Утворення ізотропних ліній та мінімальних поверхонь за допомогою плоских кривих, заданих функціями натурального параметра

Abstract

This article provides an analytical description of isotropic lines and minimal surfaces with a help of complex variable functions. To find the equation of isotropic lines we used parametric equations of a logarithmic spiral defined by natural parameter functions. To find analytical description of minimal and associated surfaces with the help of isotropic lines, Weierstrass formula was applied. When bending minimal surfaces one-parameter set of associated minimal surfaces was found. Expression of first and second coefficients of quadratic forms of generated minimal surfaces are given. It is shown that the plane curves, given by natural parameter functions, belong to formed minimal surfaces. It is possible to find an analytical description of isotropic line of zero length for any plane curve defined by parametric equations of natural parameter. Each isotropic line corresponds to the minimum isotropic surface and associated minimal surface that allow continuous bending. 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 adjoint minimal surfaces and optimization of engineering methods of technical surfaces forms design. Keywords: isotropic line, minimal surface, minimal surface , adjoint minimal surface , associated minimal surface, logarithmic spiral, quadratic form of a surface , bending of a surface , function of a complex variable , Weierstrass formula. Analytical description of minimal surfaces is an important issue of geometric modeling of technical forms and architectural designs surfaces. If some closed plane or space line is defined, the minimum surface, that passes through this line, has the smallest area. Geometrical shape of a minimal surface provides uniform distribution of forces in the shell [1, p. 43]. 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 average curvature 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, 4, 5]. To find analytical description of minimal surfaces there is another area of research connecting with use of properties of complex-variable function. 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. To find analytical description of minimal surfaces by means of complex variable functions it is necessary to define parametric equations of zero length isotropic curve [6]. In work [7] an analytical description of Bezier isotropic curves is found. The method of analytical description of isotropic curves that lie on surfaces of revolution, referred to the isometric grid lines, was realized in works [8, 9]. The research, published in article [10], is devoted to the problem of analytical description of isotropic curves on a given plane curve - their horizontal projection. It should be noted that analytical description of isotropic lines with the help of plane curves defined by functions of a natural parameter requires further research. Aim of research. To find analytical description of isotropic lines with the help of plane curves defined by natural parameter functions. To build minimal surfaces and associated minimal surfaces on the base of determined isotropic lines. Materials and methods of research. To find analytical description of minimal surfaces with the help of isotropic lines, defined by complex variable functions, Weierstrass formula was applied [6]. Results of the research and discussion. To find the equation of isotropic lines, parametric equation of a logarithmic spiral, defined by natural parameter functions was used. On the base of isotropic lines by Weierstrass formula, analytical description of minimal surfaces and adjoint minimal surfaces was found, their visualization was made. When bending surfaces one-parameter set of associated minimal surfaces was defined. An expression for coefficients of first and second quadratic forms of generated minimal surfaces was given. Conclusions and prospects. It is possible to find an analytical description of isotropic line of zero length for any high plane curve defined by parametric equations of natural parameter. Each isotropic line corresponds to the minimum isotropic surface and associated minimal surface that allow continuous bending. Prospects for future research is to study the differential characteristics of adjoint minimal surfaces and optimization of engineering methods of technical surfaces forms design.