Знаходження параметричних рівнянь ізотропних ліній за допомогою гіперболічних функцій та утворення мінімальних поверхонь

Анотація

DEFINITION OF PARAMETRIC EQUATIONS OF ISOTROPIC LINES BY HYPERBOLIC FUNCTIONS AND CONSTRUCTION OF MINIMAL SURFACES

S. Pylypaka, M. Mukvich

With the development of CAD systems in the design of surfaces of technical forms and architectural designs, use the surface compartments that are given analytically and have the required geometric properties. Geometric models, described by minimal surfaces, have the benefits of practical content. The tension at each point of the minimum surface is a constant value, so the geometric shape of the minimal surface provides uniform distribution of forces in the shell and additional rigidity. The condition for equality of the value of the average curvature of the minimum surface at all its points is a necessary condition for the minimum area of the compartment of a surface limited by a plane or a spatial curve (contour) on this surface. Known are modern studies of the acoustic properties of solid materials that form the cell structure. The architecture of these cellular materials is based on periodic minimal surfaces, which allows their use to improve sound insulation and reduce vibration.

G. Monge (1776) discovered that the condition for minimality of a surface leads to the condition  (value of the mean curvature of a surface), and therefore surfaces with  are called "minimal". In reality, it is necessary to distinguish the notions of a minimal surface and a surface of least area, since the condition  is only a necessary condition for minimality of area, which follows from the vanishing of the first variation of the surface area among all surfaces of class with the given boundary. To verify that in this class even a relative (local) minimum is attained, it is necessary to investigate the second variation of the surface area. Finding an analytic description of the minimal surface passing through a closed line reduces to solving the nonlinear differential Euler-Lagrange equation in partial derivatives, which in the general case is not integrated. Therefore current research of analytical description of minimal surfaces is to improve variational and finite-difference numerical methods for solving Euler-Lagrange differential equation.

The problem of simplifying the analytical description of minimal surfaces and obtaining their parametric equations, starting with S. Lie's works, is realized with the help of methods of the theory of functions of a complex variable. To do this, the analytical description of the minimal surfaces is found in the complex space with isotropic lines in the role of the transfer network lines.

Analysis of recent research and publications. To find the analytic description of the minimal surfaces with the help of functions of a complex variable, it is necessary to determine the parametric equations of the imaginary isotropic line of zero length. In the dissertation study an analytical description of isotropic lines is found based on Schwarz's formulas based on the spatial curve on the cylinder surface and on the basis of the slope curve. But the use of Schwarz's formulas for finding parametric equations of isotropic lines is due to the integration of complex expressions and is possible only in some cases. The minimal surface was constructed on the basis of
quasiconformal replacement of the parameter in the isotropic curve equation.

In the paper [7], the authors of this study were proposed to carry out an analytical description of isotropic lines based on functions  satisfying the condition . For these functions, analytical conditions for the formation of isotropic lines were found, and minimal surfaces were constructed using functions  and   In this case, it is necessary to study the possibility of generating parametric equations of isotropic lines and minimal surfaces using other functions satisfying the condition .

Aim of research. Find the parametric equations of the isotropic line by means of hyperbolic functions , where the imaginary unit, satisfying the condition . Use the specified isotropic line to construct minimal surfaces.

Materials and methods of research. Analytical description of minimal surfaces and associated minimal surfaces were made in complex space with isotropic lines of grid transfer.

Results of the research and discussion. The equation of isotropic lines was found. On the base of isotropic lines the analytical description of a minimal surfaces was found, their visualization was made. An expressions for coefficients of first quadratic forms of minimal surfaces was given.

Conclusions and prospects. For these functions   one can find an analytical description of two different spatial isotropic lines of zero length with the help of  functions of a complex variable. Each isotropic line corresponds to the minimal surface and the associated minimal surface, which have similar properties of the curvature of the surface. Use of functions of a complex variable allows to get a simple analytical description of minimal surfaces, investigate their design geometrical parameters. Prospects for future research is to study the differential characteristics of associated minimal surfaces and optimization of engineering methods of technical surfaces forms design.