Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space $ \mathbb{E}^{3} $

Maryam T. Aldossary; Rashad A. Abdel-Baky; Maryam T. Aldossary; Rashad A. Abdel-Baky

doi:10.3934/math.2023021

AIMS Mathematics

2023, Volume 8, Issue 1: 447-462. doi: 10.3934/math.2023021

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Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space $ \mathbb{E}^{3} $

Maryam T. Aldossary ¹,
Rashad A. Abdel-Baky ^{2
,
,}

1.
Department of Mathematics, College of Science, Imam Abdulrahman bin Faisal University, P.O. Box-1982, 31441 Damam, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Assiut, 71516 Assiut, Egypt

Received: 06 September 2022 Revised: 14 September 2022 Accepted: 22 September 2022 Published: 08 October 2022
MSC : 53A04, 53A05, 53A17

In this paper, we address a new version of Darboux frame using a common tangent vector field to a surface along a curve and call this frame the rotation minimizing Darboux frame (RMDF). Then, we give the parametric equation due to the RMDF frame for a sweeping surface and show that the parametric curves on this surface are curvature lines. Consequently, necessary and sufficient conditions for sweeping surfaces to be developable ruled surfaces are derived. Also, we analyze the conditions when the resulting developable surface is a cylinder, cone or tangential surface. We also provide some examples to illustrate the main results.
- Darboux frame,
- singularity and convexity,
- developable surface
Citation: Maryam T. Aldossary, Rashad A. Abdel-Baky. Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space $ \mathbb{E}^{3} $[J]. AIMS Mathematics, 2023, 8(1): 447-462. doi: 10.3934/math.2023021

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Abstract

In this paper, we address a new version of Darboux frame using a common tangent vector field to a surface along a curve and call this frame the rotation minimizing Darboux frame (RMDF). Then, we give the parametric equation due to the RMDF frame for a sweeping surface and show that the parametric curves on this surface are curvature lines. Consequently, necessary and sufficient conditions for sweeping surfaces to be developable ruled surfaces are derived. Also, we analyze the conditions when the resulting developable surface is a cylinder, cone or tangential surface. We also provide some examples to illustrate the main results.

References

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