On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$

Bedia Merih Özçetin Vatansever

doi:10.33434/cams.1976061

On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$

Abstract

In this paper, we introduce the $\mathcal{W}$-direction curve constructed from the unit generalized Darboux vector of a Frenet curve in $n$-dimensional Euclidean space $\mathbb{E}^n$. We derive the higher-order derivatives of the associated curve and determine its Frenet vectors and curvatures. The computational procedure is implemented via an algorithm developed in MATLAB. Finally, a representative example is provided to illustrate the applicability of the proposed method.

Keywords

Frenet curve, Associated curve, Integral curve, Darboux vector

References

A. T. Ali, Position vectors of slant helices in Euclidean $3$-space, J. Egypt. Math. Soc., 20(1) (2012), 1-6. https://doi.org/10.1016/j.joems.2011.12.005
B. Y. Chen, When does the position vector of a space curve always lie in its rectifying plane?, Amer. Math. Monthly, 10(2) (2003), 147–152. https://doi.org/10.1080/00029890.2003.11919949
L. Kula, N. Ekmekçi, Y. Yaylı, et al., Characterizations of slant helices in Euclidean $3$-space, Turkish J. Math., 34(2) (2010), 261–274. https://doi.org/10.3906/mat-0809-17
L. Kula, Y. Yaylı, On slant helix and its spherical indicatrix, Appl. Math. Comput., 169(1) (2005), 600–607. https://doi.org/10.1016/j.amc.2004.09.078
M. Babaarslan, Y.A. Tandoğan, Y. Yaylı, A note on Bertrand curves and constant slope surfaces according to Darboux frame, J. Adv. Math. Stud., 5(1) (2012), 87–96.
H. Balgetir, M. Bektaş, J. Inoguchi, Null Bertrand curves in Minkowski $3$-space and their characterizations, Note Mat., 23(1) (2004), 7-13.
S. Izumiya, N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom., 74 (2002), 97–109. https://doi.org/10.1007/PL00012543
H. Liu, F. Wang, Mannheim partner curves in $3$-space, J. Geom., 88(1-2) (2008), 120-126.
M. Turgut, S. Yilmaz, On the Frenet frame and a characterization of space-like involute-evolute curve couple in Minkowski space-time, Int. Math. Forum., 3(16) (2008), 793-801.
J. H. Choi, Y. H. Kim, Associated curves of a Frenet curve and their applications, Appl. Math. Comput., 218(18) (2012), 9116–9124. https://doi.org/10.1016/j.amc.2012.02.064

N. Macit, M. Akbıyık, S. Yüce, Some new associated curves of an admissible Frenet curve in $3$-dimensional and $4$-dimensional Galilean spaces, Rom. J. Math. Computer. Sci., 7(2) (2017), 110-122.
J. Qian, Y. H. Kim, Directional associated curves of a null curve in Minkowski $3$-space, Bull. Korean Math. Soc., 52(1) (2015), 183-200. http://doi.org/10.4134/BKMS.2015.52.1.183
J. H. Choi, Y. H. Kim, A. T. Ali, Some associated curves of Frenet non-lightlike curves in $\mathbb{E}_1^3$, J. Math. Anal. Appl., 394(2) (2012), 712-723. https://doi.org/10.1016/j.jmaa.2012.04.063
T. Körpınar, M. T. Sarıaydın, E. Turhan, Associated curves according to Bishop frame in Euclidean $3$-space, Adv. Model. Optim., 15 (2013), 713–717.
N. Macit, M. Düldül, Some new associated curves of a Frenet curve in $\mathbb{E}^3$ and $\mathbb{E}^4$, Turkish J. Math., 38(6) (2014), 1023-1037. https://doi.org/10.3906/mat-1401-85
B. Sahiner, Quaternionic direction curves, Kyungpook Math. J., 58(2) (2018), 377–388. http://doi.org/https://doi.org/10.5666/KMJ.2018.58.2.377
B. Sahiner, Direction curves of principal normal indicatrix of a curve, J. Tech. Sci., 8(2) (2018), 46-54.
B. Sahiner, Direction curves of tangent indicatrix of a curve, Appl. Math. Comput., 343 (2019), 273-284. https://doi.org/10.1016/j.amc.2018.09.021
B. M. Özçetin Vatansever, An investigation of new associated curves of a Frenet curve in Euclidean $n$-space, Turkish J. Math., 49(6) (2025), 822-837. https://doi.org/10.55730/1300-0098.3627
L. S. Moreira, Geometric analogy and products of vectors in $n$ dimensions, Adv. Linear Alg. Matrix Theory, 3(1) (2013), 1–6. https://doi.org/10.4236/alamt.2013.31001
H. Gluck, Higher curvatures of curves in Euclidean space, Amer. Math. Monthly, 73(7) (1966), 699–704. https://doi.org/10.1080/00029890.1966.11970818
H. H. Hacısalihoğlu, Differantial Geometry, Inonu University Faculty of Arts and Sciences Publications, Malatya, Turkiye, 1980.
K. İlarslan, M. Yıldırım, On Darboux Helices in Euclidean $4$-space, Math. Methods Appl. Sci., 42(16) (2019), 5184–5189. https://doi.org/10.1002/mma.5260

Details

Primary Language

English

Subjects

Algebraic and Differential Geometry

Journal Section

Research Article

Authors

Bedia Merih Özçetin Vatansever ^*
0000-0003-1424-5407
Türkiye

Early Pub Date

August 17, 2026

Publication Date

-

Submission Date

June 21, 2026

Acceptance Date

August 11, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

DOI

https://doi.org/10.33434/cams.1976061

IZ

https://izlik.org/JA88YU67UP

APA

Özçetin Vatansever, B. M. (2026). On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$. Communications in Advanced Mathematical Sciences, Advanced Online Publication, 92-99. https://doi.org/10.33434/cams.1976061

AMA

1.Özçetin Vatansever BM. On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$. Communications in Advanced Mathematical Sciences. 2026;(Advanced Online Publication):92-99. doi:10.33434/cams.1976061

Chicago

Özçetin Vatansever, Bedia Merih. 2026. “On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$”. Communications in Advanced Mathematical Sciences, no. Advanced Online Publication: 92-99. https://doi.org/10.33434/cams.1976061.

EndNote

Özçetin Vatansever BM (August 1, 2026) On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$. Communications in Advanced Mathematical Sciences Advanced Online Publication 92–99.

IEEE

[1]B. M. Özçetin Vatansever, “On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$”, Communications in Advanced Mathematical Sciences, no. Advanced Online Publication, pp. 92–99, Aug. 2026, doi: 10.33434/cams.1976061.

ISNAD

Özçetin Vatansever, Bedia Merih. “On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$”. Communications in Advanced Mathematical Sciences. Advanced Online Publication (August 1, 2026): 92-99. https://doi.org/10.33434/cams.1976061.

JAMA

1.Özçetin Vatansever BM. On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$. Communications in Advanced Mathematical Sciences. 2026;:92–99.

MLA

Özçetin Vatansever, Bedia Merih. “On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$”. Communications in Advanced Mathematical Sciences, no. Advanced Online Publication, Aug. 2026, pp. 92-99, doi:10.33434/cams.1976061.

Vancouver

1.Bedia Merih Özçetin Vatansever. On Novel Associated Curves of a Generalized Darboux Vector in $\mathbb{E}^n$. Communications in Advanced Mathematical Sciences. 2026 Aug. 1;(Advanced Online Publication):92-9. doi:10.33434/cams.1976061

The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..