Lisun I. Geometric models of spatial folded systems for different transformation schemes

In the dissertation, folded surfaces are considered when used to form architectural sheeting and other types of structures are often used. In some cases, this is due to the very architectonics of the structures, which often have parts corresponding to each other (folded in this case). In other cases – this is caused by the need to approximate a folded surface; it consists of compartments of simpler surfaces, which allows to simplify the solution of geometric problems in the design and construction. A comparable line is purposefully considered as an element of a set, which makes it possible to study the properties of such sets and the methods for isolating from them many types of folded surfaces (facets in a given vipak). This allows us to consider the method of constructing comparable surfaces as a single scheme for constructing linear and non-linear surfaces. A generic broken line is called a comparable line, which generally has curvilinear links, intersect each other at the vertices, or join them with an outstanding order of joining. The properties of generalized broken lines and their sets depend on the properties of the links (modules) and the methods of connections between them. Generalized broken lines of a general form, have zero vertices at the vertices, are formed by the method of connecting the links at the vertices without matching the related links. The methods of geometric formation of modern buildings and the newest forms and types of structures are being intensively developed, sometimes spreading with rigorous beauty and constructive improvements. Geometrical design, based on the principles of a combination of standard elements, seriality, manufacturability, coincides with the requirements of industrial production and mass use.