Nesvidomina O. Isometric grids in problems of applied geometry of surfaces

The dissertation on competition of a scientific degree of candidate of technical sciences by specialty 05.01.01 “Applied geometry, engineering graphics”. - Kyiv National University of Building and Architecture. - Kyiv, 2021. The dissertation is devoted to the development of interactive computer models for the formation of flat and space isometric grids different initial geometric conditions and their practical use. The formation of flat and space grids is necessary for solving many problems of practice, in particular, modeling the temperature distribution on curved surfaces, applying images to curved surfaces with minimal distortion, an approximation of non-developable surfaces and construction of their unfolding, and so on. There are three main directions of the formation of flat and space isometric grids: 1) use of functions of a complex variable and isotropic curves with the subsequent separation of real and imaginary parts; 2) solution of differential and integral equations with coefficients of the 1st quadratic shape of surfaces in order to assign them to isometric coordinates; 3) application of conformal mappings over isometric grids, in particular: bending of a flat isometric grid into cylindrical and conical isometric grids; spherical mapping of the original isometric grid. Using the developed interactive model of formation of flat isometric grids by separating the real and imaginary parts of the function of a complex variable, power, trigonometric, and other functions of the complex variable were analyzed. Based on this analysis, analytical and software for editing the shape and position of a flat isometric grid for any function of a complex variable, including its inversion, is proposed. Keywords: isometric grid, isotropic curve, complex variables, conformal mapping, inversion transform.