On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space

Ibrahim AL-Dayel; Emad Solouma; Meraj Khan; Ibrahim AL-Dayel; Emad Solouma; Meraj Khan

doi:10.3934/math.2022744

AIMS Mathematics

2022, Volume 7, Issue 7: 13454-13468. doi: 10.3934/math.2022744

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Research article

On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Saudi Arabia
2.
Department of Mathematics and Information Science, Faculty of Science, Beni-Suef University, Egypt
3.
Department of Mathematics, University of Tabuk, Saudi Arabia

Received: 19 December 2021 Revised: 16 April 2022 Accepted: 25 April 2022 Published: 20 May 2022
MSC : 53A05, 53A10

In Euclidean 3-space $ {\mathrm{E}}^3 $, a canonical subject is the focal surface of such a cliched space curve, which would be a two-dimensional corrosive with Lagrangian discontinuities. The tubular surfaces with respect to the B-Darboux frame and type-2 Bishop frame in $ {\mathrm{E}}^3 $ are given. These tubular surfaces' focal surfaces are then defined. For such types of surfaces, we acquire some results becoming Weingarten, flat, linear Weingarten conditions and we demonstrate that in $ {\mathrm{E}}^3 $, a tubular surface has no minimal focal surface. We also provide some examples of these types of surfaces.
- Euclidean geometry,
- focal surface,
- type-2 Bishop frame,
- tubular surface,
- B-Darboux frame
Citation: Ibrahim AL-Dayel, Emad Solouma, Meraj Khan. On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space[J]. AIMS Mathematics, 2022, 7(7): 13454-13468. doi: 10.3934/math.2022744

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Abstract

In Euclidean 3-space $ {\mathrm{E}}^3 $, a canonical subject is the focal surface of such a cliched space curve, which would be a two-dimensional corrosive with Lagrangian discontinuities. The tubular surfaces with respect to the B-Darboux frame and type-2 Bishop frame in $ {\mathrm{E}}^3 $ are given. These tubular surfaces' focal surfaces are then defined. For such types of surfaces, we acquire some results becoming Weingarten, flat, linear Weingarten conditions and we demonstrate that in $ {\mathrm{E}}^3 $, a tubular surface has no minimal focal surface. We also provide some examples of these types of surfaces.

References

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