Analytical connection between the Frenet trihedron of a direct curve and the Darboux trihedron of the same curve on the surface
Authors: Nesvidomin Andrii , Pylypaka Serhii , Volina Tetiana , Rybenko Irina , Rebrii Alla 
Published: September 2024

Frenet and Darboux trihedrons are the objects of research.
At the current point of the direction curve of the Frenet trihedron, three mutually perpendicular unit orthogonal vectors can be uniquely constructed. The orthogonal vector of the tangent is directed along the tangent to the curve at the current point. The orthogonal vector of the main normal is located in the plane, which is formed by three points of the curve on different sides from the current one when they are maximally close to the current point. It is directed to the center of the curvature of the curve. The orthogonal vector of the binormal is perpendicular to the two previous orthogonal vectors and has a direction according to the rule of the right coordinate system. Thus, the movement of the Frenet trihedron along the base curve, as a solid body, is determined.
The Darboux trihedron is also a right-hand coordinate system that moves along the base curve lying on the surface. Its orthogonal vector of the tangent is directed identically to the Frenet trihedron, and other orthogonal vectors in pairs form a certain angle ε with the orthogonal vectors of the Frenet trihedron. This is because one of the orthogonal vectors of the Darboux trihedron is normal to the surface and forms a certain angle ε with the binormal. Accordingly, the third orthogonal vector of the Darboux trihedron forms an angle ε with the orthogonal vector of the normal of the Frenet trihedron. This orthogonal vector and orthogonal vector of the tangent form the tangent plane to the surface at the current point of the curve, and the corresponding orthogonal vectors of the tangent and the normal of the Frenet trihedron form the tangent plane of the curve at the same point. Thus, when the Frenet and Darboux trihedrons move along a curve with combined vertices, there is a rotation around the common orthogonal vector point of the tangent at an angle ε between the osculating plane of the Frenet trihedron and the tangent plane to the surface of the Darboux trihedron. These trihedrons coincide in a separate case (for a flat curve) (ε=0).
The connection between Frenet and Darboux trihedrons – finding the expression for the angle ε, is considered in the article. The inverse problem – the determination of the movement of the Darboux trihedron at a given regularity of the change of the angle ε, is also considered. A partial case is considered and it is shown that for a flat base curve at ε=const, the set of positions of the orthogonal vector of normal forms a developable surface of the same angle of inclination of the generators. In addition, the inverse problem of finding the regularity of the change of the angle ε between the corresponding orthogonal vectors of the trihedrons allows constructing a ruled surface for the gravitational descent of the load, conventionally assumed to be a particle. At the same time, the balance of forces in the projections on the orthogonal vectors of the trihedron in the common normal plane of the trajectory is considered. This balance depends on the angle ε.