An Alternative Approach to Find the Position Vector of a General Helix

Year 2024, Volume: 20 Issue: 2, 54 - 60, 28.06.2024

Gizem Güzelkardeşler , Burak Şahiner

https://doi.org/10.18466/cbayarfbe.1479066

Abstract

In this paper, we introduce an alternative approach centered around an alternative moving frame for finding the position vector of a general helix given its curvature and torsion. Our methodology begins by formulating a vector differential equation, leveraging the unit principal normal vector of a general helix with the assistance of the alternative moving frame. Then, by solving this differential equation, we obtain the position vector of the general helix. This innovative technique is then applied to ascertain the position vector of a circular helix. To illustrate the effectiveness of our method, we showcase parametric representations of various general helices, each defined by unique curvature and torsion functions.

Keywords

Alternative moving frame, curvatures, general helix, position vector

Ethical Statement

There are no ethical issues after the publication of this manuscript.

References

• 1]. Eisenhart, LP. A Treatise on Differential Geometry of Curves and Surfaces; Dover, New York, 1960.
• [2]. Hartman, P., Wintner, A. 1950. On the fundamental equations of differential geometry. American Journal of Mathematics; 72(4): 757-774.
• [3]. Struik, DJ. Lectures on Classical Differential Geometry, 2nd edn.; Dover, New York, 1961.
• [4]. Euler, L. 1736. De constructione aequationum ope motus tractorii aliisque ad methodum tangentium inversam pertinentibus. Commentarii Academie Scientiarum Petropolitane; 8: 66-85.
• [5]. Ali, AT. 2011. Position vectors of general helices in Euclidean 3-space. Bulletin of Mathematical Analysis and Applications; 3(2): 198-205.
• [6]. Ali, AT. 2012. Position vectors of slant helices in Euclidean 3-space. Journal of the Egyptian Mathematical Society; 20(1): 1-6.
• [7]. Ali, AT. 2012. Position vectors of curves in the Galilean space G_3. Matematički Vesnik; 64(3): 200-210.
• [8]. Ali, AT, Mahmoud, SR. 2014. Position vector of spacelike slant helices in Minkowski 3-space. Honam Mathematical Journal; 36(2): 233-251.
• [9]. Ali, AT, Turgut, M. 2010. Position vector of a time-like slant helix in Minkowski 3-space. Journal of Mathematical Analysis and Applications; 365(2): 559-569.
• [10]. Bozok, HG, Sepet, SA, Ergüt, M. 2018. Position vectors of general helices according to type-2 Bishop frame in E^3. Mathematical Sciences and Applications E-Notes; 6(1): 64-69.
• [11]. El Haimi, A, Chahdi, AO. 2021. Parametric equations of special curves lying on a regular surface in Euclidean 3-space. Nonlinear Functional Analysis and Applications; 26(2): 225-236.
• [12]. El Haimi, A, Izid, M, Chahdi, AO. 2021. Position vectors of curves generalizing general helices and slant helices in Euclidean 3-space, Tamkang Journal of Mathematics; 52(4): 467-478.
• [13]. Öztekin, H, Tatlıpınar, S. 2014. Determination of the position vectors of curves from intrinsic equations in G_3. Walailak Journal of Science and Technology; 11(12): 1011-1018.
• [14]. Şahin, T, Dirişen, BC. 2019. Position vectors of curves with respect to Darboux frame in the Galilean space G^3. Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics; 68(2): 2079-2093.
• [15]. Yılmaz, B, Has, A. 2019. New approach to slant helix. International Electronic Journal of Geometry; 12(1): 111-115.
• [16]. Güzelkardeşler, G, Şahiner, B. 2023. An alternative method for determination of the position vector of a slant helix. Journal of New Theory; 4: 97-105.
• [17]. O’Neill, B. Elementary Differential Geometry; Academic Press, New York, 1966.
• [18]. Do Carmo, MP. Differential Geometry of Curves and Surfaces; Dover Publications, Mineola, USA, 2016.
• [19]. Chen, BY, Dillen, F, Verstraelen, L. 1986. Finite type space curves. Soochow Journal of Mathematics; 12: 1-10.
• [20]. Scofield, PD. 1995. Curves of constant precession. American Mathematical Monthly; 102(6): 531-537.
• [21]. Uzunoğlu, B, Gök, İ, Yaylı, Y. 2016. A new approach on curves of constant precession. Applied Mathematics and Computations; 275: 317-323.
• [22]. Şahiner, B. 2019. Ruled surfaces according to alternative moving frame. arXiv preprint arXiv:1910.06589.