EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5864 ISSN 1307-5543 – ejpam.com Published by New York Business Global Cα-Rectifying Curves in a New Conformable Differential Geometry Aykut Has1,∗, Beyhan Yılmaz1, Thabet Abdeljawad2 1 Department of Mathematics, Faculty of Science, Kahramanmaras Sutcu Imam University, 46100, Kahramanmaras, Turkey 2 Department of Mathematics and Sciences, Prince Sultan University, 66833, 11586 Riyadh, Saudi Arabia Abstract. In this study, we reintroduce the theory of curves by incorporating local fractional calculus. We elucidate the condition for a naturally parametrized curve to be conformable, and we define the orthonormal conformable frame of such a curve at any given point. Then, we provide a comprehensive explanation of how these newly derived conformable geometric concepts are related to their classical counterparts. Furthermore, we introduce the concept of a conformable rectifying curve and provide its characterizations in terms of this differentiation with respect to arbitrary order. Some illustrative graphs are provided. 2020 Mathematics Subject Classifications: 53A04, 26A33 Key Words and Phrases: Fractional calculus, Conformable fractional calculus, Frenet frame, Rectifying curve, Spherical curve 1. Introduction One of the most interesting topics in differential geometry is the theory of curves. The reason for this is that curves are used in modeling many problems that we encounter in real life. For example, curves are used when observing the motion of a charged particle in a magnetic field [1, 2]. In addition, many operations are performed using curves in computer- aided geometric designs [3, 4]. The first thing to do when designing these geometric models is to characterize the curve. Because curves are concepts that are characterized and studied. There are some methods used when characterizing curves. The most important of these are that the curve is characterized by its curvatures and Frenet vectors. The rectifying curves that are the subject of the article are characterized by both Frenet vectors and curvatures by B.Y. Chen [5]. A naturally parametrized curve is called a rectifying curve if it lies in the plane formed by its tangent and binormal at a point. Also, the ratio of ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5864 Email addresses: ahas@ksu.edu.tr (A. Has), beyhanyilmaz@ksu.edu.tr (B. Yılmaz), tabdeljawad@psu.edu.sa (T. Abdeljawad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 2 of 14 torsion to curvature of rectifying curves results in a linear equation. Moreover, it is known that centrodes (i.e. angular velocity vectors) play some important roles in mechanics and joint kinematics [6, 7]. In this direction B.Y. Chen and F. Dillen observed that rectifying curves can be viewed as centrodes and extremal curves in E3 [8]. In addition, rectifying curves are studied by many researchers in different spaces and different dimensions [9–13]. Fractional calculus represents a generalization of classical derivative and integral con- cepts, extensively explored by contemporary researchers. The notion of fractional deriva- tive essentially refers to derivatives of non-integer order. Remarkably, the concepts of fractional derivatives and integrals have an age parallel to that of integer derivatives and integrals, with the term ”fractional derivative” first mentioned in Leibniz’s 1695 letter to L’Hospital, as documented in various sources. In this letter, Leibniz posed the query, ”Can the notion of integer derivatives be extended to fractional derivatives?” The concept of fractional calculus has captivated the interest of numerous mathematicians, becoming a broad area of study. It has emerged as an indispensable cornerstone across various domains within basic sciences and engineering, purportedly offering more nuanced nu- merical results, particularly in the realm of differential equation solutions. Fractional calculus has gained widespread popularity, leading to diverse definitions and features pro- posed by numerous researchers. Notable among these are the Riemann-Liouville (R-L), Caputo, Grünwald-Letnikov, Wely, and Riesz fractional derivatives [14–16]. While they share common features, each fractional calculus possesses unique rules. For instance, non- local fractional derivative types diverge from satisfying the classical Leibniz and chain rules. Notably, except for the Caputo fractional derivative, the derivative of a constant is non-zero in non-local fractional derivatives [17]. On the other hand, local fractional derivatives such as Conformable, Alternative, M -fractional, and V -fractional adhere to satisfying Leibniz’s and the chain rule. Hence, local fractional derivatives hold an ad- vantageous position in algebraically constructed subjects due to their adherence to these classical rules [18–22]. The author in [23] studied more features of conformable fractional derivatives where the endpoints are allowed to appear in the weight of the conformable integral to define the concepts of left and right conformable derivatives. Recalling that, if a function f is differentiable then its conformable derivative Dαf(t) of order α ∈ (0, 1] will equal to t1−αf ′(t), we can relate conformable derivatives to a fractal type. Indeed, we have lim t→s f(t)− f(s) tα − sα = lim t→s f(t)− f(s) tα − sα . t− s t− s . (1) Then, we have lim t→s f(t)− f(s) tα − sα = lim t→s t− s tα − sα f ′(t) = 1 α t1−αf ′(t). (2) That is lim t→s f(t)− f(s) tα − sα = 1 α Dαf(t). (3) The theory of curves involves examining the movement of a point within a plane or A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 3 of 14 space using tools from linear algebra and calculus. In recent research, there’s a notable trend where fractional calculus has begun to be applied to the study of curves and sur- faces within differential geometry. This trend was initiated with the pioneering work of T. Yajima and K. Kamasaki, who conducted the first study employing fractional calculus to analyze surfaces [24]. Subsequently, T. Yajima et al. extended this exploration by de- riving Frenet formulas using fractional derivatives [25]. Another significant contribution came from K.A. Lazopoulos and A.K. Lazopoulos, who delved into the realm of fractional differentiable manifolds [26]. Additionally, M.E. Aydın et al. investigated plane curves within the context of fractional order equiaffine geometry [27]. Exploring the foundational concepts of curves and the Frenet frame within the domain of fractional order, U. Gozu- tok et al. conducted an analysis utilizing conformable local fractional derivatives [28]. Furthermore, A. Has and B. Yılmaz investigated specific curves and curve pairs within the context of fractional order, employing conformable Frenet frames [29, 30]. Moreover, the exploration of electromagnetic fields and magnetic curves under fractional derivatives has been undertaken by A. Has and B. Yılmaz [31–33]. These studies collectively show- case the burgeoning interest and application of fractional calculus in diverse aspects of curve theory, bringing forth new insights and methodologies within the field of differential geometry. In this study, algebraic and calculus-based properties of curves are reconstructed with the help of conformable local fractional derivatives. First of all, line, plane and sphere, which are the most basic concepts of geometry, are redefined in fractional order. Af- terward, the concepts of unit and orthogonality, which are the algebraic basis of curves, are defined in accordance with the fractional order. Then, the conformable frame of the conformable naturally parameterized curve is defined. Throughout this study, definitions based on conformable analysis are denoted by Cα. For example Cα−frame, Cα−naturally parameterized curve etc. It should be noted here that the conformable frame defined in this study is different from the frame discussed in the study [28]. The conformable frame mentioned in this article is completely defined by the vectors’ conformable local fractional derivative and gives different results from the classical Frenet frame. In addition, the rectifying curves defined by Chen and also called Chen curves in the article are examined with a conformable local fractional derivative and their fractional order characterizations are obtained. Finally, in the study, examples of the concepts obtained from fractional order are given and their graphs are drawn. 2. Preliminaries 2.1. Basics parametrized curves A regular natural parametrization of class Ck, with k ≥ 1 of a curve in R3 is a vector valued function x : I ⊂ R → E3, s 7→ x(s) = (x1(s),x2(s),x3(s)) defined on an interval I which satisfies x is of class Ck and x′(s) ̸= 0 for all s ∈ I where E denotes Euclidean space. A curve x is continuously differentiable if x′(s) exists for all s ∈ I and the derivative x′(s) is a continuous function; thinking dynamically, the vector x′(s) is the velocity of the curve A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 4 of 14 at time s. We call x(s) a naturally parametrized curve if xi(s) (i = 1, 2, 3) is of class Ck and ∥x′(s)∥ = 1, for each s ∈ I [34]. Let x(s) be biregular, that is, x′(s)×x′′(s) ̸= 0, for each s ∈ I. We consider a trihedron {T (s), N(s), B(s)} along x(s), so-called Frenet frame, where [34] T (s) = x′(s), N(s) = T ′(s) ∥T ′(s)∥ , B(s) = T (s)×N(s). The curvature κ, a non-negative scalar field, is defined by setting κ(s) = ∥T ′(s)∥ and torsion is defined by setting τ(s) = ⟨N ′(s), B(s)⟩. The naturally parametrized curve x has unit speed and strictly positive curvature then the following equations hold [34]T ′ N ′ B′  =  0 κ 0 −κ 0 τ 0 −τ 0 TN B  . (4) 2.2. Basics in conformable fractional calculus Given s 7→ x(s) ∈ E3, s ∈ I ⊂ R, the conformable derivative of x at s is defined by [19] Dα(x)(s) = lim ε→0 x(s+ εs1−α)− x(s) ε . Let Dx(s) = dx(s)/ds. We then notice Dαx(s) = s1−αdx(s)/ds. Denote by Dαx(s) the α-th order conformable derivative of x(s) for each s > 0, 0 < α < 1. It can be said that the conformable derivative provides some properties such as linear- ity, Leibniz rule and chain rule as in the classical derivative as follows (i) Dα(ax+ by)(s) = aDα(x)(s) + bDα(y)(s), for all a, b ∈ R, (ii) Dα(s p) = psp−α for all p ∈ R, (iii) Dα(λ) = 0, for all constant functions x(s) = λ, (iv) Dα(xy)(s) = x(s)Dαy(s) + y(s)Dαx(s), (v) Dα( x y )(s) = x(s)Dαy(s)−y(s)Dαx(s) y2(s) , (vi) Dα(y ◦ x)(s) = x(s)α−1Dαx(s)Dαy(x(s)) where x, y be conformable differentiable for each s > 0 and 0 < α < 1 [19]. The conformable integral is defined as the inverse operator to the conformable deriva- tive. Specifically, the conformable integral of a function x(s) is formally expressed as [19] Iaαf(t) = Ia1 (t α−1f) = ∫ t a f(x) x1−α dx. A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 5 of 14 The impact of conformable analysis on vector-valued functions is a subject of investi- gation, exploring both the limits and derivatives of these functions within this framework. The subsequent theorem delineates the formulation of the conformable derivative applied to vector-valued functions. Theorem 1. [35] Let x = (x1(s), x2(s), x3(s), ..., xn(s)) be a vector-valued function with n variables. So x is α−differentiable at s ∈ R, as follows Dαx(t) = (Dαx1(t), ..., Dαxm(t)). 3. Conformable parametrized curves and their conformable frame In this section, basic vector operations and parameterized curves will be reconstructed with conformable calculus. First of all, let’s define the concepts of conformable angle and conformable orthogonality, which are the most important concepts of geometry, with the help of conformable calculus as follows. The geometric interpretation of the conformable derivative is based on the notion of fractal geometry. In fractal geometry, objects exhibit self-similarity at different scales. The conformable derivative captures this self-similar behavior of a function by considering its local fractional variations. Geometrically, it can be understood as analyzing the ”zooming in” behavior of the function at that point, similar to the classical derivative capturing the local linear behavior. Overall, the geometric interpretation of the conformable derivative relates to the self-similarity and scaling properties of functions, enabling us to understand their behavior at different levels of detail and resolution. More specifically, the conformable derivative can be explained as a measure of how much a straight line and plane bends to form a curve and a surface. Figure 1 shows how a line is curved with the conformable calculus effect. Example 1. Let consider the s 7→ x(s) = (s, ∫ s1−αds), Cα−line passing through the point P = (0, 0) and whose direction is v = (s1−α, s1−α). In Figure 1 we present the graph of the conformable line for different α values. As seen in Figure 1, there is no classical line in the Cα− (fractional) system. This is only achieved when α → 1. Accordingly, it requires a new concept of angle in Cα− space. This angle is called the Cα− angle, which gives the angle between two Cα− lines. In addition, the concept of orthogonality in this Cα− space is different from the classical one. Because we cannot talk about classical directness, we cannot talk about steepness in the classical sense. We will explain this below. Notation: Along the study, expressions that are equal to 1 when α → 1 will be denoted as 1α, and expressions that are equal to 0 when α → 1 will be denoted as 0α. Suppose that x and y are Cα−unit vector that is, they are vectors of the form ∥x∥ = 1α and ∥y∥ = 1α. Then, the α−conformable radian measure of Cα−angle between x and y is defined by θα = arccos ( ⟨x,y⟩ ∥x∥∥y∥ ) . A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 6 of 14 Figure 1: Transformation from line to curve. In this sense, ⟨x,y⟩ = 0α when x and y are Cα−orthogonal. For example, the Cα− vectors u = (s1−α, 1 − α, 1 s1−α ) and v = (1−α sα , sα, 2 − 2α) are orthogonal to each other in the Cα− sense, and we present this in Figure 2. Figure 2: Cα−orthogonal vectors. In addition, vectors u, v and u×v form the fractional orthogonal system. For example, if u = (s1−α, 1−α, 1 s1−α ) and v = (1−α sα , sα, 2−2α), it becomes u×v = (2α2−4α−s2α−1+ 2, 2αs1−α − 2s1−α − α s + 1 s ,−α2s−α +2αs−α − s−α + s). The fractional orthogonal system is shown in Figure 3. A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 7 of 14 Figure 3: Cα−orthogonal system. Let x : I ⊂ R → E3 be a vector-valued function where s 7→ x(s) = (x1(s),x2(s),x3(s)). Then, Dαx(s) = (Dαx1(s), Dαx2(s), Dαx3(s)). We call x(s) Cα−naturally parametrized curve if xi(s) (i = 1, 2, 3) is of class Cα and ∥Dαx(s)∥ = s1−α, for each s ∈ I. Here α is the maximum order that we will need. In the remaining part, unless otherwise specified, we will assume that x(s) in E3 is a Cα−naturally parametrized curve. Let x(s) be Cα−biregular, that is, Dαx(s)×D2 αx(s) ̸= 0α, for each s ∈ I. We consider a trihedron {E1(s), E2(s), E3(s)} along x(s), so-called Cα−frame, where E1(s) = Dαx(s), E2(s) = DαE1(s) ∥DαE1(s)∥ , E3(s) = E1(s)× E2(s). (5) where {E1(s), E2(s), E3(s)} trihedron is called Cα−tangent, Cα−principal normal and the Cα−binormal of the Cα−curve x, respectively. Also, considering Eq. (5) Cα−tangent, Cα−principal normal and Cα−binormal of the Cα−curve x, they different from the Frenet vectors by the effect of the conformable calculus. However, these vectors turn into Frenet vectors, respectively, in case α → 1. In addition, the set {E1(s), E2(s), E3(s)} is mutually Cα−orthogonal and Cα−unit speed vectors. We call κα(s) = ∥DαE1(s)∥ Cα−curvature and τα(s) = ⟨DαE2(s), E3(s)⟩ Cα−torsion. The Cα−frame formulae are now [36]DαE1 DαE2 DαE3  =  0 κα 0 −κα 0 τα 0 −τα 0 E1 E2 E3  . (6) Conclusion 1. (What is the advantage of Cα−frame?) The concept of α-differentiability offers a distinctive perspective where functions exhibit α-differentiability at points where classical differentiability fails. For instance, consider A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 8 of 14 the function f(t) = 2 √ t. At t = 0, the classical derivative f ′(0) doesn’t exist. However, employing the conformable fractional derivative yields the result D 1 2 f(0) = 1 quite straight- forwardly. In this example, function f lacks a classical tangent at the point t = 0, but an approximation to this tangent is achievable through the conformable fractional derivative. The Frenet frame of a curve heavily relies on the existence of the curve’s tangent at a given point. Yet, the Cα-frame resolves this issue. In points within the Cα-frame where the curve’s tangent is absent, fractional values are assigned to approximate the tangent at that specific point. Moreover, when α → 1, the Cα-frame aligns with the Frenet frame. In such instances, the Cα-frame encompasses the classical Frenet frame while extending advantages to researchers dealing with points where the Frenet frame lacks definition. Theorem 2. [36] Let x = x(s) be Cα−naturally parametrized curve in the Euclidean 3−space where s measures its Cα−arc length. When α → 1, as follows κα = s1−α √ (1− α)2s−2α + s2−2ακ2. (7) and τα = s5−5ακ2 κ2α τ. (8) Proposition 1. Let the α−conforamable frame of a naturally parameterized α−conformable x curve be {E1, E2, E3} and the α−conformable curvature and torsion be κα and τα, re- spectively. If the α−conformable curve x lies on the α−conformable sphere S2α(Cα, rα), the curve x is called α−conformable spherical curve. Then ⟨x, E1⟩ = 0α and as follows x(s) = Cα + 0αλ1 − 1α κα E2 − 1α τα ( 1α κα )′ E3. Example 2. Let α−conformable spherical curve be y in S2α for rα = uα−1, Cα = (0, 0, 0) given by the parametrization y(s) = ( 1 2 uα−1 cos 2s, 1 2 uα−1 sin 2s, √ 3 2 uα−1 ) . (9) In Figure 4, we present a figures of the α−conformable spherical curve according to dif- ferent values of α. A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 9 of 14 Figure 4: α−Conformable spherical curve y(s) for for α → 1, α = 0.9, α = 0.7, α = 0.5, α = 0.3 and α = 0.1, respectively. 4. Cα−rectifying curves Let x(s) be a Cα−naturally parametrized curve in E3, s ∈ I ⊂ R and {E1(s), E2(s), E3(s)} denotes the Cα−frame. Suppose that Cα−biregular. We call x(s) Cα−rectifying curve if there is a linear relation as x(s) = λ(s)E1(s) + µ(s)E3(s). (10) Here the functions λ(s) and µ(s) are of class Cn on I called Cα−tangential and Cα−binormal components of x(s), respectively. Hence, λ(s) = ⟨x(s), E1(s)⟩ and µ(s) = ⟨x(s), E3(s)⟩. Remark also that Cα−rectifying curve x(s) holds ⟨x(s), E2(s)⟩ = 0 for each s ∈ I. Taking a conformable differentiation in Eq. (10) together with considering Cα−frame formulae, we have (Dαλ− 1)E1 + (λκα − µτα)E2 +DαµE3 = 0. (11) According to this equation, the following results are given Dαλ = 1, (12) τα κα = λ µ , (13) Dαµ = 0. (14) A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 10 of 14 Conclusion 2. From Eqs. (12) and (14) one concludes that λ(s) = sα α + c1 and µ(s) = c2 for c1, c2 ∈ R. Notice here that c2 ̸= 0 because otherwise one derives from in Eq. (13) that κα(s) is 0. This contradicts with biregularity of x(s). Analogously, τα(s) is nowhere 0. In conclusion, we deduce that a Cα−rectifying curve has to be Cα−twisted. Theorem 3. Let x(s) ⊂ E3 be a Cα−rectifying curve and κα ̸= 0 and τα is nowhere 0α. Then ρ2(s) = 1α(s 2α + csα + d) (15) where ρ(s) = ∥x(s)∥ is the distance function, c and d real numbers. Proof. Consider the a Cα−rectifying curve x(s) in Eq. (10), so the following equation exists ρ2(s) = ⟨x(s),x(s)⟩ = ∥E1∥λ2(s) + ∥E3∥µ2(s) where E1 and E3 are Cα−unit speed vectors. So this equation is edited ρ2(s) = 1α( s2α α2 + 2c1 sα α + c22) (16) or ρ2(s) = 1α α2 (s2α + 2c1αs α + c22α 2). (17) Here, when α → 1, since 1α α2 = 1 is 1α α2 = 1α can be written. Also, since α is a real number, if 2c1α = c and c22α 2 = d are selected, we get the following ρ2(s) = 1α(s 2α + csα + d). (18) Theorem 4. Let x(s) ⊂ E3 be a Cα−rectifying curve. Then the ratio of the conformable curvatures is for a, b ∈ R τα κα = asα + b. Proof. From Eq. (13), we know the following equation τα κα = λ µ . Now, if we use the results of the λ and µ conformable differentiable equations available in Conclusion 2 in the above equation, we get τα κα = sα α + c1 c2 . or τα κα = sα αc2 + c1 c2 Since c1, c2 and α are real numbers, if we select new real numbers as 1 αc2 = a and c1 c2 = b, it can be easily seen that the following equation is achieved τα κα = asα + b. A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 11 of 14 Theorem 5. Let x : I ⊂ R → E3 be a Cα−rectifying curve E3 with κα ̸= 0α. In this case, the following result applies x(s) = 1α √ f(t)2α + n2αy(s), (19) where f(t) is a conformable differentiable function and n is a positive number and y = y(s) is a Cα−unit speed spherical curve in S2 α. Proof. Let x(s) ⊂ E3 be a Cα−rectifying curve with κα ̸= 0α. Suppose that 0α lies in I and x = x(s) is Cα−unit speed curve. By the Eq. (18), the distance function ρ = ∥x∥ of the curve satisfies ρ2(s) = 1α(s 2α+csα+d) for some constant c and d. After a conformable translation in s, we may take ρ2(s) = 1α(s 2α + m), for some constant m. Since 0α ∈ I, m > 0α. Therefore, we may put m = n2α for n ∈ R+. Introduce a spherical curve y(s) as x(s) = 1α √ s2α + n2αy(s). (20) By taking the conformable derivative of this equation according to s, we get Dαx = ( 0α √ s2α + n2α + 1ααs α √ s2α + n2α ) y + 1α √ s2α + n2αDαy (21) Note that the conformable derivative of Dα1α = 0α. Also, since y is Cα−unit and y and Dαy are Cα−orthogonal, from Eq. (21) the following equation is obtained ∥Dαy∥ = √ 1αs2α(1− α2) + n2α (s2α + n2α)2 where it is clear that ∥Dαy∥ is the Cα−velocity of the Cα−spherical curve y. Then t = Is0 √ 1αs2α(1− α2) + n2α (s2α + n2α)2 = f−1(s). So, s = f(t) is obtained. Here f(t) is a function that gives the result n.tant when α → 1. If this result is written in Eq (20), we get x(s) = 1α √ f(t)2α + n2αy(s). (22) 5. Conclusion While ordinary analysis and the differential geometry are related to ordinary deriva- tives, fractional calculus provides us with the more fractional analysis which depends on differentiation and integration with respect to arbitrary order. In the last few decades, fractional analysis has been used extensively in almost all basic sciences, especially in Physics, Chemistry and Engineering. It is claimed that fractional analysis gives more nu- merical results than classical analysis. This makes fractional analysis more advantageous. A. Has, B.Yılmaz, T. Abdeljawad / Eur. J. Pure Appl. Math, 18 (2) (2025), 5864 12 of 14 Since conformable derivatives have several well-behaved properties like ordinary deriva- tives, in this work the basic concepts of geometry have been re-examined and studied in the frame of conformable fractional analysis. Our new differential geometry concepts we have studied give the readers a more general approach to deal with. The limiting case α → 1 sends us back to the classical geometry. • The Cα− frame has been constructed differently from previous similar studies and the classical Frenet frame. The advantage of this frame is that when α → 1 it gives the classical Frenet frame, it also gives the opportunity to examine the frame of the curve for all cases in the range of 0 < α < 1. In other words, it exhibits a more general situation compared to the classical Frenet frame. • Curves defined according to the Cα−frame take on a different variation of the curve for each α value. • An example of this can be seen very well in Eq. (22). For each α value in this equation, the function f(t) will give a different result. Thus, the x curve will turn into a different a Cα−rectifying curve in a conformable sense according to each α value, and when α → 1 it will turn into a classical rectifying curve. • The variation of the Cα−frame at any point of the Cα−curve and of each defined curve depending on this frame within the range of 0 < α < 1 can be examined. In addition, a curve can be generated for the resulting Cα−frame for each α value. Acknowledgements The author T.Abdeljawad would like to thank Prince Sultan University for paying the APC and for the support through TAS research lab. References [1] M. Barros, J. L. Cabrerizo, M. Fernández, and A. Romero. Magnetic vortex filament flows. Journal of Mathematical Physics, 48(8):082904, 2007. [2] Z. B. Ozdemir, I. Gok, Y. Yayli, and N. Ekmekci. Notes on magnetic curves in 3d semi-riemannian manifolds. Turkish Journal of Mathematics, 39(3):412–426, 2015. [3] M. Shareduwan, M. Kasihmuddin, M. A. Mansor, and S. Sathasivam. 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