Неперевне згинання мінімальних поверхонь, утворених за допомогою евольвенти кола, заданої функціями натурального параметра

Abstract

CONTINUOUS BENDING OF A MINIMAL SURFACES FORMED WITH A HELP OF EVOLVENT OF A CIRCLE, GIVEN BY NATURAL PARAMETER FUNCTIONS

S.F. Pylypaka, M.M. Mukvich

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.

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.

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.

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

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. The method of analytical description of isotropic curves that lie on surfaces of revolution, referred to the isometric grid lines, was realized in works [6, 7]. The research, published in article [8], 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 evolvent of a circle defined by natural parameter functions. With newfound isotropic lines define the associated one-parameter set of minimal surfaces that allow continuous bending.

Materials and methods of research. Analytical description of minimal surfaces in complex space made of isotropic lines as lines of a translation net.

Results of the research and discussion. To find the equation of isotropic lines, parametric equation of a evolvent of a circle, defined by natural parameter functions was used. On the base of isotropic lines the analytical description of a 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.