Authors describe the derivation of equations to determine the deformation parameters of the longitudinal axis of a thin rod for a given strain tensor continuum. A single wire of a steel rope is subject to complex deformation when machining, its form and internal stresses are changing as a result. Application of an equations which reflect the most general laws describing form changing of a thin rod requires sophisticated analytical transformations and engaging visual picture of spatial displacements of single points of a rod. It making difficult to conclude calculation formulas. The problem is solved in the framework of a small displacement and deformation hypothesis using tensor analysis. Formulas derived from the new equations coincide with the known results previously obtained on the basis of Clebsch equations and principle of kinematic analogy. On one hand, it confirms the validity of the proposed method and on the other hand it is an additional verification of known formulas. Some examples have been given to illustrate the efficiency of the general equations in tensor form: analyzing the sinusoid forming on a deformable plane, as well as for calculating the deformations of thin helical elements while stretching and twisting helical wire rope. Now there is no need to use a visual picture to describe the displacement of the spatially curved axis of a wire, and all the analysis is carried out by a uniform algorithm. The proposed method to calculate small deformations of a thin rod for a given strain tensor of continuum might be further applied when improving the existing and developing new software for the design of production processes in manufacturing of wire rope and cable.