HOMOLOGICAL MAPPING OF THREE-DIMENSIONAL SPACE AND RELIEF PERSPECTIVE

Abstract

The homological features of one particular type of projective mapping of three-dimensional spaceare considered. It is proved that the double plane of homology is parallel to the imaginary plane figure in the space. It is shown that by means of homology, the entire three-dimensional semi-space is mapped on the area bounded by two parallel planes. The obtained image formed by mapping satisfies the conditions of the relief perspective, therefore, a homological connection is established between the space and its relief image. Thus, the relief perspective of the space can be obtained through mapping.

Issues on the synthesis of projective conversions are considered, which allow to reflect the infinite part of the space on a limited domain.

Thus, the aim of our research is to study the very geometric transformations, which will allow to project the given space onto its boundary domain. They can serve as a basis for development of geometrical modeling algorithms of the relief perspective and the automation of their construction. The present issue is of practical importance as the existence of the computer model of the relief image will make it possible to automate the process of designing the relief image containing objects and organize their productionwith the help of digitally controlled machines (as well as 3D printers).