FORMATION OF ISOTROPIC LINES AND MINIMAL SURFACES BASED ON THE IMAGINARY FLAT LINE OF THE CONSTANT COMPLEX CURVATURE

Abstract

This article provides an analytical description of an isotropic line of zero length and minimal surfaces by means of complex analysis. 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. The parametric equations of isotropic lines are obtained using the functions   were  the imaginary unit, , satisfying the condition .

The analytical description of the minimal surfaces and the attached minimal surfaces is carried out in a complex space with isotropic lines as lines of the transfer network. The expressions of the coefficients of the first quadratic form of the formed minimal surfaces are given.

It is shown that for functions   one can find an analytical description of two different spatial isotropic lines of zero length using functions of a complex variable. Each isotropic line corresponds to a minimal surface and an attached minimal surface, having common properties of the Gaussian curvature of the surface. Minimal surfaces constructed on the basis of an analytical description of an isotropic line with opposite signs of applicate are congruent.

The method of continuous geometric modeling proposed by the authors of the article has known advantages due to the finding of the parametric equations of minimal surfaces in the form of elementary functions. Such an analytical description of minimal surfaces allows one to take into account their differential characteristics for optimization of engineering methods for designing surfaces of technical forms and architectural structures.

Key words: isotropic line, minimal surface, main surface curvature, curvature of a flat curve, function of a complex variable.