On Some Quasi-Curves in Galilean Three-Space

Elsharkawy, Ayman; Tashkandy, Yusra; Emam, Walid; Cesarano, Clemente; Elsharkawy, Noha

doi:10.3390/axioms12090823

Axioms

2023

DOI: 10.3390/axioms12090823

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On Some Quasi-Curves in Galilean Three-Space

Ayman Elsharkawy,

Yusra Tashkandy,

Walid Emam

et al.

Abstract: In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated exam… Show more

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2024

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Cited by 3 publications

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Quasi-position vector curves in Galilean 4-space

Elsharkawy,

Elsharkawy

2024

Front. Phys.

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The Frenet frame is not suitable for describing the behavior of the curve in the Galilean space since it is not defined everywhere. In this study, an alternative frame, the so-called quasi-frame, is investigated in Galilean 4-space. Furthermore, the quasi-formulas in Galilean 4-space are deduced and quasi-curvatures are obtained in terms of the quasi-frame and its derivatives. Quasi-rectifying, quasi-normal, and quasi-osculating curves are studied in Galilean 4-space. We prove that there is no quasi-normal and accordingly normal curve in Galilean 4-space.

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Quasi-position vector curves in Galilean 4-space

Elsharkawy,

Elsharkawy

2024

Front. Phys.

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Spinor Equations of Smarandache Curves in E3

İsabeyoǧlu,

Erişir,

Azak

2024

Mathematics

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This study examines the spinor representations of TN (tangent and normal), NB (normal and binormal), TB (tangent and binormal) and TNB (tangent, normal and binormal)–Smarandache curves in three-dimensional Euclidean space E3. Spinors are complex column vectors and move on Pauli spin matrices. Isotropic vectors in the C3 complex vector space form a two-dimensional surface in the C2 complex space. Additionally, each isotropic vector in C3 space corresponds to two vectors in C2 space, called spinors. Based on this information, our goal is to establish a relationship between curve theory in differential geometry and spinor space by matching a spinor with an isotropic vector and a real vector generated from the vectors of the Frenet–Serret frame of a curve in three-dimensional Euclidean space. Accordingly, we initially assume two spinors corresponding to the Frenet–Serret frames of the main curve and its (TN, NB, TB and TNB)–Smarandache curves. Then, we utilize the relationships between the Frenet frames of these curves to examine the connections between the two spinors corresponding to these curves. Thus, we give the relationships between spinors corresponding to these Smarandache curves. For this reason, this study creates a bridge between mathematics and physics. This study can also serve as a reference for new studies in geometry and physics as a geometric interpretation of a physical expression.

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Involute-evolute curves with modified orthogonal frame in Galilean space G 3

Elsharkawy,

Turan,

Bozok

2024

Ukr. Mat. Zhurn.

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UDC 514 We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space G 3 . The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

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On Some Quasi-Curves in Galilean Three-Space

Cited by 3 publications

References 11 publications

Quasi-position vector curves in Galilean 4-space

Quasi-position vector curves in Galilean 4-space

Spinor Equations of Smarandache Curves in E3

Involute-evolute curves with modified orthogonal frame in Galilean space G 3

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