Abstract
Helical surfaces are deployable and non-deployable. The first is the surface of a helical conoid widely used in technology, known as the screw. The second one is the surface of an unfolding helicoid or torso-helicoid. In both cases, the surface is formed from a blank - a sweep in the form of a flat ring. If in the first case the sweep can be calculated approximately, since the exact one does not exist due to the properties of non-expanded surfaces, then in the second case it is calculated exactly. However, this does not mean that it is just as possible to form a ready-made product from it - a twist of a helicoid torso. In contrast to non-expanded surfaces, during the manufacture of which the workpiece is subjected to complex stretching and compression deformations, an expanded surface can be obtained by bending with minimal plastic deformations, the magnitude of which depends on the thickness of the sheet. Bending occurs along rectilinear generators, which are theoretically located on the workpiece. In the process of bending, the location of these generators should not change. In the theory of differential geometry, such a process can be described analytically and is called continuous bending. With regard to the manufacture of the twist of the torso-helicoid, this means a gradual increase in the step to the desired value. The work contains parametric equations that describe this method of bending. At the same time, not only stretching of the workpiece along the axis of the coil occurs, but also its twisting around the axis. A comparison of these two movements was made and it was found that the relationship between them is not linear. The work gives a formula describing this dependence. According to it, when the workpiece is uniformly stretched along the axis, the angle of its twist around the axis increases according to a dependence close to quadratic. In the work, examples of the use of the torso-helicoid are given, the surface is visualized, and graphs are constructed. Confirmation of the reliability of the obtained results is the first found quadratic form of the torso-helicoid, which does not change when the surface is bent