On Timelike Bertrand Curves in Minkowski 3-Space

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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“…In this case, any timelike anti-Salkowski curve is a timelike B-Bertrand curve. provided in [14]. It is clear that γ is a timelike curve.…”

Section: Preliminariesmentioning

confidence: 93%

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

BİLGİN>

Camcı

2022

Journal of New Theory

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“…Let us consider a timelike general helix in E 3 1 with the equation Remark 1. If we take the functions u and w as u = w = 0 in Theorem 1, we obtain the theorems in [29]. In addition to [29], in this paper, we give the necessary and sufficient conditions for timelike curves in Minkowski 3-space to have a Cartan null Bertrand mate curve given by β ⋆ (s ⋆ ) = β ⋆ f (s) = β(s) + v(s)N(s).…”

Section: Examplementioning

confidence: 97%

“…In what follows, we give the examples for timelike general helices, which are Bertrand curves. We know that the timelike general helices do not satisfy the conditions of the theory for classical Bertrand curves (see [29]). So these examples are so important for Bertrand curves.…”

Section: Examplementioning

confidence: 99%

“…Some known results related to Bertrand curves and Razzaboni surfaces in Euclidean 3-space generalized to Minkowski 3-space E 3 1 by C. Xu et al in [33]. In [29], the authors studied the timelike Bertrand curves in Minkowski 3-space. They obtained the necessary and sufficient conditions for timelike curves to have a timelike, spacelike or Cartan null Bertrand mate curve, separately.…”

Section: Introductionmentioning

confidence: 99%

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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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“…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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On Timelike Bertrand Curves in Minkowski 3-Space

Cited by 4 publications

References 5 publications

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

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

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

A New Generalization of Some Curve Pairs

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