Developing an integrated approach to formation of curves and surfaces of real space based on isotropic characteristics

Authors

DOI:

https://doi.org/10.15587/2312-8372.2014.25274

Keywords:

isotropic curve, isotropic curvature, isotropic torsion, isotropic network, zero characteristics

Abstract

The principle of forming curves and surfaces based on imaginary properties is considered in the paper. The main objective lies in developing a new integrated approach to modeling three-dimensional real space objects at preset zero characteristics. An algorithm of forming curves and surfaces is proposed, and isotropic characteristics are considered. For modeling isotropic curves isotropic intervals, polygons, an isotropic curve length, isotropic curvature and torsion are used. For modeling isotropic networks on a plane and a surface in three-dimensional space, the kinematic construction method is used. Curves with assigned isotropic characteristics are selected as directors and generators. For representing the obtained objects, real and imaginary parts are designated, and then obtained abstractions are studied. The approach was developed for obtaining objects with prescribed metric and differential properties. The studies can be used for managing algebraic functions at conformal mappings, in the theory of a thin-walled structure bending.

Author Biography

Наталія Миколаївна Аушева, National Technical University of Ukraine 37 ave. Pobedy, Kiyv , 03056 Ukraine

Ph.D., associate professor, assistant professor

Department of Energy Processes and Systems

References

  1. Александров, П. С. Энциклопедия элементарной математики [Текст]/ П. С. Александров, А. И. Маркушевич, А. Я. Хинчин. – Кн. 5: Геометрия. – Москва: Государственное издательство физ.-мат. литературы, 1966. – 624 с.
  2. Дубровин, Б. А. Современная геометрия: Методы и приложения [Текст]/ Б. А. Дубровин, С. П. Новиков, А. Т. Фоменко. – 2-е изд., перераб. – М.: Наука, Гл. ред. физ.-мат. лит., 1986. – 760 с.
  3. Lie, S. Über Integralinvarianten und Differentialgleichungen [Теxt]/ S. Lie // Cornell University Digital Collections. – 2008. – 84 p.
  4. Бляшке, В. Дифференциальная геометрия и геометрические основы теории относительности Эйнштейна [Текст]/ В. Бляшке. – Главная редакция общетехнической литературы и номографии, 1935. – 330 с.
  5. Пилипака, С. Ф. Конструювання мінімальної поверхні гвинтовим рухом просторової кривої [Текст]/ С. Ф. Пилипака, І. О. Коровіна // Праці Таврійського державного агротехнологічного університету. Прикладна геометрія та інженерна графіка. – Т. 39, Вип.4. – Мелітополь: ТДАТУ, 2008. – С. 30–36.
  6. Картан, Э. Теория конечных непрерывных групп и дифференциальная геометрия, изложенные методом подвижного репера [Текст]/ Э. Картан. - Изд-во Московского университета, 1963. - 366 c.
  7. Wang, Z. Singularities of Focal Surfaces of Null Cartan Curves in Minkowski 3–Space [Теxt]/ Z. Wang, D. Pei, L. Chen, L. Kong, Q. Han // Abstract and Applied Analysis. –Vol. 2012. – Article ID 823809. – 20 p.
  8. Pekmen, Üm. On minimal space curves in the sense of Bertrand curves [Теxt]/ Üm. Pekmen //Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. – № 10. – 1999. – P. 3–8.
  9. Hwang, J. M. Geometry of minimal rational curves on Fano manifolds [Теxt]/ J. M. Hwang// Lecture given at the School on vanishing theorems and effective results in algebraic geometry. – Trieste, 2000. – P. 344–392.
  10. Beban-Brkic, J. Butterflies in the Isotropic Plane [Теxt]/ J. Beban-Brkic, V. Volenec// KoG. – 2004. – Number 8. – P. 29–35.
  11. Andrews, B. Classification of limiting shapes for isotropic curve flows [Теxt]/ B. Andrews // Journal of the American mathematical society. – Vol. 16, № 2. – 2002. – P. 443–459.
  12. Yılmaz, S. Some Characterizations of Isotropic Curves In the Euclidean Space [Теxt]/ S. Yılmaz, M. Turgut // International Journal of Computational and Mathematical Sciences. – № 2. – Spring, 2008. – P. 107–109.
  13. Sachs, H. Isotrope Geometrie des Raumes [Теxt]/ H. Sachs.  Vieweg, Braunschweig/Wiesbaden, 1990. – 317 p.
  14. Pottmann, H. Discrete Surfaces in Isotropic Geometry [Теxt]/ H. Pottmann, Y. Liu; Eds. R. Martin, M. Sabin, J. Winkler// Mathematics of Surfaces 2007, LNCS 4647. – 2007. – P. 341–363.
  15. Sipuˇs, Z. M. Surfaces of Constant Curvature in the Pseudo–Galilean Space [Теxt]/ Z. M. Sipuˇs, B. Divjak // International Journal of Mathematics and Mathematical Sciences. – 2002. – Vol. 2012. – Article ID 375264. – 28 p.
  16. Sipuˇs, Z. M. Hyperspherical curves in n–dimensional k–isotropic space Ikn [Теxt]/ Z. M. Sipuˇs // Mathematical Communications. – 2001. – №6. – Р. 39-45.
  17. Аушева, Н. М. Моделювання плоских сіток на основі дробово-раціональних ізотропних кривих [Текст]: матеріали конференції "Наукові підсумки 2013"/ Н. М. Аушева // Технологічний аудит та резерви виробництва. – 2013. – №4/6(14). – С. 41–43.
  18. Ausheva, N. Modeling of minimal surfaces based on isotropic curves and quasiconformal change of parameter [Теxt]/ N. Ausheva// Science and Education a New Dimension: Natural and Technical Sciences. – I(2), Issue 15. – Budapest, 2013. – P.67–70.
  19. Aleksandrov, P., Markushevich, A., Xinchin, A. (1966). Enciklopediya elementarnoj matematiki. Kn. 5 Geometriia. M.: Gosudarstvennoe izdatelstvo fiz.-mat. literatury, 624.
  20. Dubrovin, B., Novikov, S., Fomenko A. (1986). Sovremennaya geometriya: Metody i prilozheniya. M.: Nauka, Gl. red. fiz.–mat. lit.,760.
  21. Lie, S. (2008). Über Integralinvarianten und Differentialgleichungen. Cornell University Digital Collections, 84.
  22. Blyashke, V. (1935). Differencialnaya geometriya i geometricheskie osnovy teorii otnositelnosti Ejnshtejna. Glavnaya redakciya obshhetehnicheskoj literatury i nomografii, 330.
  23. Pilipaka, S., Korovіna, І. (2008). Konstruyuvannya mіnіmal'noi poverhnі gvintovim ruhom prostorovoi krivoi. Prikladna geometrіya ta іnzhenerna grafіka, 39, 30–36.
  24. Kartan, E. (1963). Teoriya konechnyh nepreryvnyh grupp i differencialnaya geometriya, izlozhennye metodom podvizhnogo repera. Izdatelstvo Moskovskogo universiteta, 366.
  25. Wang, Z., Pei, D., Chen, L., Kong, L., Han, Q. (2012). Singularities of Focal Surfaces of Null Cartan Curves in Minkowski 3-Space. Publishing Corporation Abstract and Applied Analysis, 20.
  26. Pekmen, Üm. (1999). On minimal space curves in the sense of Bertrand curves. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 10, 3–8.
  27. Hwang, J. (2000). Geometry of minimal rational curves on Fano manifolds. Lecture given at the School on vanishing theorems and effective results in algebraic geometry. Trieste, 344–392.
  28. Beban-Brkic, J. (2004). Butterflies in the Isotropic Plane. KoG, 8, 29–35.
  29. Andrews, B. (2002). Classification of limiting shapes for isotropic curve flows. Journal of the American mathematical society,16, № 2, 443–459.
  30. Yılmaz, S. (2008). Some Characterizations of Isotropic Curves In the Euclidean Space. International Journal of Computational and Mathematical Sciences, 2, 107–109.
  31. Sachs, H. (1990). Isotrope Geometrie des Raumes. Vieweg, Braunschweig/Wiesbaden, 317.
  32. Pottmann, H., Liu, Y. (2007). Discrete Surfaces in Isotropic Geometry. Mathematics of Surfaces 2007, LNCS 4647, 341–363.
  33. Sipuˇs, Z., Divjak, B. (2012). Surfaces of Constant Curvature in the Pseudo–Galilean Space. International Journal of Mathematics and Mathematical Sciences, Article ID 375264, 28.
  34. Sipuˇs, Z. (2001). Hyperspherical curves in n–dimensional k–isotropic space Ikn. Mathematical Communications, 6, 39–45.
  35. Ausheva, N. (2013). Modeling of planar networks based on the fractional-rational curves isotropic. Technology Audit And Production Reserves, 6(4(14)), 41-43.
  36. Ausheva, N. (2013). Modeling of minimal surfaces based on isotropic curves and quasiconformal change of parameter. Science and Education a New Dimension: Natural and Technical Sciences, 15, 67–70.

Downloads

Published

2014-05-29

How to Cite

Аушева, Н. М. (2014). Developing an integrated approach to formation of curves and surfaces of real space based on isotropic characteristics. Technology Audit and Production Reserves, 3(1(17), 17–20. https://doi.org/10.15587/2312-8372.2014.25274

Issue

Vol. 3 No. 1(17) (2014): Technology audit

Section

Technology audit

License

Copyright (c) 2016 Наталія Миколаївна Аушева

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.