A method is given for determining the characteristics of geodesic lines on the surfaces of shells of revolution and the winding machine characteristics that will ensure geodesic winding of the shell. The mathematical description aims at a computer solution of the derived relations and differential equations. The required formulas are presented in such a way that in the general solution for any surface of revolution the variables are the function ρ(z) and its derivatives, which depend on the nature of the specific surface. Therefore, in the computer program only the subroutine for these variables needs to be changed. The winding scheme considered ensures geodesic winding provided that only two characteristics-the angle of rotation of the mandrel and the reciprocating motion of the winding head-are suitably related. Values of the function ρ(z) and its derivatives are given for several surfaces (cone, ellipsoid of revolution, torus) and examples of the determination of the characteristics are presented for an elliptical shell.