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Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis
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MATHEMATICAL DESCRIPTION FOR A PARTICULAR CASE OF ELLIPSE FOCUS QUASI-ROTATION AROUND AN ELLIPTICAL AXIS
UDK 51 Математика
Antonova I. V. 1
Solomonova E. 2
Kadykova Nina S. 3
Authors:
1.  MIREA – Russian Technological University (kafedra vysshey i prikladnoy matematiki, Associate Professor)

Russian Federation
2.  MIREA – Russian Technological University (Institute of Fine Chemical Technologies named after M.V. Lomonosov, Senior Lecturer)

Russian Federation
3.  MIREA — Russian Technological University (Department of Engineering Graphics, Associate Professor)

Russian Federation
Type:
Article
DOI:
https://doi.org/10.12737/2308-4898-2021-9-1-39-45
Pages:
from 39 to 45
Status:
Published
Received:
06.07.2021
Accepted:
22.07.2021
Published:
22.07.2021
Subject area:
UDK 51 Математика
Language:
Russian
Keywords:
ellipse; focus; rotation center; rotation trajectory; parametric equations
Abstract and keywords
Abstract (English):
In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Keywords:
ellipse; focus; rotation center; rotation trajectory; parametric equations
References

1. Antonova I.V. Matematicheskoe opisanie vrascheniya tochki vokrug ellipticheskoy osi v nekotoryh chastnyh sluchayah [Tekst] / I.V. Antonova, I.A Beglov, E.V. Solomonova // Geometriya i grafika. - 2019. - T. 7. - № 3. - S. 36-50. - DOI:https://doi.org/10.12737/article_5dce66dd9fb966.59423840.

2. Beglov I.A. Metod vrascheniya geometricheskih ob'ektov vokrug krivolineynoy osi [Tekst] / I.A. Beglov, V.V Rustamyan // Geometriya i grafika. - 2017. - T. 5. - № 3. - S. 45-50. - DOI:https://doi.org/10.12737/article_59bfa4eb0bf488.99866490.

3. Beglov I.A. Matematicheskoe opisanie metoda vrascheniya tochki vokrug krivolineynoy osi vtorogo poryadka [Tekst] / I.A. Beglov, V.V Rustamyan, I.V. Antonova // Geometriya i grafika. - 2018. - T. 6. - № 4. - S. 39-46. - DOI:https://doi.org/10.12737/article_5c21f6e832b4d2.25216268.

4. Beklemishev D.V. Kurs analiticheskoy geometrii i lineynoy algebry [Tekst] / D.V. Beklemishev. - M.: Fizmatlit, 2009. - 320s.

5. Bermant A.F. Geometricheskiy spravochnik po matematike (Atlas krivyh). Ch. 1. [Tekst] / A.F. Bermant. - M.-L.: ONGIZ NKTP, 1937. - 209 s.

6. Voloshinov D.V. Edinyy konstruktivnyy algoritm postroeniya fokusov krivyh vtorogo poryadka obrazov [Tekst] / D.V. Voloshinov // Geometriya i grafika. - 2018. - T. 6. - №. 2. - S. 47-54. - DOI: 10.12737/ article_5b559dc3551f95.26045830.

7. Vygodskiy M.Ya. Spravochnik po vysshey matematike [Tekst] / M.Ya. Vygodskiy. - M.: ACT: Astrel', 2006. - 991 s.

8. Girsh A.G. Vzaimnye zadachi s konikami [Tekst] / A.G. Girsh // Geometriya i grafika. - 2020. - T. 8. - № 1. - S. 15-24. - DOI:https://doi.org/10.12737/2308-4898-2020-15-24.

9. Girsh A. G. Fokusy algebraicheskih krivyh [Tekst] / A.G. Girsh // Geometriya i grafika. - 2015. - T. 3. - № 3. - C. 4-17. - DOI:https://doi.org/10.12737/14415.

10. Grafskiy O.A. Obosnovanie postroeniya kasatel'noy k okruzhnosti i ellipsu [Tekst] / O.A. Grafskiy, O.V. Saenko // Nauchno-tehnicheskie problemy transporta, promyshlennosti i obrazovaniya: trudy Vserossiyskoy nauchno-praktich. konferencii, 20-22 aprelya 2011 g. - Habarovsk: Izd-vo DVGUPS, 2011. - S. 14-18.

11. Gryaznov Ya.A. Otsek kanalovoy poverhnosti kak obraz cilindra v rassloyaemom obrazovanii [Tekst] / Ya.A. Gryaznov // Geometriya i grafika. - 2013. - T. 1. - № 3. - C. 17-19. - DOI:https://doi.org/10.12737/6518.

12. Zhiharev L.A. Otrazhenie ot krivolineynyh zerkal v ploskosti [Tekst] / L.A. Zhiharev // Geometriya i grafika. - 2019. - T. 7. - № 1. - S. 46-54. - DOI:https://doi.org/10.12737/article_5c9203adb22641.01479568.

13. Ivanov G.S. Konstruktivnyy sposob issledovaniya svoystv parametricheski zadannyh krivyh [Tekst] / G.S. Ivanov // Geometriya i grafika. - 2014. - T. 2. - № 3. - C. 3-6. - DOI:https://doi.org/10.12737/6518.

14. Kokareva Ya.A. Konstruirovanie kanalovyh poverhnostey s peremennoy obrazuyuschey i ploskost'yu parallelizma na osnove ekviaffinnyh preobrazovaniy ploskosti [Tekst] / Ya.A. Kokareva // Geometriya i grafika. - 2017. - T. 5. - № 1. - S. 12-20. - DOI:https://doi.org/10.12737/25119.

15. Kokareva Ya.A. Sintez uravneniy lineychatyh poverhnostey s dvumya krivolineynymi i odnoy pryamolineynoy napravlyayuschimi [Tekst] / Ya.A. Kokareva // Geometriya i grafika. - 2018. - T. 6. - № 3. - S. 3-12. - DOI:https://doi.org/10.12737/article_5bc454948a7d90.80979486.

16. Panchuk K.L. Ciklograficheskaya interpretaciya i komp'yuternoe reshenie odnoy sistemy algebraicheskih uravneniy [Tekst] / K.L. Panchuk, E.V. Lyubchinov // Geometriya i grafika. - 2019. - T. 7. - № 3. - S. 3-14. - DOI:https://doi.org/10.12737/article_5dce5e528e4301.77886978.

17. Sal'kov N.A. Nachertatel'naya geometriya - baza dlya geometrii analiticheskoy [Tekst] / N. A. Sal'kov // Geometriya i grafika. - 2016. - T. 4. - №. 1. - C. 44-54. - DOI:https://doi.org/10.12737/18057.

18. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 1. [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2015. - T. 3. - № 1. - S. 16-25. - DOI:https://doi.org/10.12737/10454.

19. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch.2. [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2015. - T. 3. - № 2. - S. 9-22. - DOI:https://doi.org/10.12737/12164.

20. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 3. [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2015. - T. 3. - № 4. - S. 3-14. - DOI:https://doi.org/10.12737/17345.

21. Sal'kov N.A. Svoystva ciklid Dyupena i ih primenenie. Ch. 4. [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2016. - T. 4. - № 1. - S. 21-33. - DOI:https://doi.org/10.12737/18055.

22. Sal'kov N.A. Ellips: kasatel'naya i normal' [Tekst] / N.A. Sal'kov // Geometriya i grafika. - 2013. - T. 1. - № 1. - C. 35-37. - DOI:https://doi.org/10.12737/470.

23. Beglov I.A. Generation of the surfaces via quasi-rotation of higher order [Tekst] / I A Beglov // Journal of Physics: Conference Series. - 2020. - V. 1546 - 012032. - DOIhttps://doi.org/10.1088/1742-6596/1546/1/012032.

24. Beglov I.A. Mass-centering characteristics of solids within quasi-rotation surfaces [Tekst] / I A Beglov // Journal of Physics: Conference Series. - 2021. - V. 1791 - 012035. - DOI:https://doi.org/10.1088/1742-6596/1791/1/012035.

25. Beglov I.A. N-n-digit interrelations between the sets within the R 2 plane generated by quasi-rotation of R 3 space [Tekst] / I A Beglov // Journal of Physics: Conference Series. - 2020. - V. 1546 - 012033 - DOIhttps://doi.org/10.1088/1742-6596/1546/1/012033.

26. Beglov I.A., Panchuk K.L. Plane tangent to quasi-rotation surface [Tekst] / Ivan Beglov and Konstantin Panchuk // CEUR Workshop Proceedings - 2020. - V. 2744. - 59. - DOI:https://doi.org/10.51130/graphicon-2020-2-3-59.

27. Sal’kov N.A. Application of the Dupin cyclide in temple architecture [Tekst] / N.A. Sal’kov // Journal of Physics: Conference Series. - 2020. - V. 1546 - 012042 DOIhttps://doi.org/10.1088/1742-6596/1546/1/012042.

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