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Characterizations of Framed Curves in Four-Dimensional Euclidean Space

Year 2021, Volume: 4 Issue: 4, 125 - 131, 30.12.2021
https://doi.org/10.32323/ujma.1008148

Abstract

Framed curves in Euclidean space are used to investigate singular curves and are important
for singularity theory. In this study, framed curves in four-dimensional Euclidean space are
introduced and new results are obtained. The relation of framed curves with Frenet curves
in four-dimensional Euclidean space is given and Bishop-type frame of framed curves is
introduced with the help of Euler angles. In addition, by using Bishop-type framed curves in
four-dimensional Euclidean space, framed rectifying curves, framed osculating curves and
framed normal curves are introduced. Also, some characterizations depending on framed
curvatures are obtained.

References

  • [1] S. Honda, M. Takahashi, Framed curves in the Euclidean space, Adv. Geo., 16(3) (2016), 265-276, https://doi.org/10.1515/advgeom-2015-0035.
  • [2] T. Fukunaga, M. Takahashi, Existence conditions of framed curves for smooth curves, J. Geo., 108 (2017), 763-774, https://doi.org/10.1007/s00022-017- 0371-5.
  • [3] S. Honda, M. Takahashi, Evolutes and focal surfaces of framed immersions in the Euclidean space, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 150(1) (2020), 497-516, https://doi.org/10.1017/prm.2018.84.
  • [4] Y. Wang Y, D. Pei, R. Gao, Generic properties of framed rectifying curves, Mathematics, 7(1)(2019), 37.
  • [5] B. D. Yazıcı, S. O . Karakus¸, M. Tosun, On the classification of framed rectifying curves in Euclidean space, Math. Meth. Appl. Sci., (2021), 1-10, https://doi.org/10.1002/mma.7561.
  • [6] B. D. Yazıcı, S. O. Karakus,M. Tosun, Framed normal curves in Euclidean space, TbilisiMath. J., (2021), 27-37, https://doi.org/10.2478/9788395793882- 003.
  • [7] S. Honda, M. Takahashi, Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turkish Journal of Mathematics, 44 (2020), 883-899, https://doi.org/10.3906/mat-1905-63.
  • [8] L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82(3) (1975), 246-251, https://doi.org/10.2307/2319846.
  • [9] F. Gokcelik, Z. Bozkurt, I. Gok, F. N. Ekmekc¸i, Y. Yaylı, Parallel transport frame in 4-dimensional Euclidean space, Caspian J. of Math. Sci., 3 (2014), 91-103.
  • [10] F. Ates¸, I. Gok, F.N. Ekmekc¸i, Y. Yaylı, Characterizations of inclined curves according to parallel transport frame in E4 and Bishop Frame in E3, Konuralp Journal of Mathematics, 7 (2019), 16-24.
  • [11] T. Korpınar, E. Turhan, Biharmonic curves according to parallel transport frame in E4, Bol. Soc. Paran. Mat., 31 (2)(2013), 213-217, https://doi.org/10.5269/bspm.v31i2.17669.
  • [12] Z. Ozdemir, I. Gok, F. N. Ekmekci, Y. Yaylı, A new approach on type-3 slant helix in E4, Gen. Math. Notes, 28(1) (2015), 40-49.
  • [13] A. J. Hanson, H. Ma, Parallel Transport Approach to Curve Framing, Tech. Math. Rep. 425, Indiana University Computer science Department, 1995.
  • [14] M. Z. Williams, F.M. Stein, A triple product of vectors in four-space, Math Mag., 37 (1964), 230-235.
Year 2021, Volume: 4 Issue: 4, 125 - 131, 30.12.2021
https://doi.org/10.32323/ujma.1008148

Abstract

References

  • [1] S. Honda, M. Takahashi, Framed curves in the Euclidean space, Adv. Geo., 16(3) (2016), 265-276, https://doi.org/10.1515/advgeom-2015-0035.
  • [2] T. Fukunaga, M. Takahashi, Existence conditions of framed curves for smooth curves, J. Geo., 108 (2017), 763-774, https://doi.org/10.1007/s00022-017- 0371-5.
  • [3] S. Honda, M. Takahashi, Evolutes and focal surfaces of framed immersions in the Euclidean space, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 150(1) (2020), 497-516, https://doi.org/10.1017/prm.2018.84.
  • [4] Y. Wang Y, D. Pei, R. Gao, Generic properties of framed rectifying curves, Mathematics, 7(1)(2019), 37.
  • [5] B. D. Yazıcı, S. O . Karakus¸, M. Tosun, On the classification of framed rectifying curves in Euclidean space, Math. Meth. Appl. Sci., (2021), 1-10, https://doi.org/10.1002/mma.7561.
  • [6] B. D. Yazıcı, S. O. Karakus,M. Tosun, Framed normal curves in Euclidean space, TbilisiMath. J., (2021), 27-37, https://doi.org/10.2478/9788395793882- 003.
  • [7] S. Honda, M. Takahashi, Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space, Turkish Journal of Mathematics, 44 (2020), 883-899, https://doi.org/10.3906/mat-1905-63.
  • [8] L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Monthly, 82(3) (1975), 246-251, https://doi.org/10.2307/2319846.
  • [9] F. Gokcelik, Z. Bozkurt, I. Gok, F. N. Ekmekc¸i, Y. Yaylı, Parallel transport frame in 4-dimensional Euclidean space, Caspian J. of Math. Sci., 3 (2014), 91-103.
  • [10] F. Ates¸, I. Gok, F.N. Ekmekc¸i, Y. Yaylı, Characterizations of inclined curves according to parallel transport frame in E4 and Bishop Frame in E3, Konuralp Journal of Mathematics, 7 (2019), 16-24.
  • [11] T. Korpınar, E. Turhan, Biharmonic curves according to parallel transport frame in E4, Bol. Soc. Paran. Mat., 31 (2)(2013), 213-217, https://doi.org/10.5269/bspm.v31i2.17669.
  • [12] Z. Ozdemir, I. Gok, F. N. Ekmekci, Y. Yaylı, A new approach on type-3 slant helix in E4, Gen. Math. Notes, 28(1) (2015), 40-49.
  • [13] A. J. Hanson, H. Ma, Parallel Transport Approach to Curve Framing, Tech. Math. Rep. 425, Indiana University Computer science Department, 1995.
  • [14] M. Z. Williams, F.M. Stein, A triple product of vectors in four-space, Math Mag., 37 (1964), 230-235.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Bahar Doğan Yazıcı 0000-0001-5690-4840

Sıddıka Özkaldı Karakuş 0000-0002-2699-4109

Murat Tosun 0000-0002-4888-1412

Publication Date December 30, 2021
Submission Date October 11, 2021
Acceptance Date December 29, 2021
Published in Issue Year 2021 Volume: 4 Issue: 4

Cite

APA Doğan Yazıcı, B., Özkaldı Karakuş, S., & Tosun, M. (2021). Characterizations of Framed Curves in Four-Dimensional Euclidean Space. Universal Journal of Mathematics and Applications, 4(4), 125-131. https://doi.org/10.32323/ujma.1008148
AMA Doğan Yazıcı B, Özkaldı Karakuş S, Tosun M. Characterizations of Framed Curves in Four-Dimensional Euclidean Space. Univ. J. Math. Appl. December 2021;4(4):125-131. doi:10.32323/ujma.1008148
Chicago Doğan Yazıcı, Bahar, Sıddıka Özkaldı Karakuş, and Murat Tosun. “Characterizations of Framed Curves in Four-Dimensional Euclidean Space”. Universal Journal of Mathematics and Applications 4, no. 4 (December 2021): 125-31. https://doi.org/10.32323/ujma.1008148.
EndNote Doğan Yazıcı B, Özkaldı Karakuş S, Tosun M (December 1, 2021) Characterizations of Framed Curves in Four-Dimensional Euclidean Space. Universal Journal of Mathematics and Applications 4 4 125–131.
IEEE B. Doğan Yazıcı, S. Özkaldı Karakuş, and M. Tosun, “Characterizations of Framed Curves in Four-Dimensional Euclidean Space”, Univ. J. Math. Appl., vol. 4, no. 4, pp. 125–131, 2021, doi: 10.32323/ujma.1008148.
ISNAD Doğan Yazıcı, Bahar et al. “Characterizations of Framed Curves in Four-Dimensional Euclidean Space”. Universal Journal of Mathematics and Applications 4/4 (December 2021), 125-131. https://doi.org/10.32323/ujma.1008148.
JAMA Doğan Yazıcı B, Özkaldı Karakuş S, Tosun M. Characterizations of Framed Curves in Four-Dimensional Euclidean Space. Univ. J. Math. Appl. 2021;4:125–131.
MLA Doğan Yazıcı, Bahar et al. “Characterizations of Framed Curves in Four-Dimensional Euclidean Space”. Universal Journal of Mathematics and Applications, vol. 4, no. 4, 2021, pp. 125-31, doi:10.32323/ujma.1008148.
Vancouver Doğan Yazıcı B, Özkaldı Karakuş S, Tosun M. Characterizations of Framed Curves in Four-Dimensional Euclidean Space. Univ. J. Math. Appl. 2021;4(4):125-31.

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