Дослідження аналітичних залежностей для утворення ізотропних ліній та конструювання мінімальних поверхонь

С. Ф. Пилипака, М. М. Муквич



S. Pylypaka, M. Mukvich



The research of analytical conditions for the formation of parametric equations of isotropic lines is due to the problem of analytical description of minimal surfaces. The use of CAD systems in geometric models described by the minimal surfaces is due to the advantages of practical content when designing the surfaces of technical forms and architectural constructions.

The geometrical shape of minimal surface provides even distribution of efforts in the shell of surface and extra rigidity [1, p. 72]. The main curvature at all points of a minimal surface equals zero, which is a prerequisite for solving the problem of finding the smallest surface area, which passes through a given plane or spatial curve.

Finding analytical description of minimal surface passing through the closed

plane line, is reduced to solution of  Euler-Lagrange nonlinear differential equation in partial derivatives, which generally is not integrated [2, p. 683].

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 [2, p. 683].

Therefore current research of analytical description of minimal surfaces is to improve variational and finite-difference numerical methods for solving Euler-Lagrange differential equation [3].

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 [2, p. 685]. Use of complex-variable function allows to get a parametric equation of minimal surfaces, investigate their differential characteristics, optimize engineering design methods of technical surface forms.

Analysis of recent research and publications. For analytical description of minimal surfaces by means of complex variable must define parametric equations isotropic line of zero length [5]. In the thesis [6] found analytic description of isotropic lines by formulas Schwartz (H. Schwarz) based on the spatial curve on the surface of the cylinder and on the basis of the slope of the curve. But the formula for finding Schwartz parametric equations of isotropic lines associated with the integration of complex expressions and can only in rare cases. In [7] have been considered isolated cases analytical description of isotropic lines based on an imaginary plane curve equations of the form:  , where  і  real or complex variable functions. Despite the variety of known origin parametric equations of isotropic lines, simplify their analytical description remains an important problem in modeling minimal surfaces.

Aim of research. To reseach the analytical conditions for the formation of spatial isotropic lines of zero length on the basis of a plane curve given by parametric equations , satisfying the condition . Using the parametric equations of an isotropic line, determine the analytical description of the minimal surfaces.

Materials and methods of research. Isotropic lines parametric equations are obtained from the condition of differential of the arc of spatial curve equality to zero. Analytical description of minimal surfaces and connected 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 and second quadratic forms of generated minimal surfaces was given.

Conclusions and prospects. It is research that for a plane curve given by parametric equations , satisfying a condition , one can find an analytical description of two 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 function 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.


Повний текст:



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