The surface of non-linear rotation

Abstract

The paper considers a geometric scheme, a mathematical model and an algorithm for shaping a non-linear rotation surface. It is known that in Euclidean geometry and mechanics the transformation of rotation is linear, while distance and angle are its invariants. The authors proposed a geometric scheme of non-linear rotation, in which the axis of rotation is a smooth spatial curve and the object of rotation is a smooth line. Several propositions, a lemma and a theorem are proved, which allow one to form the initial data in the problem of nonlinear rotation, the solution of which is the parametric equations of smooth surfaces. The research results make it possible to expand the variety of cyclic surfaces in the existing classification of analytic surfaces. They can also be useful in the creation of CAD, which provides for the design of surface forms of products for mechanical engineering, construction, architecture and other practical areas based on cyclic surfaces.

Keywords

• smooth curve
• movable trihedron
• non-linear rotation surface
• shaping algorithm
• cyclic surface