Ruled surfaces constructed by quaternions

Aslan, Selahattin; Bekar, Murat; Yaylı, Yusuf

doi:10.1016/j.geomphys.2020.104048

Cited by 14 publications

(9 citation statements)

References 6 publications

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“…We give split semi‐quaternionic and matrix representations of GTs in pseudo‐Galilean 3‐space. Similar ideas were already considered for Euclidean and Minkowski 3‐spaces (see the literature 29–36 ). As an application, we introduce magnetic flux tubes in pseudo‐Galilean 3‐space.…”

Section: Introductionmentioning

confidence: 74%

Generalized tubes in pseudo‐Galilean 3‐space: Split semi‐quaternionic representations and an application to magnetic flux tubes

Tuncer

2021

Math Methods in App Sciences

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This paper deals with generalized tube surfaces (GTs) and their geometric properties in pseudo‐Galilean 3‐space. We classify these surfaces into two types. We firstly compute the first and second fundamental forms to investigate geometric properties of a GT. Then, we obtain the condition for such a surface to be minimal and present some results which express the conditions for which parameter curves on a GT are geodesics, asymptotics, or lines of curvature. Furthermore, we show how to form GTs by using split semi‐quaternions or their matrix representations. Finally, as an application, we introduce generalized magnetic flux tubes in pseudo‐Galilean 3‐space and obtain the local magnetic field components of such surfaces. The theory studied in the paper is supported by illustrated examples.

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Section: Introductionmentioning

confidence: 74%

Generalized tubes in pseudo‐Galilean 3‐space: Split semi‐quaternionic representations and an application to magnetic flux tubes

Tuncer

2021

Math Methods in App Sciences

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“…Finally, they deal with the first and second Laplace–Beltrami operators of 2‐ruled hypersurfaces in

{normal𝔼}^{4}

. Aslan et al [9, 10] characterize the ruled surface through quaternions in

{normal𝔼}^{3}

and

{normal𝔼}_{1}^{3}

. In three dimensions, the quaternions can be used to characterize the ruled surfaces.…”

Section: Introductionmentioning

confidence: 99%

The 2‐ruled hypersurfaces in Minkowski 4‐space and their constructions via octonions

Ndiaye,

Özdemir

2023

Math Methods in App Sciences

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In this paper, we define three types of 2‐ruled hypersurfaces in the Minkowski 4‐space . We obtain Gaussian and mean curvatures of the 2‐ruled hypersurfaces of type‐1 and type‐2 and some characterizations about its minimality. We also deal with the first Laplace–Beltrami operators of these types of 2‐ruled hypersurfaces in . Moreover, the importance of this paper is the definition of these surfaces by using the octonions in . Thus, this is a new idea and makes the paper original. We give an example of 2‐ruled hypersurface constructed by octonion, and we visualize the projections of the images with MAPLE program. Furthermore, the optical fiber can be defined as a one‐dimensional object embedded in the four‐dimensional Minkowski space . Thus, as a discussion, we investigate the geometric evolution of a linearly polarized light wave along an optical fiber by means of the 2‐ruled hypersurfaces in a four‐dimensional Minkowski space.

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“…The basics related to this type of surfaces are given in [7,23,25]. And there are various other studies on ruled surfaces, [2,3,10,12,13,27,33,36]. On the other hand, the theory of curves also occupies an important place in differential geometry.…”

Section: Introductionmentioning

confidence: 99%

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

ŞENYURT,

CANLI,

ÇAN

et al. 2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this study, we first define the Smarandache curves derived from the Frenet vectors and the Darboux vector of any curve. Then, we construct new ruled surfaces along these Smarandache curves with the direction vectors obtained from the Frenet vectors and the Darboux vector, and give the equations of these surfaces. In addition, we calculate the Gaussian and mean curvatures of these surfaces separately and present the conditions to be minimal and developable for these surfaces. Finally, as an example, we obtain ruled surfaces whose base curves are Viviani’s curves and plot the graphics of these surfaces.

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Ruled surfaces constructed by quaternions

Cited by 14 publications

References 6 publications

Generalized tubes in pseudo‐Galilean 3‐space: Split semi‐quaternionic representations and an application to magnetic flux tubes

Generalized tubes in pseudo‐Galilean 3‐space: Split semi‐quaternionic representations and an application to magnetic flux tubes

The 2‐ruled hypersurfaces in Minkowski 4‐space and their constructions via octonions

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

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