Analytical description minimal surfaces on the basis of isotropic lines with the help of integral dependencies

Abstract

In this article, an analytical description of isotropic zero-length lines and minimal surfaces is made using the functions of a complex variable. The integral dependences of the formation of parametric equations of imaginary isotropic lines are used, obtained from the equality of the differential of the arc of the spatial line to zero. Analytical description of isotropic lines is made on the basis of the functions of the complex variable obtained by differentiating the expressions of parametric equations from the natural parameter of an imaginary flat line with a constant complex value of curvature The analytical description of the minimal surfaces and the associated minimal surfaces is made in a complex space with isotropic lines as the lines of a transfer grid. The expressions of the coefficients of the first and second quadratic forms of the formed minimal surfaces are given. A comparative analysis of the differential properties of minimal surfaces constructed on the basis of isotropic lines with different integral dependences is carried out.It has been investigated that for specified functions that satisfy the condition , one can find an analytical description of two different spatial isotropic lines of zero length using the functions of a complex variable. Each isotropic line corresponds to the minimal surface and the associated minimal surface, which have similar properties to the Gaussian curvature of the surface. It is shown that the parametric equations of the formed minimal surfaces and the expressions of their quadratic forms, constructed on the basis of isotropic lines with different integral dependences, differ only in signs. The minimal surfaces constructed on the basis of the analytical description of the isotropic line with opposite signs of the aplicate expressions are congruent.The method of continuous geometric modeling of isotropic lines proposed by the authors of the article by means of complex analysis has the advantages caused by finding parametric equations of the corresponding minimal surfaces in the form of elementary functions.