BENDING OF THE SURFACES OF THE BINORMALS OF THE SLOPE LINES

The bending of the surfaces of the binormals by deformation of the directing curve, which is the slope line, is considered. Such deformation changes the curvature of the curve, but it remains a slope line. The rectilinear generators of the surfaces at deformation of the curve coincide with the unit binormal of the accompanying trihedral at all its points. Parametric equations of continuous bending of these surfaces have been made by changing the angle of inclination of the directional slope line. The initial slope curve is given by the angle of its rise and the conformity of change of curvature as a function of arc length. The partial case where the curvature of the initial curve is constant is considered. In this case, it is shown that helicoids, including the screw conoid, are the set of surface bends. Specific examples are considered. According to the obtained equations, the surfaces that are the bending of the initial are constructed. The doctrine of bending is an important part of the differential geometry of surfaces. The most studied in theoretical terms is the bending of the developable surfaces, because the particular case of bending such surfaces on the plane, that is, obtaining an involute, is applied. It is a well-known theorem on the possibility of bending screw surfaces into the surfaces of revolution. A classic example is the bending of a screw conoid into a catenoid. Screw lines of a surface at its bending gradually deform, their step decreases, until they turn into the parallels of a surface of revolution. Nondevelopable ruled surfaces can also be bent like developable ones, that is, deform the directing curve in such a way that rectilinear generators remain straight. This article is devoted to this bending of ruled non developable surfaces, where the slope line is a directing curve of the surface. Bending of linear unbranched surfaces can be carried out in the same way as unfolded, that is, with the preservation of rectilinear twists. A partial case of such bending is the bending of the surface of the binormals of the spatial curve guide. When the surface is bent, the guide curve is deformed so that its twist as a function of the arc length remains unchanged. Using the example of slope curves, which are directional curves of the surface of binormals, generalized parametric equations of bending such surfaces by changing the angle of their rise are obtained. When this angle increases to a limit value of 90°, the surface is transformed so that the slope curve turns into a straight line. The result of the bending of the surface of the binormals of the helical line is the helicoids and the helical conoid with a maximum lifting angle of 90 °.