Construction of a family of flat curves according to the equations of isometric grids
DOI:
https://doi.org/10.31548/energiya2021.02.146Abstract
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 problem 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 drawing images on 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 they are performed in the environment of symbolic algebra Maple. For this purpose, the corresponding software of the interactive model of 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 integer values of the power of the exponential function of the complex variable, the values of the abscissa and the ordinate will be represented by algebraic surfaces in the explicit form. The projections of the cross 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. 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.
Key words: isometric grids, functions of a complex variable, families of orthogonal lines, geometric flow modeling
References
Nesvidomina, O. V. (2017). Pobudova ploskykh izometrychnykh sitok za napered zadanymy ploskymy kryvymy. [Construction of flat isometric grids according to predefined flat curves]. Visnyk Khersonskoho natsionalnoho tekhnichnoho universytetu, 3(62), Vol. 2, 298-302.
Pylypaka, S. F., Kremets, T. S., Nesvidomina, O. V. (2018). Konformne vidobrazhennia rastrovykh napysiv na ploski kryvoliniini oblasti [Conformal mapping of raster inscriptions on flat curved areas]. Suchasni problemy modeliuvannia: zb. nauk. prats MDPU im. B. Khmelnytskoho. Melitopol: MDPU, 13, 124-130.
Shoman, O. V., Danylenko, V. Ya. (2014). Rozv'iazannia zadach formoutvorennia dvovymirnykh heometrychnykh mnozhyn u tryvymirnomu prostori [Solving problems of forming two-dimensional geometric sets in three-dimensional space]. Suchasni problemy modeliuvannia. Melitopol, 3, 147-152.
Downloads
Published
Issue
Section
License
Relationship between right holders and users shall be governed by the terms of the license Creative Commons Attribution – non-commercial – Distribution On Same Conditions 4.0 international (CC BY-NC-SA 4.0):https://creativecommons.org/licenses/by-nc-sa/4.0/deed.uk
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
