Surface Family Pair with Bertrand Pair as Common Geodesic Curves in Galilean 3-Space 𝔾3

Almoneef, Areej A.; Abdel-Baky, Rashad A.

doi:10.3390/math11102391

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…In addition, using the Cartan frame, a surfaces family was constructed with an asymptotic curve where the surface was presented as a linear combination of this frame [24,25]. Furthermore, the focus of the concept of a Bertrand pair appears in recent studies more in [26,27], where both studies were carried out in the Galilean space, but the first used a Bertrand pair as common asymptotic curves and the other used a Bertrand pair as common geodesic curves.…”

Section: Introductionmentioning

confidence: 99%

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

Mofarreh

2024

Symmetry

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A curve on a surface is a geodesic curve if its principal normal vector is anywhere aligned with the surface normal. Using the Serret–Frenet frame, a timelike surface couple (TLSC) with the symmetry of a Bertrand couple (BC) can be specified in terms of linear combinations of the components of the local frames in Minkowski 3-space E13. With these parametric representations, the necessary and sufficient conditions for the specified BC are derived to be the geodesic curves defining these surfaces. Afterward, the definition of a TL ruled surface (RS) is also provided. Furthermore, the application of the method to some significant models is given.

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Section: Introductionmentioning

confidence: 99%

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

Mofarreh

2024

Symmetry

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“…Jiang et al [20] considered a surface pencil pair with a Bertrand pair as joint asymptotic curves. Almoneef and Abdel-Baky [21] designed a surface family with Bertrand curves to be geodesic curves. AL-Jedani and Abdel-Baky [22] inspected a surface family and developable surface family with a joint geodesic curve.…”

Section: Introductionmentioning

confidence: 99%

Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space

Al-Jedani,

Abdel-Baky

2023

Mathematics

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The primary objective of this work is to discuss a surface family with the similarity of Bertrand curves in 3D Galilean space. Subsequently, by applying the Serret–Frenet frame, we estimate the sufficient and necessary statuses of a surface family with Bertrand curves as joint asymptotic curves. The dilation to ruled surfaces is also summarized. Meanwhile, the epitomes are illustrated to provide an explanation of the theoretical results.

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Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

Alluhaibi,

Abdel-Baky

2023

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A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space G3. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreover, the necessary and sufficient condition for the given Bertrand couple to satisfy the principal curves and the geodesic requirements are also analyzed. As implementations of our main consequences, we expound upon some models to confirm the method.

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Surface Family Pair with Bertrand Pair as Common Geodesic Curves in Galilean 3-Space 𝔾3

Cited by 3 publications

References 16 publications

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space

Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

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