Generalized method for forming plane isotropic curves
DOI:
https://doi.org/10.31548/energiya2020.04.109Abstract
The process of modeling the temperature distribution on surfaces, applying an image to curved areas with minimal distortion requires the formation of isometric grids on the plane and on the surface. One of the common ways to form planar isometric networks is to use the functions of a complex variable and planar isotropic curves, followed by separation of the real and imaginary parts. The development of computer models for the interactive search and analysis of isometric networks according to various initial geometric conditions provides a generalized method for their formation with the possibility of varying their shape and position.
It is proposed to use an isotropic vector for the formation of flat isotropic curves, which ensured a single sequence of analytical calculations according to the following initial conditions: 1) selection of an arbitrary function of a real argument; 2) a given parametric equation of a plane curve; 3) a given polar equation of a plane curve.
Since the analytical calculations of the derivation of the parametric equation of a plane isotropic curve and the corresponding isometric grid are rather laborious, their execution is carried out in the environment of the Maple symbolic algebra. To this end, the corresponding software has been created, which interactively allows you to select the function of a real argument, a parametric or polar equation of a plane guide curve. All subsequent stages of analytical transformations to form an isotropic curve and the corresponding isometric grid are carried out automatically. An interactive model for the formation and analysis of plane isotropic curves with various initial conditions has been created, which has shown its effectiveness, which is confirmed by the given examples of plane isometric grids for specific functions of the real parameter, plane curves in the parametric and polar form of their job.
Key words: isotropic vector, plane isotropic curve, plane isometric grid, real and complex planes, equation of curves in the complex plane
References
Alad’yev, V. Z., Boyko, V. K., Rovba, E. A. (2007). Programmirovaniye i razrabotka prilozheniy v Maple [Programming and application development in Maple]. Grodno-Tallin, 458.
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. Kherson: KhNTU, 3(620), Vol. 2, 196-199.
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: zbirnyk nauk. prats MDPU im. B. Khmelnytskoho. Melitopol: MDPU, 13, 124 – 130.
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).
