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Peer-reviewedCC BY 4.0Axioms

Inextensible Curve Flows in Dual Space D3: A Quasi-Frame Approach

  • Ayman Elsharkawy1
  • Murat Turan2
  • Eman Ghareeb Rezk3
  • Hülya Gün Bozok2
  • Sıddıka Özkaldı Karakuş4
4 affiliations
  1. 1Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
  2. 2Department of Mathematics, Faculty of Engineering and Natural Sciences, Osmaniye Korkut Ata University, Osmaniye 80000, Türkiye
  3. 3Mathematical Science Department, College of Science, Princess Nourah bint Abdlrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
  4. 4Department of Mathematics, Faculty of Science, Bilecik Seyh Edebali University, Bilecik 11100, Türkiye
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Abstract

This paper develops a comprehensive quasi-frame theory for inextensible curve flows in the dual space D3. While the Frenet frame has been the traditional tool for studying curve evolution, it encounters singularities when curvature vanishes. We introduce a quasi-frame (q-frame) formulation that overcomes these limitations and provides a robust framework for analyzing inextensible flows, flows that preserve arc length throughout evolution. We derive the fundamental evolution equations for quasi-curvatures under inextensible flow conditions and establish necessary and sufficient conditions for inextensibility in dual space. In contrast to the Euclidean, Minkowski, and Galilean settings, the quasi-curvatures in dual space are dual-valued, so they simultaneously encode the direction of the oriented line and the evolution of its moment vector. Our approach extends inextensible flows using the Frenet frame by providing a more general framework that remains well-defined even at singular points. We demonstrate the advantages of our method through detailed worked examples, including complete computations of time-evolving curves with explicit verification of arc-length preservation. These results contribute to the theoretical foundations of curve dynamics in dual spaces.

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Cite

BibTeXRIS
BibTeX
@article{elsharkawy2026inextensible,
  title = {{Inextensible Curve Flows in Dual Space D3: A Quasi-Frame Approach}},
  author = {Ayman Elsharkawy and Murat Turan and Eman Ghareeb Rezk and Hülya Gün Bozok and Sıddıka Özkaldı Karakuş},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100767},
  url = {https://www.mdpi.com/2075-1680/15/10/767},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Inextensible Curve Flows in Dual Space D3: A Quasi-Frame Approach
AU  - Ayman Elsharkawy
AU  - Murat Turan
AU  - Eman Ghareeb Rezk
AU  - Hülya Gün Bozok
AU  - Sıddıka Özkaldı Karakuş
PY  - 2026
DA  - 2026/10/14
JO  - Axioms
DO  - 10.3390/axioms15100767
UR  - https://www.mdpi.com/2075-1680/15/10/767
AB  - This paper develops a comprehensive quasi-frame theory for inextensible curve flows in the dual space D3. While the Frenet frame has been the traditional tool for studying curve evolution, it encounters singularities when curvature vanishes. We introduce a quasi-frame (q-frame) formulation that overcomes these limitations and provides a robust framework for analyzing inextensible flows, flows that preserve arc length throughout evolution. We derive the fundamental evolution equations for quasi-curvatures under inextensible flow conditions and establish necessary and sufficient conditions for inextensibility in dual space. In contrast to the Euclidean, Minkowski, and Galilean settings, the quasi-curvatures in dual space are dual-valued, so they simultaneously encode the direction of the oriented line and the evolution of its moment vector. Our approach extends inextensible flows using the Frenet frame by providing a more general framework that remains well-defined even at singular points. We demonstrate the advantages of our method through detailed worked examples, including complete computations of time-evolving curves with explicit verification of arc-length preservation. These results contribute to the theoretical foundations of curve dynamics in dual spaces.
SN  - 2075-1680
ER  -