A study of the inextensible flows of tube-like surfaces associated with focal curves in Galilean 3-space G&lt;sub&gt;3&lt;/sub&gt;

Sorour, Adel H.

doi:10.24193/subbmath.2017.3.07

Cited by 2 publications

(2 citation statements)

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“…A novel approach to these flows was expressed using Frenet and Darboux frames with the help of Fermi-Walker derivatives. Sorour [24] studied the inextensible flows of focal curves associated with tube-like surfaces in Galilean three-dimensional space G 3 and gave some characterizations for the curvatures of the focal curves associated with tube-like surfaces.…”

Section: Introductionmentioning

confidence: 99%

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

Gaber

Sorour

2023

Symmetry

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The study of the flows of curves is one of the most fascinating research areas in differential geometry. In this paper, we investigate the geometry of the flows of timelike curves according to the quasi-frame in Minkowski space R2,1 (In this paper, we refer to these curves as “quasi-timelike curves”). We investigate the evolution of quasi-timelike curves using the velocity functions and obtain the necessary and sufficient conditions for inextensibility. Additionally, we obtain the explicit forms of the time evolution equations for the quasi-orthonormal frames (tangent, quasi-normal, and quasi-binormal vectors) of the quasi-timelike curve as well as the time evolution equations of their quasi-curvatures. We present a new application for motion with velocities equal to the quasi-curvatures of the quasi-timelike curve. In this application, the time evolution equations of the quasi-curvatures arise as a system of partial differential equations with the form of the heat equation, and by solving this system, we visualize the evolution of quasi-curvatures and the evolution of the quasi-timelike curve. In addition, the acceleration functions are used to investigate the flows of inextensible quasi-timelike curves, and an application for accelerations equal to the quasi-curvatures is given. Through this application, the position vector of the quasi-timelike curve satisfies the one-dimensional wave equation, and the time evolution equations of the quasi-curvatures arise as a system of transport equations. We obtain the solutions and graph them using Wolfram Mathematica 12.

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Section: Introductionmentioning

confidence: 99%

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

Gaber

Sorour

2023

Symmetry

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“…S. Gaber [13], constructed new models of normal motions of inextensible curves in R 3 that move according to the type−1 Bishop Frame. Adel H. Sorour [14], studied the inextensible flows of focal curves associated with tube-like surfaces in Galilean 3−space G 3 and gave some characterizations for the curvatures of focal curves associated with tube-like surfaces. The present work is organized as follows: In section 2, some geometric concepts of timelike curves in Minkowski space are given according to Frenet frame and quasi frame.…”

Section: Introductionmentioning

confidence: 99%

The Geometry of Inextensible Flows of Timelike Curves according to Quasi Frame in Minkowski space $\mathbb{R}^{2,1}$

Gaber

Sorour

2022

Preprint

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In this paper, we study the geometry of the motion of timelike curves by quasi-frame according to the velocity and acceleration fields in Minkowski space $\mathbb{R}^{2,1}$. We study the timelike curves and get sufficient conditions to be inextensible. Also, we obtain the explicit form of the evolution equations for quasi orthonormal vectors (tangent, quasi-normal, and quasi binormal) of the quasi-timelike curve and the evolution equations of the quasi curvatures as a system of partial differential equations. We give new applications to the motion of quasi-timelike curves according to quasi-frame by velocity fields and acceleration fields.Mathematics subject classification 2020: 53A04;53A05; 53E10; 53Z05.

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A study of the inextensible flows of tube-like surfaces associated with focal curves in Galilean 3-space G<sub>3</sub>

Cited by 2 publications

References 5 publications

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

The Geometry of the Inextensible Flows of Timelike Curves according to the Quasi-Frame in Minkowski Space R2,1

The Geometry of Inextensible Flows of Timelike Curves according to Quasi Frame in Minkowski space $\mathbb{R}^{2,1}$

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