Involute-evolute curves with modified orthogonal frame in Galilean space $G_{3}$

Ayman Elsharkawy; Murat Turan; Hülya Gün Bozok

doi:10.3842/umzh.v76i10.7822

Ayman Elsharkawy Department of Mathematics, Faculty of Science, Tanta University, Egypt
Murat Turan Department of Mathematics, Faculty of Engineering and Natural Sciences, Osmaniye Korkut Ata University, Turkey
Hülya Gün Bozok Department of Mathematics, Faculty of Engineering and Natural Sciences, Osmaniye Korkut Ata University, Turkey

DOI: https://doi.org/10.3842/umzh.v76i10.7822

Keywords: Galilean space, involute-evolute curve pair, modified orthogonal frame.

Abstract

UDC 514

We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space $G_3$. The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

References

M. Akyigit, K. Eren, H. H. Kosal, Tubular surfaces with modified orthogonal frame in euclidean 3-space, Honam Math. J., 143, № 3, 453–463 (2021).

M. Akyigit, A. Z. Azak, S. Ersoy, Involute-evolute curves in Galilean space $G_{3}$, Sci. Magna, 16, 75–80 (2010).

A. Z. Azak, Involute-evolute curves according to modified orthogonal frame, J. Sci. and Arts, 121, № 2, 385–394 (2021). DOI: https://doi.org/10.46939/J.Sci.Arts-21.2-a06

B. Bukcu, M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3-space, Math. Sci. and Appl. E-Notes, 14, 184–188 (2016). DOI: https://doi.org/10.36753/mathenot.421429

H. K. Elsayied, A. A. Altaha, A. Elsharkawy, Bertrand curves with the modified orthogonal frame in Мinkowski 3-space $E_{1}^{3}$, Rev. Educ., 1392, № 6, 43–55 (2021).

H. K. Elsayied, A. A. Altaha, A. Elsharkawy, On some special curves according to the modified orthogonal frame in Minkowski 3-space $E_{1}^{3}$, Kasmera, 149, № 1, 2–15 (2021).

A. Elsharkawy, Generalized involute and evolute curves of equiform spacelike curves with a timelike equiform principal normal in $E_1^ 3$, J. Egyptian Math. Soc., 128, № 1, Article 26 (2020). DOI: https://doi.org/10.1186/s42787-020-00086-4

A. Elsharkawy, Y. Tashkandy, W. Emam, C. Cesarano, N. Elsharkawy, On some quasi-curves in Galilean three-space, Axioms, 112, № 9, 823 (2023). DOI: https://doi.org/10.3390/axioms12090823

I. Kamenarovic, Existence theorems for ruled surfaces in the Galilean space $G_{3}$, Rad Hrvat. Akad. Znan. Umjet. Math., 110, 183–196 (1991).

S. Kiziltug, A. Cakmak, T. Erisir, G. Mumcu, On tubular surfaces with modified orthogonal frame in the Galilean space $G_{3}$, Thermal Sci., 126, Special Issue 2, S571–S581 (2022). DOI: https://doi.org/10.2298/TSCI22S2571K

M. S. Lone, H. Es, M. K. Karacan, B. Bukcu, On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. and Technol. Trans. A Sci., 143, 1905–1916 (2019). DOI: https://doi.org/10.1007/s40995-018-0661-2

M. S. Lone, H. Es, M. K. Karacan, B. Bukcu, Mannheim curves with modified orthogonal frame in Euclidean 3-space, Turk. J. Math., 143, № 2, 648–663 (2019). DOI: https://doi.org/10.3906/mat-1807-177

O. Röschel, Die Geometrie des Galileischen Raumes, Habilitationsschrift, Leoben (1984).

T. Sasai, The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math J., 136, 17–24 (1984). DOI: https://doi.org/10.2748/tmj/1178228899

M. Sevinc, H. Kusak Samanci, Characterizations of the ruled surfaces with modified orthogonal frame, Erzincan University J. Sci. and Technol., 115, № 2, 420–441 (2022). DOI: https://doi.org/10.18185/erzifbed.997998