Construction and Analysis of Smarandache Curves for Integral Binormal Curves in Euclidean 3-Space

Year 2025, Volume: 8 Issue: 3, 149 - 157, 17.09.2025

Ayman Elsharkawy , Clemente Cesarano , Hasnaa Baizeed

https://doi.org/10.32323/ujma.1739984

Abstract

This paper presents a detailed geometric analysis of Smarandache curves generated from integral binormal curves within three-dimensional Euclidean space. We provide a complete derivation of the Frenet apparatus encompassing tangent, normal, and binormal vectors, alongside curvature and torsion functions for four distinct types of Smarandache curves: $TN$, $TB$, $NB$, and $TNB$. Furthermore, we establish the necessary and sufficient criteria for these curves to be characterized as general helices or Salkowski curves. A significant outcome of our work is the demonstration that helical characteristics are transmitted from the original curve to its Smarandache derivatives. The theoretical framework is substantiated with numerical examples, including a circular helix and other spatial curves.

Keywords

Frenet frame , General helices , Integral curves , Salkowski curves , Smarandache curves

References

• [1] A. Elsharkawy, M. Turan, H. Bozok, Involute-evolute curves with modified orthogonal frame in Galilean space $G_3$, Ukr. Math. J., 76(10) (2024), 1625–1636. https://doi.org/10.1007/s11253-025-02411-5
• [2] A. Elsharkawy, A. M. Elshenhab, Mannheim curves and their partner curves in Minkowski 3-space $E_1^3$, Demonstratio Math., 55 (2022), 798–811. https://doi.org/10.1515/dema-2022-0163
• [3] 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., 28 (2020), Article ID 26. https://doi.org/10.1186/s42787-020-00086-4
• [4] S. P. Chimienti, M. Bencze, Smarandache anti-geometry, Smarandache Notions J., 9(1-3) (1998), 48–52.
• [5] A. T. Ali, Special Smarandache curves in the Euclidean space, Int. J. Math. Combin., 2 (2010), 30–36.
• [6] M. Turgut, S. Yilmaz, Smarandache curves in Minkowski space-time, Int. J. Math. Combin., 3 (2008), 51–55.
• [7] C. Ekici, M. B. Göksel, M. Dede, Smarandache curves according to $q$-frame in Minkowski 3-space, Conf. Proc. Sci. Technol., 2(2) (2019), 110–118.
• [8] O. Bektaş, S. Yuce, Special Smarandache curves according to Darboux frame in Euclidean 3-space, Rom. J. Math. Comput. Sci., 3(1) (2013), 48–59.
• [9] M. Cetin, Y. Tuncer, M. K. Karacan, Smarandache curves according to Bishop frame in Euclidean 3-space, Gen. Math. Notes, 20(2) (2014), 50-66.
• [10] H. K. Elsayied, A. M. Tawfiq, A. Elsharkawy, Special Smarandache curves according to the quasi frame in 4-dimensional Euclidean space $E^4$, Houston J. Math., 47(2) (2021), 467–482.
• [11] S. K. Nurkan, İ. A. Güven, A new approach for Smarandache curves, Turk. J. Math. Comput. Sci., 14(1) (2022), 155–165. https://doi.org/10.47000/tjmcs.1004423
• [12] M. Barros, General helices and a theorem of Lancret, Proc. Amer. Math. Soc., 125(5) (1997), 1503–1509. https://www.jstor.org/stable/2162098?seq=1&cid=pdf
• [13] C. Coleman, A certain class of integral curves in 3-space, Ann. of Math., 69(3) (1959), 678–685. https://www.jstor.org/stable/1970031
• [14] A. Elsharkawy, H. Baizeed, Some integral curves according to quasi-frame in Euclidean 3-space, Sci. Afr., 27 (2025), Article ID e02583. https://doi.org/10.1016/j.sciaf.2025.e02583
• [15] İ. A. Güven, Some integral curves with a new frame, Open Math., 18(1) (2020), 1332–1341. https://doi.org/10.1515/math-2020-0078
• [16] M. Düldül, Integral curves connected with a framed curve in 3-space, Honam Math. J., 45(1) (2023), 130–145.
• [17] S. Şenyurt, K. Eren, Some Smarandache curves constructed by a spacelike Salkowski curve with timelike principal normal, Punjab Univ. J. Math., 53(9) (2021), 679–690. https://doi.org/10.52280/pujm.2021.530905
• [18] S. Şenyurt, K. Eren, Smarandache curves of spacelike anti-Salkowski curve with a spacelike principal normal according to Frenet frame, Gümüşhane Univ. J. Sci., 10(1) (2020), 251–260. https://doi.org/10.17714/gumusfenbil.621363
• [19] H. Baizeed, A. Elsharkawy, C. Cesarano, A. A. Ramadan, Smarandache curves for the integral curves with the quasi frame in Euclidean 3-space, Azerb. J. Math., 15(2) (2025), 219-239. https://doi.org/10.59849/2218-6816.2025.2.219
• [20] A. Elsharkawy, H. Elsayied, A. Refaat, Quasi ruled surfaces in Euclidean 3-space, Eur. J. Pure Appl. Math., 18(1) (2025), Article ID 5710. https://doi.org/10.29020/nybg.ejpam.v18i1.5710
• [21] N. Macit, M. Düldül, Some new associated curves of a Frenet curve in E3 and E4, Turk. J. Math., 38(6) (2014), 1023–1037. https://doi.org/10.3906/mat-1401-85
• [22] J. Monterde, Salkowski curves revisited: a family of curves with constant curvature and non-constant torsion, Comput. Aided Geom. Design, 26(3) (2009), 271–278. https://doi.org/10.1016/j.cagd.2008.10.002
• [23] A. Elsharkawy, H. Hamdani, C. Cesarano, N. Elsharkawy, Geometric properties of Smarandache ruled surfaces generated by integral binormal curves in Euclidean 3-space, Part. Differ. Equ. Appl. Math., 15 (2025), Article ID 101298. https://doi.org/10.1016/j.padiff.2025.101298