Computer geometric modeling of quasi-rotation surfaces

Abstract

Abstract The paper presents an analytic description of quasi-rotation correspondence. The formulas for calculation of values characterizing geometric objects are provided on the basis of constructive description of the quasi-rotation correspondence. The values characterizing circles generated through quasi-rotation of the initial point are calculated. A system of parametric equations describing a generic quasi-rotation surface is provided. Every type of curvilinear second-degree axis – circular, elliptic, parabolic, and hyperbolic – is considered. An algorithm for surface modeling through computer algebra systems has been developed on the basis of the analytic description and tested by means of Maple software. The algorithm allows one to acquire plots of separate sheets of the surface as well as all four sheets simultaneously. The capability to acquire 3D models of quasi-rotation surfaces given the desired shape and mutual location of the axis and the generatrix is demonstrated. The developed algorithm is applicable to any flat generating curve belonging to the axis plane and defined by a single-parameter system of equations. The resultant 3D model polygon count is user-defined and only limited by computer’s processing capability assuring high surface quality. The images of several four-sheet quasi-rotation surfaces are provided. The developed algorithm allows one to design surfaces of pre-defined properties.