Mathematical description of bending a surface of revolution into a helical conoid

The object of this study is the process of theoretical gradual bending of a catenoid into a helical conoid coil. A helical conoid or a straight closed helicoid is formed by the helical motion of a segment around an axis, and this segment intersects the axis at a right angle during movement. It cannot be bent into a plane, but by gradually reducing the pitch it can be transformed into a known surface of revolution – a catenoid. With such deformation, the lengths of the lines and the area of the coil as a whole do not change, that is, the deformation occurs similarly to unfolded surfaces. Such deformation is based on the theory of bending surfaces of a separate section of differential geometry. According to it, any helical surface can be bent into a surface of revolution and vice versa. Bending the non-folded surface of a helical conoid into a catenoid is a classic example of differential geometry. This approach makes it possible to find an approximate flat workpiece for manufacturing a screw coil. This task is resolved by approximating the obtained catenoid by a truncated cone. The sweep of the truncated cone will be the approximate sweep of the screw turn. This is the peculiarity of finding the approximate sweep, which in engineering practice is calculated by other formulas. This is also the essence of the reported results. In the work, parametric equations were derived that describe a one-parameter set of intermediate surfaces during bending of a screw conoid due to a gradual decrease in the surface pitch to zero. In the given example, one turn of the screw is considered, put on a shaft with a radius r = 0.125 m and limited by an external radius R = 0.25 m with a surface pitch H = 0.5 m. The dimensions of the truncated cone, which replaces the catenoid, are r = 0.148 m for the smaller base, R = 0.262 m for the larger base, and H = 0.05 m for the height of the cone. The specified dimensions of the cone are sufficient to find its exact sweep, which will be approximate for the turn of the screw conoid