Cyclic surfaces accompanying non-ruled quadrics of rotation

Abstract

The paper considers the shaping of cyclic surfaces based on nonlinear rotation, in which the axis of rotation and the generatrix in the general case are three-dimensional smooth curves. As a tool for shaping surfaces of non-linear rotation, the method of the accompanying Frenet trihedron, known in the differential geometry of curved lines, is used. The geometric scheme of surface shaping is based on a construction that includes: a curvilinear axis of rotation and a one-parameter set of its normal planes; a generatrix whose points describe in normal planes circular trajectories centered on a curvilinear axis. A mathematical model of shaping the surface of non-linear rotation for the general case of specifying the axis of rotation and the generatrix is given. On the basis of this model, test examples of the formation of surfaces of nonlinear rotation, which are cyclic surfaces, each of which accompanies the corresponding nonlinear quadric of rotation, are considered. In the examples of shaping, the original rectilinear axis of a non-linear quadric of revolution and its generating line, a second-order curve, are functionally interchanged: the second-order curve becomes the rotation axis, and the rectilinear axis becomes the generatrix.

The resulting family of surfaces of non-linear rotation belongs to the well-known class in the theory of analytic surfaces "Normal cyclic surfaces". It complements this class and fundamentally differs in the method of shaping.