Kinematic operation of designing geometric solids using point calculus

This article continues the series of works by the authors on the development of a mathematical apparatus for modeling geometric solids in the point calculus. It is devoted to the description of a kinematic operation obtained by means of the movable simplex method. The principles of modeling geometric solids in the point calculus are based on the concept of their representation as a three-parameter set of points belonging to the three-dimensional space. The modeling task consists of two parts: determination of the guiding trajectory of a plane simplex in space; and the determination of the closed area formations in the plane of the moving simplex. Instead of a closed area, two curves bounding a certain area or a guiding curve with a wall of constant thickness can be used. As an example, we describe the process of modeling geometric solids with a parallelism plane, including a computational algorithm for its formation in the form of a sequence of parametric equations. As a result, geometric and computer models of solids with a generatrix in the form of a circle of a given radius, and with a generatrix in the form of a closed curve of the “sine wave” type were obtained. The methods of parametrization of geometrical objects and their analytical description in the point calculus, presented in the article, can find wide application as effective tools of modern systems for solid modeling and computer aided design. The idea of determining equidistant curves by means of the point formula of parallel transfer, to extend the capabilities of existing geometric modeling tools, is also proposed.