CONSTRUCTION OF A FAMILY OF FLAT CURVES ACCORDING TO THE EQUATIONS OF ISOMETRIC GRIDS

Abstract

The article reveals an analytical description of the formation of families of orthogonal flat curved lines in the implicit form based on the analysis of the parametric equation of a flat isometric grid constructed by separating the real and imaginary parts of the function of a complex variable. This formulation is due to the fact that flat isometric grids, as two families of orthogonal coordinate lines with square cells, are used in conformal mappings, for example, when applying images to curved surfaces with the least distortion. At the same time, families of flat parallel lines are widely used in geometric modeling of heat transfer, electric fields, fluid flow, etc. There is a connection between these geometric images, which is explained by specific examples. Analytical calculations of deriving the parametric equation of an isometric grid are quite time-consuming, so their implementation is carried out in the environment of symbolic algebra Maple. For this purpose, the corresponding software of the interactive model of the derivation of parametric equations of isometric grids for any initial function of a complex variable with the subsequent separation of its real and imaginary parts was created. It was found that the values of the abscissa and ordinates of the parametric equation of a flat isometric grid can be represented as explicit surface equations. For the exponential function of the complex variable, the abscissa and ordinate values will be represented by algebraic surfaces explicitly. The projections of the sections of the abscissa and ordinate surfaces by horizontal cutting planes on the horizontal plane form two families of curved lines, the equations of which can be obtained only implicitly. By the example of the quadratic function of a complex variable, it is proved that these families of lines are mutually perpendicular. Depending on the complexity of the function of the variable, families of lines can have different shapes. The practical application of building a family of lines for geometric modeling of fluid flow lines that flow around the barrier in the form of a semicircle is shown.

Keywords: isometric grids, functions of a complex variable, families of orthogonal lines, geometric flow modeling.