This study aims to investigate the geometric properties of $ V_i $-helices in the four-dimensional Euclidean space $ \mathbb{E}^4 $, considering Frenet vector fields as instances of $ F $-constant vector fields. For each $ i \in \{1, 2, 3, 4\}, $ the necessary and sufficient conditions for $ V_i $-helices are derived in terms of their curvatures, and a generalization of these helices is presented. Examples of these structures are provided, and their projections onto three-dimensional (3D) spaces are visualized using Python. Furthermore, the corresponding Python codes are included in the Appendix.
Citation: Derya Sağlam, Umut Selvi, Faik Babadağ, Ali Atasoy. Helices with $ F $-constant vector fields in the Euclidean space $ \mathbb{E}^4 $[J]. AIMS Mathematics, 2026, 11(2): 4299-4334. doi: 10.3934/math.2026173
This study aims to investigate the geometric properties of $ V_i $-helices in the four-dimensional Euclidean space $ \mathbb{E}^4 $, considering Frenet vector fields as instances of $ F $-constant vector fields. For each $ i \in \{1, 2, 3, 4\}, $ the necessary and sufficient conditions for $ V_i $-helices are derived in terms of their curvatures, and a generalization of these helices is presented. Examples of these structures are provided, and their projections onto three-dimensional (3D) spaces are visualized using Python. Furthermore, the corresponding Python codes are included in the Appendix.
| [1] |
A. Mennınger, Characterization of the slant helix as successor curve of the general helix, Int. Electron. J. Geom., 7 (2014), 84–91. https://doi.org/10.36890/iejg.593986 doi: 10.36890/iejg.593986
|
| [2] | A. Şenol, E. Ziplar, Y. Yayli, İ. Gök, A new approach on helices in Euclidean n-space, Math. Commun., 18 (2013), 241–256. |
| [3] |
A. T. Ali, Position vectors of spacelike general helices in Minkowski 3-space, Nonlinear Anal.-Theor., 73 (2010), 1118–1126. https://doi.org/10.1016/j.na.2010.04.051 doi: 10.1016/j.na.2010.04.051
|
| [4] | A. T. Ali, R. López, Some characterizations of inclined curves in Euclidean $\mathbb{E}^n$ space, Novi Sad J. Math., 40 (2010), 9–17. |
| [5] |
Ç. Camcı, K. İlarslan, L. Kula, H. H. Hacısalihoğlu, Harmonic curvatures and generalized helices in $\mathbb{E}^n$, Chaos Soliton. Fract., 40 (2009), 2590–2596. https://doi.org/10.1016/j.chaos.2007.11.001 doi: 10.1016/j.chaos.2007.11.001
|
| [6] |
D. Sağlam, On dual slant helices in $\mathbb{D}^3$, Adv. Math., 11 (2022), 577–589. https://doi.org/10.37418/amsj.11.7.2 doi: 10.37418/amsj.11.7.2
|
| [7] | E. Ziplar, A. Şenol, Y. Yayli, On Darboux helices in Euclidean 3-space, Global Journal of Science Frontier Research Mathematics and Decision Sciences, 12 (2012), 73–80. |
| [8] |
H. A. Hayden, On a generalized helix in a Riemannian n-space, Proce. Lond. Math. Soc., s2-32 (1931), 337–345. https://doi.org/10.1112/plms/s2-32.1.337 doi: 10.1112/plms/s2-32.1.337
|
| [9] | İ. Gök, C. Camci, H. Hacısalihoğlu, $ V_ {n} $-slant helices in Euclidean $ n $-space $ E^{n} $, Math. Commun., 14 (2009), 317–329. |
| [10] | M. Barros, General helices and a theorem of Lancret, Proce. Amer. Math. Soc., 125 (1997), 1503–1509. |
| [11] | M. Barros, A. Ferrandez, P. Lusas, M. A. Merono, General helices in the three-dimensional Lorentzian space forms, The Rocky Mountain Journal of Mathematics, 31 (2001), 373–388. |
| [12] |
L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput., 169 (2005), 600–607. https://doi.org/10.1016/j.amc.2004.09.078 doi: 10.1016/j.amc.2004.09.078
|
| [13] |
P. Lucas, J. A. Ortega-Yagües, Slant helices in the Euclidean 3-space revisited, Bull. Belg. Math. Soc. Simon Stevin, 23 (2016), 133–150. https://doi.org/10.36045/bbms/1457560859 doi: 10.36045/bbms/1457560859
|
| [14] |
P. Lucas, J. A. Ortega-Yagües, A generalization of the notion of helix, Turk. J. Math., 47 (2023), 1158–1168. https://doi.org/10.55730/1300-0098.3418 doi: 10.55730/1300-0098.3418
|
| [15] |
U. Öztürk, Z. B. Alkan, Darboux helices in three dimensional Lie groups, AIMS Mathematics, 5 (2020), 3169–3181. https://doi.org/10.3934/math.2020204 doi: 10.3934/math.2020204
|
| [16] |
X. Yang, High accuracy approximation of helices by quintic curves, Comput. Aided Geom. D., 20 (2003), 303–317. https://doi.org/10.1016/S0167-8396(03)00074-8 doi: 10.1016/S0167-8396(03)00074-8
|
| [17] |
Y. Ünlütürk, T. Korpmar, M. Çimdiker, On k-type pseudo null slant helices due to the Bishop frame in Minkowski 3-space $\mathbb{E}_1^3$, AIMS Mathematics, 5 (2019), 286–299. https://doi.org/10.3934/math.2020019 doi: 10.3934/math.2020019
|
| [18] |
A. Macdonald, A survey of geometric algebra and geometric calculus, Adv. Appl. Clifford Algebras, 27 (2017), 853–891. https://doi.org/10.1007/s00006-016-0665-y doi: 10.1007/s00006-016-0665-y
|
| [19] |
E. Özdamar, Characterizations of spherical curves in euclidean n-space, Commun. Fac. Sci. Univ., 23 (1974), 110–125. https://doi.org/10.1501/Commua1_0000000619 doi: 10.1501/Commua1_0000000619
|
| [20] | F. G. Montoya, J. Ventura, F. M. Arrabal-Campos, A. Alcayde, A. H. Eid, Frequency generalization via Darboux bivector and electrical curves in multi-phase power systems, TechRxiv. https://doi.org/10.36227/techrxiv.22829084.v1 |
| [21] |
J. Monterde, Salkowski curves revisited: a family of curves with constant curvature and non-constant torsion, Comput. Aided Geom. D., 26 (2009), 271–278. https://doi.org/10.1016/J.CAGD.2008.10.002 doi: 10.1016/J.CAGD.2008.10.002
|
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Derya Sağlam, Umut Selvi, Faik Babadağ, Ali Atasoy. Helices with $ F $-constant vector fields in the Euclidean space $ \mathbb{E}^4 $[J]. AIMS Mathematics, 2026, 11(2): 4299-4334. doi: 10.3934/math.2026173
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