Elements of spatial cyclography

Abstract

In modern CAD/CAM systems and in CAGD (Computer Aided Geometric Design), models of geometric objects (lines, surfaces) of space R4 in space R3 and vice versa are used. The analysis of geometric models used in such systems allows us to conclude that research is relevant to the development in R3 of analytical models of lines and surfaces of the space R4. In this paper, we show the possibility of obtaining of constructiveanalytical model using the three-dimensional drawing of the space R4, proposed by N. V. Naumovich. Based on this drawing, a constructive interpretation of the proposed analytical model is given and its implementation is realized in a virtual electronic 3D-space.The model of the R4 space curve proposed in this paper is based on the theoretical positions of the spatial cyclography realized in Naumovich’s three-dimensional drawing. This proposed model differs from existing ones using the analytical method of modeling. The essence of the proposed constructive-analytical modeling consists in the geometric representation and interpretation in the three-dimensional drawing of the cyclographic images of points, lines, the set of points and lines of the space R4. The constructive-analytical modeling of the R4 space curve on the basis of the cyclographic mapping and the possibility of its realization on Naumovich’s three-dimensional drawing allow to get a full picture of the interrelation and mutual influence of all the elements of the model. Such a representation is based on the implementation of a 3D-drawing in virtual electronic 3D-space with the means of modern graphical CAD and allows solving the optimization of the developed models of geometric objects in relation to the requirements of modern CAD/CAM systems and CAGD.