ANALYTICAL RELATION BETWEEN THE FRENET TRIHEDRON OF A DIRECT CURVE AND THE DARBOUX TRIHEDRON OF THE SAME CURVE ON A SURFACE

The Frenet trihedron plays an extremely important role in the theory of differential geometry. It concerns curved lines, which we will call direction lines. At the current point of the direction curve, three mutually perpendicular unit orthogonals of this trihedron can be uniquely constructed. The tangent ort is directed along the tangent to the curve at the current point. The ort of main normal is located in the plane formed by three points of the curve on different sides of the current one when they approach the current point as close as possible. It is directed towards the center of curvature of the curve. The ort of binormal is perpendicular to two orts and and has a direction according to the rule of the right-hand coordinate system. Thus, the motion of the Frenet trihedron along the direction curve, as a rigid body, will be definite. It can be described in a fixed coordinate system through nine direction cosines or through three Euler angles in a spherical coordinate system, which are determined through direction cosines. The direction cosines themselves are determined through the first and second derivatives of the equations of the direction curve. The Darboux trihedron is also a right coordinate system that moves along the direction curve lying on the surface. Its tangential ort is also directed along the tangent to the curve at the current point, and the other orts pairwise form a certain angle ε with the Frenet trihedron orts. This is due to the fact that one of the orts of the Darboux trihedron is normal to the surface and forms a certain angle ε with the binormal. Accordingly, the third ort of the Darboux trihedron forms an angle ε with ort of the Frenet trihedron. The orts and of the Darboux trihedron form a tangent plane to the surface at the current point of the curve, and the corresponding orts and of the Frenet trihedron form a tangent plane to the curve at the same point. Thus, when the Frenet and Darboux trihedra move along a curve with aligned vertices, they rotate around a common orthogonal ort by an angle ε between the tangent plane of the Frenet trihedra and the tangent plane to the surface of the Darboux trihedra. In a special case (for example, for a plane curve), these trihedra coincide, i.e. ε=0. The article considers the analytical realtion between the Frenet and Darboux trihedra, i.e. finding an expression for the angle ε. The inverse problem is considered - determining the motion of the Darboux trihedra for a given pattern of change in the angle ε. A special case is considered and it is shown that for a plane directed curve at ε=const the set of ort positions forms a developed surface of the same slope of the generators. In particular, at ε=90° such a surface will be a cylinder for which the flat directed curve is an ort section, and for ε=0° – the plane in which the directed curve is located.