EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5875 ISSN 1307-5543 – ejpam.com Published by New York Business Global Some Characterizations of Quasi-Curves in Galilean 3-Space Ayman Elsharkawy1,∗, Noha Elsharkawy1 1 Department of Mathematics, Faculty of Science, University of Tanta, Tanta, Egypt Abstract. This study investigates the theoretical basis of the quasi-frame in three-dimensional Galilean geometry. We derive mathematical expressions for the position vectors of curves defined in relation to this quasi-frame and establish the quasi equations that govern their behavior. Our findings demonstrate the absence of normal curves in Galilean 3-space, challenging existing theories in the field and providing new insights into the geometric structure of the Galilean 3-space. We explore the geometric properties of quasi-rectifying and quasi-osculating curves, establishing the necessary and sufficient conditions for their classification. A curve is identified as quasi-rectifying if its position vector can be represented as a linear combination of its tangent and quasi-binormal vectors. In contrast, a curve is classified as quasi-osculating if it remains entirely within its quasi-osculating plane, determined by its tangent and quasi-normal vectors. The quasi-frame serves as a generalization of the classical Frenet frame, particularly useful in scenarios where the curvature vanishes and the Frenet frame becomes undefined. By introducing the quasi curvatures, we provide a robust framework for analyzing curves in Galilean 3-space. We derive explicit expressions for the position vectors of curves with respect to the Quasi frame and solve for their components under specific conditions. Furthermore, we prove that normal curves cannot exist in Galilean space, a result that clarifies the limitations of certain geometric classifications in this context. 2020 Mathematics Subject Classifications: 51A05, 53A35 Key Words and Phrases: Quasi-frame, Galilean 3-space, quasi-normal curves, quasi-rectifying curves, quasi-osculating curves. 1. Introduction Galilean geometry, as articulated by Cayley and Klein, encompasses transformations that are fundamental to both classical and modern physics. The group of Galilean trans- formations is pivotal within these theoretical frameworks [1]. Notably, the conventional Frenet frame becomes inapplicable at points where curvature approaches zero, specifically at locations where the normal and binormal vectors are undefined [2–4]. In response to ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5875 Email addresses: ayman_ramadan@science.tanta.edu.eg (A. Elsharkawy) noha_elsharkawy@science.tanta.edu.eg (N. Elsharkawy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 2 of 15 this limitation, various researchers have devised alternative frames that effectively han- dle such scenarios in Euclidean space [5, 6], Minkowski space [7–9], and Galilean space [10–14]. These include the equiform frame [15, 16], the Bishop frame [17], the Darboux frame [17, 18], the modified frame [2, 3, 12, 13], and the quasi frame [5, 6, 10, 11]. In the context of plane curves, three primary classifications can be identified: osculat- ing, normal, and rectifying curves. An osculating curve is characterized by the tangent and normal vectors residing within the same plane defined by its position vector at all instances. In contrast, a normal curve is defined by a position vector that consistently maintains a normal orientation. The plane in question is formed by the curve’s normal and binormal vectors. Previous studies have explored the characteristics of normal, oscu- lating, and rectifying curves across various geometric frameworks [19–24]. A recent study by Dede et al. [25] introduced an innovative approach by constructing an adapted frame that precisely follows a space curve, moving beyond reliance on the traditional Serret Frenet frame. This newly developed framework termed the quasi-frame (Q-frame), en- hances precision and applicability, thereby serving as an expanded interpretation of the Frenet frame. The Q-frame is distinguished by a fixed vector and the angle between the quasi-normal vector and the principal normal of the Frenet frame. At points where the curvature is zero, this frame undergoes rotation by the specified angle, establishing the Q-normal as orthogonal to both the tangent vector and the fixed vector. The Q-binormal vector is defined as the unit vector orthogonal to both the tangent and Q-normal vectors. Numerous studies have examined the Q-frame within Euclidean and Minkowski spaces [26–30], while more recent investigations have focused on position vectors in Galilean three- and four-dimensional spaces using the Frenet frame [31–34]. The structure of this paper is organized as follows: Section 2 details the Q-frame and its relationship to the Frenet frame. Section 3 delves into the analysis of quasi- formulas within Galilean 3-space. Section 4 investigates position vectors in Galilean 3-space and determines coefficients under specific conditions, covering quasi-rectifying and quasi-osculating curves. Furthermore, we establish the non-existence of Q-normal curves in Galilean 3-space, outlining the necessary and sufficient criteria for classifying a curve as either quasi-rectifying or quasi-osculating. 2. Preliminaries In this section, we present essential concepts and definitions that will be crucial for our subsequent analysis. The three-dimensional Galilean space, denoted as G3, is a real vector space structured according to the Cayley-Klein model, characterized by a projective metric with signature (0, 0,+,+). The absolute structure of this three- dimensional Galilean space can be represented by an ordered triple {Ω, L, J}, where Ω signifies the absolute plane within G3, L denotes the absolute line contained in Ω, and J represents a fixed elliptic involution of the points along L. A vector p = (p1, p2, p3) in G3 is classified as non-isotropic if p1 ̸= 0; otherwise, it is termed isotropic [31, 32]. Vectors of the form p = (1, p2, p3) are considered unit non-isotropic vectors. The Galilean metric g for vectors p and q in G3 is defined as A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 3 of 15 follows: g(p,q) = { p1q1, if p1 ̸= 0 or q1 ̸= 0, p2q2 + p3q3, if p1 = 0 and q1 = 0. Consequently, the Galilean norm of the vector q is given by: ∥q∥ = { |q1|, if q1 ̸= 0,√ q22 + q23, if q1 = 0. The Galilean vector product of vectors p and q is defined as: p× q =  ∣∣∣∣∣∣∣ 0 e2 e3 p1 p2 p3 q1 q2 q3 ∣∣∣∣∣∣∣ , if p1 ̸= 0 or q1 ̸= 0, ∣∣∣∣∣∣∣ e1 e2 e3 p1 p2 p3 q1 q2 q3 ∣∣∣∣∣∣∣ , if p1 = 0 and q1 = 0, where (e1, e2, e3) denotes the standard basis of R3 [11, 35]. In G3, a curve is defined as a mapping from an open interval J in R to G3, represented as: γ : J → G3, t 7→ γ(t) = (x(t), y(t), z(t)). A curve is considered admissible if it has no inflection points (i.e., γ̇(t) × γ̈(t) ̸= 0) and does not possess isotropic tangents (i.e., ẋ(t) ̸= 0 for all t ∈ J) [11, 31]. For an admissible differentiable curve γ : J ⊂ R → G3, parameterized by the Galilean invariant arc length s, the curve can be expressed as: γ(s) = (s, y(s), z(s)). The curvature κ(s) and torsion τ(s) of the curve γ(s) are given by the formulas: κ(s) = ∥γ′′(s)∥ = √ y′′2(s) + z′′2(s), τ(s) = det(γ′(s), γ′′(s), γ′′′(s)) κ2(s) . The moving Frenet frame {T (s), N(s), B(s)} for the curve γ(s) is defined as: T (s) = γ′(s) = (1, y′(s), z′(s)), N(s) = 1 κ(s) γ′′(s) = 1 κ(s) (0, y′′(s), z′′(s)), A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 4 of 15 B(s) = T (s)×N(s) = 1 κ(s) (0,−z′′(s), y′′(s)). Finally, the Frenet derivative formulas can be represented in matrix form as follows: T ′ N ′ B′  = 0 κ(s) 0 0 0 τ(s) 0 −τ(s) 0 T N B  . (2.1) 3. Quasi-Frame and Quasi Equations This section delves into the concept of the quasi-frame (Q-frame) and its relationship with the classical Frenet frame, as well as an examination of the quasi equations within the context of Galilean three-dimensional space. Let α(s) represent a curve in G3. The Q-frame is established using three orthonormal vectors: T (s), the unit tangent vector; Nq(s), the unit Q-normal vector; and Bq(s), the unit Q-binormal vector. The Q-frame {T (s), Nq(s), Bq(s)} is defined as follows: T = α′ ∥α′∥ , Nq = T × z ∥T × z∥ , Bq = T ×Nq, (3.1) where z is a projection vector that can be chosen from (1, 0, 0), (0, 1, 0), or (0, 0, 1). Denote the standard Frenet frame by {T,N,B}, and let θ(s) denote the angle between the vectors N and Nq. We can express Nq and Bq in terms of N and B as follows: Nq = cos(θ)N + sin(θ)B, (3.2) Bq = − sin(θ)N + cos(θ)B. (3.3) From these relationships, it is also possible to rewrite N and B in terms of the Q- frame: N = cos(θ)Nq − sin(θ)Bq, (3.4) B = sin(θ)Nq + cos(θ)Bq. (3.5) Utilizing the Frenet formulas, we have: T ′ = κN = κ(cos(θ)Nq − sin(θ)Bq). By defining K1 = κ cos(θ) and K2 = κ sin(θ), we can express the derivative of T as: T ′ = K1Nq −K2Bq. (3.6) Using the established relationships, the derivatives N ′ q and B′ q can be expressed as: A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 5 of 15 N ′ q = K3Bq, B′ q = −K3Nq, (3.7) where K3 = θ′ + τ . Consequently, the quasi equations can be presented in a matrix form: T ′ N ′ q B′ q  = 0 K1 −K2 0 0 K3 0 −K3 0  T Nq Bq  . (3.8) The quasi curvatures K1, K2, and K3 are associated with the Q-frame as follows: Corollary 1. If α(s) is a curve in G3, then the quasi curvatures K1,K2, and K3 in terms of the Q-frame are given by K1 = g(T ′, Nq), K2 = −g(T ′, Bq), K3 = g(N ′ q, Bq) = −g(B′ q, Nq). Corollary 2. If K2 = 0, the Q-frame coincides with the Frenet frame, demonstrating that the Q-frame generalizes the Frenet frame in G3. 4. Generalized position vector This section examines the spatial coordinates of position vectors within the Galilean three-dimensional manifold. We derive the components associated with the tangential, Q-normal, and Q-binormal vectors under specific conditions. Additionally, we investigate the properties of Q-rectifying and Q-osculating curves in G3 with respect to the Q-frame of reference. Furthermore, we demonstrate the absence of normal curves within the Galilean spatial framework. Let α = α(s) be a unit speed curve in G3. We can write the position vector concerning the Q-frame in G3 as α = α(s) = m1(s)T +m2(s)Nq +m3(s)Bq, (4.1) for some differentiable functions m1(s),m2(s) and m3(s). By differentiating Equation (4.1) and using Equation (3.8), we have m′ 1 = 1, (4.2) m1K1 +m′ 2 −m3K3 = 0, (4.3) −m1K2 +m2K3 +m′ 3 = 0. (4.4) From Equation (4.2), we get: m1 = C + s, (4.5) where C is constant. A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 6 of 15 By substitution from Equation (4.5) into Equation (4.3), we obtain: m3 = (C + s)K1 +m′ 2 K3 . (4.6) By differentiate Equation (4.6), we have m′ 3 = K3[sK ′ 1 +K1 + CK ′ 1 +m′′ 2]−K ′ 3[sK1 + CK1 +m′ 2] (K3)2 . (4.7) No general solution has been found for this system. Because of this, we give the solution in some special cases. Case 1: By substituting from Equation (4.7) into Equation (4.4), and substituting K1 = K2 = 0,K3 = constant = a ̸= 0, we obtain a non-homogeneous second linear differential equation K3m ′′ 2 +K3 3m2 = 0. So, m2 = C1 cos as+ C2 sin as, (4.8) By taking the derivative of (4.8) and substituting K1 = 0,K3 = a, into Equation (4.6), we obtain: m3 = −C1 sin as+ C2 cos as. (4.9) Therefore, we can write the position vector as α(s) = (C + s)T + (C1 cos as+ C2 sin as)Nq + (−C1 cos as+ C2 sin as)Bq, where C,C1, C2, A and a are constants. Case 2: Let m2 = C3 ̸= 0, from Equation (4.3), we obtain: m3 = (C + s) K1 K3 . Therefore, in this case, we can write the position vector as α(s) = (C + s)T + C3Nq + (C + s) K1 K3 Bq. From Equation (4.4), we have a linear first-order differential equation ( K1 K3 )′ + 1 C + s ( K1 K3 ) = K2 − C3 C + s K3. (4.10) The integrating factor is given by µ = e ∫ 1 c+s ds = (c + s), therefore the solution of Equation(4.10) is given by K1 K3 = 1 C + s ∫ [(C + s)K2 − C3K3]ds+ C4. (4.11) A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 7 of 15 Case 3: Let m3 = C5 ̸= 0 from Equation (4.4), we obtain: m2 = (C + s) K2 K3 . (4.12) Therefore, in this case, we can write the position vector as α(s) = (C + s)T + (C + s) K2 K3 Nq + C5Bq. Substituting Equation (4.12) into Equation (4.3) yields a linear first-order differential equation: ( K2 K3 )′ + 1 C + s ( K2 K3 ) = C5 C + s K3 +K1. So, K2 K3 = 1 C + s ∫ (C5K3 + (C + s)K1)ds+ C6. (4.13) Corollary 3. In the case of the Frenet curve, we can put K2 = 0,K3 = τ , and K1 = κ. Then, Equations (4.2), (4.3), and (4.4) become m′ 1 = 1, (4.14) m1κ+m′ 2 −m3τ = 0, (4.15) m2τ +m′ 3 = 0. (4.16) Therefore, m1 = C + s. In case 2, if m2 = C3, Thus m3 = (C + s)κτ . Also, Equation (4.11), becomes −C3 C + s ∫ [τds+ C4] = κ τ . In case 3, From Equation (4.16), we obtain τ = 0. From Equation (4.15), we obtain m2 = − ∫ (c+ s)κds. Also, Equation (4.13) becomes C6 = − ∫ ((C + s)κ)ds. These results are consistent with those in [34]. A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 8 of 15 5. Q-Normal Curves In this section, we demonstrate the absence of normal curves within the context of Galilean 3-space, as defined by both the Frenet frame and the Q-frame. A curve α(s) is classified as a Q-normal curve in G3 if it resides entirely within its Q-normal plane. Formally, this means that the curve α satisfies the equation α(s) = λ(s)Nq(s) + π(s)Bq(s), where Nq(s) and Bq(s) represent the Q-normal and Q-binormal vectors, respectively. Theorem 1. For any admissible differentiable curve parameterized by s, there do not exist quasi-normal curves in G3. Proof. Let β(s) denote an admissible differentiable curve parameterized by the Galilean invariant arc length s in G3. We can express β(s) in the following form: β(s) = (s, y(s), z(s)) . Differentiating with respect to s, we obtain the tangent vector: T = (1, y′, z′). From the Galilean metric, we find that g(β, T ) = s ̸= 0, which implies that β cannot be classified as a Q-normal curve. Corollary 4. If β(s) is an admissible differentiable curve parameterized by the Galilean invariant arc length s, then β(s) does not constitute a normal curve in G3 or Gn . Consequently, the results presented in [23] are invalid. 6. Quasi-rectifying curves In this subsection, we establish the fundamental criteria for characterizing a Q-curve with position vector Γ(s) as a Q-rectifying curve within G3. The definition of a Q- rectifying curve is predicated on its geometric relationship to the Q-rectifying plane. Specifically, a curve Γ is classified as Q-rectifying if and only if its position vector can be expressed as a linear combination of its tangent vector T (s) and Q-binormal vector Bq(s), such that Γ(s) = γ(s)T (s) + ϵ(s)Bq(s), where γ(s) and ϵ(s) are scalar functions. Theorem 2. Let Γ(s) be a Q-rectifying curve in G3. Then the tangential and the binor- mal components of the position vector Γ(s) are given, respectively, by g(Γ, T ) = C + s, g(Γ, Bq) = (C + s) K1 K3 . A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 9 of 15 Proof. Suppose that Γ(s) is a Q-rectifying curve, then Γ(s) = γ(s)T (s) + ϵ(s)Bq(s), (6.1) for some differentiable functions γ(s) and ϵ(s). We can deduce that γ(s) = C + s, ϵ(s) = (C + s) K1 K3 . Therefore, g(Γ, T ) = C + s, g(Γ, Bq) = (C + s) K1 K3 . Thus, Γ(s) = (C + s)T + (C + s) K1 K3 Bq. Corollary 5. Let Γ(s) be a Frenet rectifying curve in G3. Then the Frenet tangential and the binormal components of the position vector are given, respectively, by g(Γ, T ) = C + s, g(Γ, B) = (C + s) κ τ . Theorem 3. Let Γ(s) be a quasi curve with position vector in G3. The curve is Q- rectifying if and only if K1 K3 = 1 C + s ∫ (C + s)K2ds+ C7, where C7 is constant. Proof. Suppose that Γ(s) is a Q-rectifying curve. So, we can put m2 = 0 in Equation (4.3) and Equation (4.4), we obtain: (C + s)K1 − ϵK3 = 0, −(C + s)K2 + ϵ′ = 0. By solving these equations we get a linear first-order differential equation ( K1 K3 )′ + 1 C + s K1 K3 = K2. So, K1 K3 = 1 C + s ∫ (C + s)K2ds+ C7. (6.2) Conversely, Let the condition (6.2) be satisfied. Let a vector r be as follows r = Γ(s)− (C + s)Tq − (C + s) K1 K3 Bq. (6.3) A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 10 of 15 By differentiating Equation (6.3) with respect to s and using Equation (6.2), we deduce r′ = 0. Thus Γ(s)− r = (C + s)Tq + (C + s) K1 K3 Bq. Up to a transformation with r, we find Γ is a Q-rectifying curve in G3. Corollary 6. let Γ(s) be curve with position vector in G3. The curve is a Frenet rectifying curve if and only if κ τ = C∗ 7 C + s , where C∗ 7 is constant. 7. Quasi-Osculating Curves In this section, we establish the necessary and sufficient conditions for a curve with position vector η(s) to be classified as a Q-osculating curve within the framework of G3. A curve η(s) is deemed a Q-osculating curve if it remains contained within its Q-osculating plane. In mathematical terms, the curve η satisfies the following representation: η(s) = ϱ(s)T (s) + ε(s)Nq(s), where T (s) and Nq(s) denote the tangent and Q-normal vectors, respectively. Theorem 4. Let η(s) be a Q-osculating curve in G3. Then the tangential and Q-normal components of the position vector η can be expressed as follows: g(η, T ) = C + s, g(η,Nq) = − ∫ (C + s)K1 ds+ C8, where C and C8 are constants. Proof. Assuming that η(s) is a Q-osculating curve, we can write: η(s) = ϱ(s)T (s) + ε(s)Nq(s). From this representation, we derive: ϱ(s) = C + s, ε(s) = − ∫ (C + s)K1 ds+ C8. Thus, we find that: g(η, T ) = C + s, g(η,Nq) = − ∫ (C + s)K1 ds+ C8. Consequently, we can express η(s) as: η(s) = (C + s)T + [ − ∫ (C + s)K1 ds+ C8 ] Nq, where C and C8 are constants. A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 11 of 15 Corollary 7. If the curve η(s) is a Frenet osculating curve, then its tangential and normal components are given by: g(η, T ) = C + s, g(η,N) = − ∫ (C + s)κ ds+ C∗, where C∗ is a constant. Theorem 5. Consider a quasi curve with position vector η(s) in G3. The curve is classified as Q-osculating if and only if the following condition holds: K2 K3 = 1 C + s ∫ [−(C + s)K1] ds+ C9, where C9 is a constant. Proof. Suppose that η(s) is a Q-osculating curve. Then, setting m3 = 0 in Equations (4.3) and (4.4), we obtain: (C + s)K1 + ϵ′ = 0, (7.1) −(C + s)K2 + ϵK3 = 0. (7.2) From equation (7.1), we deduce m2 = − ∫ (C + s)K1ds + C9, and by solving these equations we get a linear first-order differential equation ( K2 K3 )′ + 1 C + s ( K2 K3 ) = −K1. So, K2 K3 = 1 C + s ∫ −(C + s)K1ds+ C10. (7.3) Conversely, Let the condition (7.3) be satisfied. Let a vector p be as follows p = η(s)− (C + s)Tq − (C + s) K2 K3 Nq. (7.4) By differentiating Equation(7.4), we deduce p′ = 0, therefore p is constant. Thus η(s)− p = (C + s)Tq + (C + s) K2 K3 Nq. Up to a transformation with p, we find η is a Q-osculating curve in G3. Corollary 8. let η(s) be curve with position vector in G3. The curve is a Frenet oscu- lating curve if and only if ∫ (C + s)κds = C∗∗, where C and C∗∗ are constants. A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 12 of 15 8. Conclusion In this paper, we have investigated the theoretical foundations of the quasi-frame (Q-frame) in three-dimensional Galilean space (G3) and explored its applications to the study of curves in this geometric framework. Our work has provided a comprehensive analysis of the quasi-frame, its relationship with the classical Frenet frame, and the geometric properties of curves defined in relation to this frame. The quasi-frame serves as a generalization of the Frenet frame, particularly useful in scenarios where the curvature vanishes and the Frenet frame becomes undefined. By establishing the quasi equations in matrix form, we derived the quasi curvatures K1, K2, and K3, which govern the behavior of curves in G3. This framework not only extends the applicability of the Frenet frame but also provides a more robust tool for analyzing curves in Galilean space. We derived explicit expressions for the position vectors of curves in G3 with respect to the Q-frame. By examining specific cases, we obtained solutions for the components of the position vector under various conditions. This analysis allowed us to classify curves into quasi-rectifying and quasi-osculating categories based on their geometric properties. A curve is quasi-rectifying if its position vector lies in the plane spanned by its tan- gent and quasi-binormal vectors, while a curve is quasi-osculating if it remains entirely within its quasi-osculating plane, defined by the tangent and quasi-normal vectors. These classifications provide a deeper understanding of the geometric behavior of curves in G3. One of the most significant findings of this study is the demonstration that normal curves do not exist in G3. This result challenges existing theories and provides new insights into the geometric structure of Galilean space. By proving the absence of normal curves, we have clarified the limitations of certain geometric classifications in this context. Furthermore, our results generalize several well-known properties of curves in Euclidean and Minkowski spaces to the Galilean setting. For instance, the conditions for quasi- rectifying and quasi-osculating curves reduce to their Frenet counterparts when the quasi curvatures are appropriately specialized. This demonstrates the versatility of the Q-frame and its ability to unify various geometric frameworks. The findings of this study have several important implications for both theoretical and applied mathematics. The quasi-frame provides a powerful tool for analyzing curves in Galilean space, particularly in cases where the Frenet frame fails. This has potential applications in physics, engineering, and computer graphics, where Galilean geometry is often used to model motion and spatial relationships. Future research could explore extending the quasi-frame to higher-dimensional Galilean spaces (Gn), providing new insights into the geometry of curves and surfaces in these settings. Additionally, the quasi-frame could be applied to problems in classical and relativistic mechanics, where Galilean transformations play a central role. Investigating the properties of surfaces generated by quasi-curves, such as quasi-ruled surfaces, could also lead to new geometric constructions and classifications. A. Elsharkawy, N. Elsharkawy / Eur. J. Pure Appl. Math, 18 (2) (2025), 5875 13 of 15 References [1] I. M. Yaglom. A Simple Non-Euclidean Geometry and Its Physical Basis. Springer- Verlag, New York, NY, 1979. [2] H. K. Elsayied, A. A. Altaha, and A. Elsharkawy. On some special curves according to the modified orthogonal frame in Minkowski 3-space E3 1 . Kasmera, 49(1):2–15, 2021. [3] H. K. Elsayied, A. A. Altaha, and A. Elsharkawy. Bertrand curves with the modified orthogonal frame in Minkowski 3-space E3 1 . Revista de Educacion, 392(6):43–55, 2022. [4] H. K. Elsayied, M. Elzawy, and A. Elsharkawy. Equiform timelike normal curves in Minkowski space E3 1 . 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