On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion

Yüksel, Nural; Saltık, Burçin

doi:10.3934/math.2023573

Cited by 4 publications

(2 citation statements)

References 27 publications

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“…In [19][20][21][22], the features and applications of generated surfaces across various mathematical fields have been specified. The investigation of equiform Bishop spherical image governed surfaces in Minkowski 3-space yields important minimality and developability requirements, with consequences for computer-aided geometric design and physics.…”

Section: Introductionmentioning

confidence: 99%

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

Gaber,

Sorour,

Abdel-Salam

2024

Symmetry

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Ruled surfaces are considered one of the significant aspects of differential geometry. These surfaces are formed by the motion of a straight line called a generator, and every curve that intersects all the generators is called a directrix. In the present research paper, we explore a family of ruled surfaces constructed from circular helices (W-curve) using the Frenet frame in the Euclidean space E3. We derive the explicit formulas for the second mean curvature and second Gaussian curvature. We present some ruled surfaces, and we describe their properties. In addition, we determine the sufficient conditions for these surfaces to be minimal, flat, II-minimal, and II-flat. Also, we obtain sufficient conditions for the base curve for these ruled surfaces to be a geodesic curve, an asymptotic line, and a principal line. Furthermore, we present an application for a ruled surface whose base curve is a circular helix, we compute some quantities for this surface such as the mean curvature and Gaussian curvatures and we plot the ruled surface with its base curve, and at symmetric points and along a symmetry axis.

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Section: Introductionmentioning

confidence: 99%

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

Gaber,

Sorour,

Abdel-Salam

2024

Symmetry

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“…Eren et al [7] employed the modified orthogonal frame to investigate the evolution of space curves and some special ruled surfaces. Yuksel et al [8], studied the inextensibility of tangential, normal, and binormal ruled surfaces generated by a Salkowski curve. Some theorems related to the inextensibility of ruled surfaces in R 3 were proved.…”

Section: Introductionmentioning

confidence: 99%

A Study on Normal Motion of the Torus of Revolution in ℝ³

Gaber,

Alfadhli,

Mahmoud

2023

Applied Mathematics and Nonlinear Sciences

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In the present research paper, we investigate the motion of surfaces in R3 according to their curvatures. We study the motion of the torus of revolution via the normal velocity. We consider two cases: when the normal velocity is a function of both the time and the coordinates of the torus, and when it is a function of time only. We also study how the torus moves under different types of curvature flows, such as inverse mean curvature flow, inverse Gaussian curvature flow, and harmonic mean curvature flow. Moreover, we present some new applications of these flows.

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Alternative View of Inextensible Flows of Curves and Ruled Surfaces via Alternative Frame

Savić,

Eren,

Ersoy

et al. 2024

Mathematics

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In this paper, we present the evolutions of ruled surfaces generated by the principal normal, the principal normal’s derivative, and the Darboux vector fields along a space curve that are the elements of an alternative frame. The comprehension of an object’s rotational behavior is crucial knowledge relevant to various realms, and this can be accomplished by analyzing the Darboux vector along the path of a point on the object as it moves through space. In that regard, examining the evolutions of the ruled surfaces based on the changes in their directrices, including the Darboux vector in the alternative frame along a space curve, is significant. As the first step of this study, we express the evolution of the alternative frame elements of a space curve. Subsequently, the conditions for the ruled surfaces generated by them to be minimal, developable, and inextensible are investigated. These findings can allow some physical phenomena to be well understood through surface evolutions satisfying these conditions. In the final step, we provide graphical representations of some examples of inextensible ruled surfaces and curve evolutions.

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On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion

Cited by 4 publications

References 27 publications

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

A Study on Normal Motion of the Torus of Revolution in ℝ³

Alternative View of Inextensible Flows of Curves and Ruled Surfaces via Alternative Frame

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On inextensible ruled surfaces generated via a curve derived from a curve with constant torsion

Cited by 4 publications

References 27 publications

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

Construction of Ruled Surfaces from the W-Curves and Their Characterizations in E3

A Study on Normal Motion of the Torus of Revolution in ℝ3

Alternative View of Inextensible Flows of Curves and Ruled Surfaces via Alternative Frame

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Product

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A Study on Normal Motion of the Torus of Revolution in ℝ³