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Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications

Year 2020, Volume: 13 Issue: 1, 116 - 128, 30.01.2020

Fırat Yerlikaya İsmail Aydemir

https://doi.org/10.36890/iejg.621588
Cited By: 1

Abstract

We analyze integrability for the derivative formulas of the rotation minimizing frame in the Euclidean 3-space from a viewpoint of rotations around axes of the natural coordinate system. We give a theorem that presents only one component of the indirect solution of the rotation minimizing formulas. Using this theorem, we find a lemma which states the necessary condition for the indirect solution to be a steady solution. As an application of the lemma, the natural representation of the position vector field of a smooth curve whose the rotation minimizing vector field (or the Darboux vector field) makes a constant angle with a fixed straight line in space is obtained. Also, we realize that general helices using the position vector field consist of slant helices and Darboux helices in the sense of Bishop.

Keywords

Bishop slant helix Darboux helix rotation minimizing frame Euclidean 3-space

References

  • [1] Ali, A.T., Turgut, M.: Position vector of a time-like slant helix in Minkowski 3-space. J. Math. Analysis and Appl. 365(2), 559-569 (2010).
  • [2] Ali, A.T.: Position vectors of slant helices in Euclidean 3-space. J. Egyptian Math. Soc. 20(1), 1-6 (2012).
  • [3] Ali, A.T.: Position vectors of general helices in Euclidean 3-space. Bull. Math. Anal. Appl. 3(2), 198-205 (2010).
  • [4] Ali, A.T., Turgut, M.: Position vectors of timelike general helices in Minkowski 3-space. Glo. J. Adv. Res. Class. Mod. Geom. 2(1), 2-10 (2013).
  • [5] Ali, A.T.: Position vectors of spacelike general helices in Minkowski 3-space. Nonlinear Analysis: Theory, Methods and Applications. 73(4), 1118-1126 (2010).
  • [6] Ali, A.T.: Position vectors of curves in the Galilean space G3. Matematicki Vesnik. 64(3), 200-210 (2012).
  • [7] Bishop, R.L.: There is more than one way to frame a curve. The American Mathematical Monthly. 82(3), 246-251 (1975).
  • [8] Bükcü B., Karacan, M.K.: The slant helices according to Bishop frame. I. J. Computational and Math. Sci. 3(2), 67-70 (2009).
  • [9] Carmo, M.D.: Differential Geometry of Curves and Surfaces. Prentice–Hall. New Jersey (1976).
  • [10] Choi, J.H., Kim, Y.H.: Associated curves of a frenet curve and their applications. Applied Mathematics and Computation, 218(18), 9116-9124 (2012).
  • [11] Kızıltuğ, S., Önder, M.: Associated curves of frenet curves in three dimensional compact lie group. Miskolc Mathematical Notes. 16(2), 953-694 (2015).
  • [12] Kim, Y.H., Choi J.H., Ali, A.T.: Some associated curves of frenet non-lightlike curves in E31. J. Math. Analy. Appl. 394(2), 712-723 (2012).
  • [13] Lucas, P., Ortega-Yagues, J.A.: Slant helices in the euclidean 3-space revisited. Bull. Belgian Math. Soc. Simon Stevin. 23(1), 133-150 (2016).
  • [14] Macit, N., Akbıyık, M., Yüce, S.: Some new associated curves of an admissible frenet curve in 3-dimensional and 4-dimensional Galilean spaces. Romanian J. Math. Computer Sci. 7(2), 110-122 (2017).
  • [15] Mak, M., Altınbaş, H.: Some special associated curves of non-degenerate curve in anti de sitter 3-space. Math. Sci. Appl. E-Notes. 5(2), 89-97 (2017).
  • [16] Öztekin, H., Tatlıpınar, S.: Determination of the position vectors of curves from intrinsic equations in G3.Walailak J. Sci. Tech. 11(12), 1011-1018 (2014).
  • [17] Reich, K.: Die geschichte der differential geometrie von gauss bis Riemann. Archive for History of Exact Sciences. 11(4), 273-376 (1973).
  • [18] Savcı, U.Z., Yılmaz, S., Mağden, A.: Position vector of some special curves in Galilean 3-spaces G3. Glo. J. Adv. Res. Class. Mod. Geom. 3(1), 7-11 (2014).

Year 2020, Volume: 13 Issue: 1, 116 - 128, 30.01.2020

Fırat Yerlikaya İsmail Aydemir

https://doi.org/10.36890/iejg.621588
Cited By: 1

Abstract

References

  • [1] Ali, A.T., Turgut, M.: Position vector of a time-like slant helix in Minkowski 3-space. J. Math. Analysis and Appl. 365(2), 559-569 (2010).
  • [2] Ali, A.T.: Position vectors of slant helices in Euclidean 3-space. J. Egyptian Math. Soc. 20(1), 1-6 (2012).
  • [3] Ali, A.T.: Position vectors of general helices in Euclidean 3-space. Bull. Math. Anal. Appl. 3(2), 198-205 (2010).
  • [4] Ali, A.T., Turgut, M.: Position vectors of timelike general helices in Minkowski 3-space. Glo. J. Adv. Res. Class. Mod. Geom. 2(1), 2-10 (2013).
  • [5] Ali, A.T.: Position vectors of spacelike general helices in Minkowski 3-space. Nonlinear Analysis: Theory, Methods and Applications. 73(4), 1118-1126 (2010).
  • [6] Ali, A.T.: Position vectors of curves in the Galilean space G3. Matematicki Vesnik. 64(3), 200-210 (2012).
  • [7] Bishop, R.L.: There is more than one way to frame a curve. The American Mathematical Monthly. 82(3), 246-251 (1975).
  • [8] Bükcü B., Karacan, M.K.: The slant helices according to Bishop frame. I. J. Computational and Math. Sci. 3(2), 67-70 (2009).
  • [9] Carmo, M.D.: Differential Geometry of Curves and Surfaces. Prentice–Hall. New Jersey (1976).
  • [10] Choi, J.H., Kim, Y.H.: Associated curves of a frenet curve and their applications. Applied Mathematics and Computation, 218(18), 9116-9124 (2012).
  • [11] Kızıltuğ, S., Önder, M.: Associated curves of frenet curves in three dimensional compact lie group. Miskolc Mathematical Notes. 16(2), 953-694 (2015).
  • [12] Kim, Y.H., Choi J.H., Ali, A.T.: Some associated curves of frenet non-lightlike curves in E31. J. Math. Analy. Appl. 394(2), 712-723 (2012).
  • [13] Lucas, P., Ortega-Yagues, J.A.: Slant helices in the euclidean 3-space revisited. Bull. Belgian Math. Soc. Simon Stevin. 23(1), 133-150 (2016).
  • [14] Macit, N., Akbıyık, M., Yüce, S.: Some new associated curves of an admissible frenet curve in 3-dimensional and 4-dimensional Galilean spaces. Romanian J. Math. Computer Sci. 7(2), 110-122 (2017).
  • [15] Mak, M., Altınbaş, H.: Some special associated curves of non-degenerate curve in anti de sitter 3-space. Math. Sci. Appl. E-Notes. 5(2), 89-97 (2017).
  • [16] Öztekin, H., Tatlıpınar, S.: Determination of the position vectors of curves from intrinsic equations in G3.Walailak J. Sci. Tech. 11(12), 1011-1018 (2014).
  • [17] Reich, K.: Die geschichte der differential geometrie von gauss bis Riemann. Archive for History of Exact Sciences. 11(4), 273-376 (1973).
  • [18] Savcı, U.Z., Yılmaz, S., Mağden, A.: Position vector of some special curves in Galilean 3-spaces G3. Glo. J. Adv. Res. Class. Mod. Geom. 3(1), 7-11 (2014).
There are 18 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Fırat Yerlikaya 0000-0003-2360-1522

İsmail Aydemir 0000-0002-0238-2079

Publication Date January 30, 2020
Acceptance Date December 7, 2019
Published in Issue Year 2020 Volume: 13 Issue: 1

Cite

APA Yerlikaya, F., & Aydemir, İ. (2020). Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. International Electronic Journal of Geometry, 13(1), 116-128. https://doi.org/10.36890/iejg.621588
AMA Yerlikaya F, Aydemir İ. Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. Int. Electron. J. Geom. January 2020;13(1):116-128. doi:10.36890/iejg.621588
Chicago Yerlikaya, Fırat, and İsmail Aydemir. “Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications”. International Electronic Journal of Geometry 13, no. 1 (January 2020): 116-28. https://doi.org/10.36890/iejg.621588.
EndNote Yerlikaya F, Aydemir İ (January 1, 2020) Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. International Electronic Journal of Geometry 13 1 116–128.
IEEE F. Yerlikaya and İ. Aydemir, “Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications”, Int. Electron. J. Geom., vol. 13, no. 1, pp. 116–128, 2020, doi: 10.36890/iejg.621588.
ISNAD Yerlikaya, Fırat - Aydemir, İsmail. “Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications”. International Electronic Journal of Geometry 13/1 (January 2020), 116-128. https://doi.org/10.36890/iejg.621588.
JAMA Yerlikaya F, Aydemir İ. Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. Int. Electron. J. Geom. 2020;13:116–128.
MLA Yerlikaya, Fırat and İsmail Aydemir. “Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications”. International Electronic Journal of Geometry, vol. 13, no. 1, 2020, pp. 116-28, doi:10.36890/iejg.621588.
Vancouver Yerlikaya F, Aydemir İ. Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications. Int. Electron. J. Geom. 2020;13(1):116-28.

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https://doi.org/10.21597/jist.702398
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