AN ANALOGUE OF FRENET FORMULAS FOR THE DARBOUX TRIEDRON CURVE ON THE SURFACE

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Frenet and Darboux trihedrons are accompanying for the guide curve. The position of the Frenet trihedron on the curve is uniquely determined by its first and second derivatives. The Darboux trihedron is the accompanying to the curve on the surface. Both trihedrons move along the curve so that one of the orthos is tangent to the curve. If in the Frenet trihedron one of the faces is the tangent plane of the curve, then in the Darboux trihedron the corresponding face is tangent to the surface, that is, there is a certain angle between these faces of the trihedrons, which can change when they move along the curve. Accordingly, this angle exists in pairs between two other vertices of trihedrons. If the surface is a plane, then this angle is zero and both trihedrons coincide. The position of the Frenet trihedron is determined by the parametric equations of the curve, and the position of the Darboux trihedron is determined by the parametric equations of the surface and the curve on it. The curve can be specified in the function of an arbitrary parameter. On the surface, the curve can be obtained by establishing a certain relationship between the independent variables of the surface. If the curve on the surface is defined not as a function of an arbitrary parameter, but as a function of a natural parameter, that is, as a function of the length of its own arc, then in this case the Frenet formulas are valid, which play a very important role in differential geometry. In this case, the movement of the Darboux trihedron will also depend on the natural parameter. When the trihedrons move, they rotate around the orts of the tangents, which coincide, and an angle is formed in pairs between the other vertices, the value of which depends on the length of the arc of the curve, that is, the value of the angle is a function of the natural parameter. The paper uses the dependence of the angle as a function of the natural parameter between the trihedrons to obtain the formulas for the Darboux trihedron, which are analogous to the Frenet formulas for the Frenet trihedron. For this, the kinematics of both trihedrons as they move along a curve are used. This movement can be decomposed into translational in the direction of the tangent to the curve and rotational around the axis of instantaneous rotation. The translational motion along the common orts of the tangent is the same for both trihedrons, but the rotational motion around them differs. Based on the mathematical description of these movements, formulas for the Darboux trihedron are obtained in the article, which are analogous to Frenet's formulas for the Frenet trihedron.