Generalized Bertrand Curves with Spacelike $\left(1,3\right) $-Normal Plane in Minkowski Space-Time

Uçum, Ali; Keçilioğlu, Osman; İlarslan, Kazım

doi:10.3906/mat-1503-12

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…3-space and Minkowski space-time (see [1,2,7,11,[24][25][26]28]) as well as in Euclidean space. In addition, in [27,30], the authors studied (1, 3)-type Bertrand curves in semi-Euclidean 4-space with index 2.…”

Section: Introductionmentioning

confidence: 99%

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

Erdem,

Uçum,

İlarslan

et al. 2023

Carpathian Math. Publ.

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In the theory of curves in Euclidean $3$-space, it is well known that a curve $\beta $ is said to be a Bertrand curve if for another curve $\beta^{\star}$ there exists a one-to-one correspondence between $\beta $ and $\beta^{\star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Minkowski $3$-space. Related examples that meet these conditions are given. Moreover, thanks to this new approach, timelike, spacelike and Cartan null Bertrand mates of a timelike general helix have been obtained.

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Section: Introductionmentioning

confidence: 99%

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

Erdem,

Uçum,

İlarslan

et al. 2023

Carpathian Math. Publ.

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“…Some of these curves mates have been generalized to larger dimensions and have been studied by many authors [11,17,19,28,35]. Also, these pairs of curves have been studied by many authors in the Lorentzian space [3,5,16,18,21,24,34,40,41,43].…”

Section: Introductionmentioning

confidence: 99%

A New Generalization of Some Curve Pairs

ÇELİK

Özdemir²

2022

International Electronic Journal of Geometry

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In this study, we give a new curve pair that generalizes some of the famous pairs of curves as Bertrand and constant torsion curves. This curve pair is defined with the help of a vector obtained by the intersection of the osculating planes such that this vector makes the same angle $\gamma$ with the tangents of the curves. We examine the relations between torsions and curvatures of these curve mates. Also, We have seen that the unit quaternion corresponding to the rotation matrix between the Frenet vectors of the curves is $q=\cos (\theta/2)-\mathbf{i}\sin (\theta/2)\cos \gamma -\mathbf{j}\sin (\theta/2)\sin \gamma$, where $\theta$ is the angle between the reciprocal binormals of the curves. Finally, we show in which specific case which well-known pairs of curves will be obtained.

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“…İlarslan et al have defined null Cartan and pseudo null Bertrand curves in Minkowski 3-space E 3 1 [6]. Further, (1,3)-Bertrand curves in a timelike (1,3)-normal plane in Minkowski space-time E 4 1 have been examined [7]. Also, Matsuda and Yorozu have shown that there is no Bertrand curve in Euclidean n-space E n such that n ≥ 4 and have defined (1,3)-Bertrand curves in Euclidean 4-space E 4 [8].…”

Section: Introductionmentioning

confidence: 99%

Timelike V-Bertrand Curves in Minkowski 3-Space $E_{1}^{3}$

BİLGİN>

Camcı

2022

Journal of New Theory

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In this paper, the timelike V-Bertrand curve, a new type Bertrand curve in Minkowski 3-Space $E_{1}^{3}$, is characterized. Based on the timelike V-Bertrand curve, the properties of the timelike T, N, and B Bertrand curves are obtained. From the timelike V-Bertrand curve, f-Bertrand curves and Bertrand surfaces are defined. We support the existence of these new curves and surfaces with examples. Finally, we discuss the results for further research.

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Generalized Bertrand Curves with Spacelike $\left(1,3\right) $-Normal Plane in Minkowski Space-Time

Cited by 5 publications

References 11 publications

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

New approach to timelike Bertrand curves in 3-dimensional Minkowski space

A New Generalization of Some Curve Pairs

Timelike V-Bertrand Curves in Minkowski 3-Space $E_{1}^{3}$

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