Forming of Curvilinear Structures Based on Polyhedrons by the Projectivographical Method

The possibilities of forming curvilinear structures by the projectivographic method for use in architecture and design are considered. As is known, the projectivographic method allows new polyhedral structures to be obtained on the basis of trace epures of polyhedrons taken as a basis. The faces of these shapes are flat polygons, and the edges are line segments. More extensive sets of shapes can be obtained by moving from strictly straight-line elements of a projective drawing (PD) to nonlinear ones using various deformations. For this purpose, a polyhedral three-dimensional object obtained in a standard way in Euclidean space by using PD is subjected to deformation, which leads to the appearance of curved faces. In this article, we used deformations that are described by the simplest analytical expressions: spherical deformation (surfaces acquire positive Gaussian curvature) and hyperbolic deformation (surfaces acquire negative Gaussian curvature).