Abstract
This study investigates the conformable differential geometry of some special curves defined in the equi-affine space. The conformable derivative, a generalization of fractional calculus, is a flexible operator controlled by a parameter $\alpha$ that enables the modeling of nonlocal behavior in functions. This paper aims to offer a new perspective by combining the modern concept of derivative with equi-affine differential geometry. First, this paper introduces the conformable equi-affine arc length parameter and the corresponding conformable Frenet frame in the equi-affine space. The main focus is on characterizing special classes of curves, such as helices, slant helices, and rectifying curves, within the conformable equi-affine frame. These results expand the geometric applications of conformable calculus and provide a broader theoretical framework for curve analysis in equi-affine geometry. Finally, the accuracy of the results is observed with an example, and the curvatures are plotted as a function of $\alpha$ with MATLAB R2022b.
References
- M. E. Aydın, A. Mihai, A. Yokuş, Applications of fractional calculus in equiaffine geometry: Plane curves with fractional order, Mathematical Methods in the Applied Sciences 44 (2021) 13659–13669.
- M. Takahashi, Equi-affine plane curves with singular points, Journal of Geometry 113(1) (2022) 16.
- M. Ögrenmiş, A. O. Ögrenmis, Lancret-type theorems in equiaffine space, Journal of Geometry 115(2) (2024) 25.
- M. Öğrenmiş, Fractional curvatures of equiaffine curves in three-dimensional affine space, Journal of New Theory (46) (2024) 11–22.
- M. Ögrenmiş, A. O. Ögrenmiş, Curves in multiplicative equiaffine space, Mathematics 13 (2025) 1107.
- R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014) 65–70.
- U. Gözütok, H. A. Çoban, Y. Sağıroğlu, Frenet frame with respect to conformable derivative, Filomat 33(6) (2019) 1541–1550.
- B. Yılmaz, A new type electromagnetic curves in optical fiber and rotation of the polarization plane using fractional calculus, Optik 247 (2021) 168026.
- A. Has, B. Yılmaz, A. Akkurt, H. Yıldırım, Conformable special curves in Euclidean $\mathbb{R}^3$-space, Filomat 36(14) (2022) 4687–4698.
- A. Has, B. Yılmaz, Measurement and calculation on conformable surfaces, Mediterranean Journal of Mathematics 20(5) (2023) 274.
- E. Karaca, A. Altınkaya, Minkowski geometry of special conformable curves, International Journal of Geometric Methods in Modern Physics 22(05) (2025) 2450318.
- D. Davis, Generic affine differential geometry of curves in $\mathbb{R}^n$, Proceedings of the Royal Society of Edinburgh 136A (2006) 1195–1205.
Details
| Primary Language | English |
|---|---|
| Subjects | Algebraic and Differential Geometry |
| Journal Section | Research Article |
| Authors | |
| Early Pub Date | September 30, 2025 |
| Publication Date | September 30, 2025 |
| Submission Date | August 30, 2025 |
| Acceptance Date | September 30, 2025 |
| Published in Issue | Year 2025 Issue: 52 |
Cite
- Article Files