Synthesis of kinematic surfaces based on elliptic rotation of the plane
Abstract
Synthesis of kinematic surfaces based on elliptic rotation of the plane
Incoming article date: 07.05.2018The algorithm of forming of kinematic surfaces with a constant cross-sectional area based on a complex equiaffine transformation of the plane (elliptic rotation) is considered. Functional dependences of elliptic rotation parameters for the formation of a one-parameter family of equiaffine lines on a plane are determined. At the same time the received families of lines can not include the line prototype and the line image at the set constant parameters of rotation. Conditions of receiving the line prototype and the line image at the set parameter of family and the parametrical equations of shift of the geometrical center of a curve are defined. The conditions for the formation of central surfaces are determined. It is established that the surfaces obtained can be periodic. The domains of admissible values of the parameters and functions included in the parametric equations of one-parameter families of curves are determined. It is shown that as a closed loop one can use not only an analytically determined curve, but also a polyline (for example, a polygon). Examples of kinematic surfaces are given.
Keywords: kinematic surface, equiaffine transformations, algorithm, elliptic rotation, affine-like curves, parametric equations, line-image, prototype line