Continuous bending of minimal surfaces, formed by means of plane curves of complex curvature

Abstract

Analytical description of an оne-parameter set of an associated minimal surfaces formed under their continuous bending, using complex variable was made. To find the equation of isotropic lines, parametric equations of logarithmic spiral and evolvent of circle defined by functions of natural parameter with complex curvature were used. Isotropic lines parametric equations are obtained from the condition of differential arc of spatial curve equality to zero. Analytical description of minimal surfaces and connected minimal surfaces were made in complex space with isotropic lines of grid transfer. Expressions of the first and second quadratic forms of generated minimal surfaces coefficients were found. It is shown that the mean curvature of formed minimal surfaces equals zero at all points. We investigated that minimum surface and connected minimal surface, which are formed on the base of isotropic line with the help of a logarithmic spiral with curvature of complex value share common properties with appropriate curvature surfaces built using logarithmic spiral curvature of actual value. Using various methods of analytical description of isotropic lines with evolvent of circle with curvature of complex value, minimal surfaces that with the replacement of variable have common properties with a curvature, but different metric characteristics were constructed. Analytical description of one-parameter set of associated minimal surfaces allows to control their shape for solving various applications. Parametric equations of minimal surfaces were found in the form of elementary functions, allowing to explore their geometric properties and differential characteristics to optimize engineering methods of technical forms and architectural constructions design. Key words: оne-parameter set of associated minimal surfaces, isotropic line, logarithmic spiral, evolvent of the circle, function of a complex variable, сomplex curvature, mean curvature of a surface.