the vector V а (11) of the absolute velocity of the particle:…”