Machinery & Energetics

In this article an analytical description of isotropic lines of zero length and minimal surfaces is carried out with the help of function of a complex variable. The integral dependences of the formation of imaginary isotropic lines, found from the condition of equality of zero differential of the arc of the spatial line, are used. Parametric equations of isotropic lines are found using functions u t cth k t ; v t i csch k t . Analytical description of minimal surfaces and associated minimal surfaces in complex space made of isotropic lines as lines of a translation net. The expressions of the coefficients of the first quadratic form of the formed minimal surfaces are given. It is explored that for the indicated functions satisfying 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 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 further research are to determine the differential characteristics of the created minimal surfaces for optimization of engineering methods of designing surfaces of technical forms.

isotropic line, minimal surface, associated minimal surface, hyperbolic functions, quadratic form of a surface, function of a complex variable