Surfaces Family with Common Smarandache Geodesic Curve According to Bishop Frame in Euclidean Space

Atalay, Gülnur Şaffak; Kasap, Emi̇n

doi:10.36753/mathenot.421425

Cited by 16 publications

(13 citation statements)

References 5 publications

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“…In Galilean 3-space G 3 , there exist only three types of ruled surfaces, specified as follows [22,23]:…”

Section: Ruled Surface Pencil Couple With Bertrand Couple As Joint Pr...mentioning

confidence: 99%

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

Alluhaibi,

Abdel-Baky

2023

Axioms

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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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“…In Galilean 3-space G 3 , there exist only three types of ruled surfaces, specified as follows [22,23]:…”

Section: Ruled Surface Pencil Couple With Bertrand Couple As Joint Pr...mentioning

confidence: 99%

“…They demonstrated the necessary and sufficient conditions for the coefficients to be satisfied with both the isoparametric and the geodesic demands. This scheme has been used by many scholars (see, for example, [12][13][14][15][16][17][18][19][20][21][22][23]).…”

Section: Introductionmentioning

confidence: 99%

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

Alluhaibi,

Abdel-Baky

2023

Axioms

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“…They demonstrated the necessary and sufficient conditions for the coefficients to be satisfied by both the iso-parametric and geodesic demands. A variety of studies have investigated the issue of surface pencils with distinctive curves [12][13][14][15][16][17][18][19][20][21][22][23][24]. The similarity among curves is a popular topic in curve theory.…”

Section: Introductionmentioning

confidence: 99%

A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space

Nazra,

Abdel-Baky

2023

Symmetry

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The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.

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“…It was Bayram et al (2012) who followed the same idea and constructed the parametric form of surfaces with a common asymptotic curve [4]. There have been other studies characterizing surfaces on which a given specific curve lies on as geodesic, asymptotic and line of curvature ( [1], [2], [3], [5]). Motivated by these, we present the necessary and sufficient conditions to formulate a family of surfaces having both the involute and evolute curves as of each geodesic, asymptotic and curvature line.…”

Section: Introductionmentioning

confidence: 99%

Family of Surfaces with a Common Special Involute and Evolute Curves

Şenyurt

Ayvacı

Canlı

2022

International Electronic Journal of Geometry

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In this paper, we define the necessary and sufficient conditions for a parametric surface on which both the involute and evolute of any given curve lie to be geodesic, asymptotic and curvature line. Then, the first and second fundamental forms of these surfaces are calculated. By using the Gaussian and mean curvatures, the developability and minimality assumptions are drawn, as well. Moreover we extended the idea to the ruled surfaces. Finally, we provide a set of examples to illustrate the corresponding surfaces.

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Surfaces Family with Common Smarandache Geodesic Curve According to Bishop Frame in Euclidean Space

Cited by 16 publications

References 5 publications

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

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

A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space

Family of Surfaces with a Common Special Involute and Evolute Curves

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