Mathematical model of the formation of cyclic and channel surfaces based on nonlinear rotation

Abstract

This work is a continuation of the series of works by the authors devoted to the issues of shaping surfaces of nonlinear rotation. The geometric scheme for the formation of surfaces of this class includes: an axis of nonlinear rotation, which is a smooth, generally spatial curve, and a forming line, also a smooth spatial curve. When the generating line rotates relative to the curvilinear axis, each point of the generating line describes a circumferential trajectory in the corresponding normal plane of the rotation axis. As a result, a surface of nonlinear rotation is formed, which is a normal cyclic surface. In this work, in order to develop the research results previously obtained by the authors in the field of shaping surfaces of nonlinear rotation, the solution to the inverse problem of shaping is considered and a mathematical justification is given for the possibility of shaping a channel surface based on solutions to the direct and inverse problems. The work provides numerical examples of the formation of the surfaces under consideration, accompanied by mathematical models of surfaces, their computer implementation and visualization. The research results can be useful in the development of CAD systems that involve the design of surface forms of products based on cyclic and channel surfaces in mechanical engineering, construction, architecture and other practical fields.