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

Al-Dayel, İbrahim; Solouma, E. M.; Khan, Meraj Ali

doi:10.3934/math.2022744

Cited by 4 publications

(3 citation statements)

References 16 publications

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“…Its double-helical structure has come to represent the remarkable advances achieved in molecular biology within the last century. Even though Watson and Crick's model-building study predicted the double-helix structure more than seven decades ago, little is known about DNA's detailed structure, dynamics, and energetics, despite its pivotal role in all living systems [21]. The three-dimensional structure of DNA was not obtained until 1981 from single-crystal diffraction [22].…”

Section: Double Helix Dna Surfacementioning

confidence: 99%

Generating and Characterization Loft Surface by Type-2-Bishop Frame in Isotropic Space

Mahmoud,

Soliman,

Moukhtar

2023

ijmst

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This paper aims to design and generate a loft surface using three distinct curves (Hermitee cubic curve, straight line, and quadratic Bezier curve) and analyze its characteristics in isotropic space using type-2-Bishop. Moreover, the loft surface's 1st and 2nd - fundamental forms in the loft surface were investigated. In addition, their Gaussian, mean, and second Gaussian curvatures in I3 were computed using the first and second type-2-Bishop curvatures. Furthermore, the required and adequate conditions for the surface with type 2-Bishop curvatures that are developable, minimal, and Weingarten surfaces (W-surface) were furnished. Besides, based on our results, two applications were provided to create and design a loft surface in biology: a double helix DNA surface and a cusp surface and all the above properties (I, II, H, K, K11, developable, minimal, and W-surface) were studied.

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Section: Double Helix Dna Surfacementioning

confidence: 99%

Generating and Characterization Loft Surface by Type-2-Bishop Frame in Isotropic Space

Mahmoud,

Soliman,

Moukhtar

2023

ijmst

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“…So if you do a vector multiplication using the ( 6) and ( 11) equations, * ∧ * = 2 For this expression to be zero, the coefficients must be zero. 3 It is developable from theorem (6) and theorem (7), but not minimal. The parameter curves are asymptotic and curvature lines from theorem (8) and theorem (10).…”

Section: On Geometry Of Focal Surfaces Due To Flc Frame In Euclidean ...mentioning

confidence: 99%

“…Based on this study, Dayel et al by defining the tube surfaces according to the B-Darboux frame and the type-2 Bishop frame, the focal surfaces of these tube surfaces are discussed. Moreover, by examining the Weingarten, planar and linear Weingarten states of the defined tube surfaces, he proved that the tube surface does not have a minimal focal surface, 7 . In this study, tube surfaces will be defined according to the Flc frame and their geometrical properties will be examined, and then the focal surfaces of these tube surfaces will be defined.…”

Section: Introductionmentioning

confidence: 99%

On geometry of focal surfaces due to Flc frame in Euclidean 3-space

Şenyurt

2022

Preprint

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In this study, firstly, tube surfaces given with Flc frame are introduced. The Gaussian and mean curvature of these surfaces were found and their singular points were determined. By investigating the conditions for parameter curves on the surface to be asymptotic, geodesic and curvature lines, it has been shown that θ parameter curves of the surface can never be asymptotic, but always geodesic. In particular, it is concluded that the parameter curves t and θ are curvature lines if they are planar curves. Afterwards, the focal surfaces of the tube surfaces given with the Flc frame are defined. By calculating the Gaussian and mean curvatures of these focal surfaces, it was found that they are always developable and never minimal. Afterwards, the conditions for parameter curves on the focal surface to be asymptotic, geodesic and curvature lines were investigated, and it was seen that the t parameter curves of the surface could never be asymptotic and the θ parameter curves were always asymptotic curves and curvature lines. Finally, examples of these surfaces are given using the Maple program.

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Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space

Solouma

Abdelkawy

2023

MATH

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<abstract><p>The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.</p></abstract>

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

Cited by 4 publications

References 16 publications

Generating and Characterization Loft Surface by Type-2-Bishop Frame in Isotropic Space

Generating and Characterization Loft Surface by Type-2-Bishop Frame in Isotropic Space

On geometry of focal surfaces due to Flc frame in Euclidean 3-space

Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space

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