On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space

Grbović, Milica; Nešović, Emilija

doi:10.1016/j.jmaa.2018.01.014

Cited by 26 publications

(23 citation statements)

References 12 publications

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“…The Bishop frame or relatively parallel adapted frame {T, N 1 , N 2 } of a regular curve in Euclidean 3-space contains a tangential vector field T and two normal vector fields N 1 and N 2 , which can be obtained by rotating the Frenet vectors N and B in the normal plane T ⊥ of the curve, in such a way that they become relatively parallel. This means that their derivatives N 1 and N 2 with respect to the arc-length parameter s of the curve are collinear with the tangential vector field T [10]. Remark 1.…”

Section: The Bishop Framementioning

confidence: 99%

“…1 is positively oriented pseudo orthonormal frame consisting of the tangential vector field T 1 and two relatively parallel lightlike normal vector fields N 1 and N 2 . Bishop vector N 1 (of the first Bishop frame) can be obtained by applying the hyperbolic rotation to the principal normal vector N, while the normal Bishop vector N 2 (of the first Bishop frame) can be obtained by applying the composition of three rotations about two lightlike and one spacelike axis to the binormal vector B [10] . Theorem 1.…”

Section: The Bishop Frame Of a Pseudo Null Curve Inmentioning

confidence: 99%

“…But, the curve might not be continuous at some points, which is undefined when the second derivative of the curve vanishes [9]. In 1975, Richard Lawrence Bishop first introduced the parallel frame as a new frame which is well defined even if the curve has vanishing second derivative, then the parallel frame came to be called the Bishop frame [8,9,10]. Bishop frame contains the tangential vector field T and two normal vector fields N 1 and N 2 .…”

Section: Introductionmentioning

confidence: 99%

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Special Curves According to Bishop Frame in Minkowski 3-Space

Isah

Külahçı

2020

Applied Mathematics and Nonlinear Sciences

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Pseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space [E_1^3 . We define helix and slant helix according to Bishop frame in [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

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Section: The Bishop Framementioning

confidence: 99%

Section: The Bishop Frame Of a Pseudo Null Curve Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Special Curves According to Bishop Frame in Minkowski 3-Space

Isah

Külahçı

2020

Applied Mathematics and Nonlinear Sciences

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“…It can be verified that the Darboux frame of α coincides with its Bishop frame if and only if τ = 0. For the Bishop frame of a pseudo null curve, we refer to [5].…”

Section: Proposition 32mentioning

confidence: 99%

On k-type spacelike slant helices lying on lightlike surfaces

Öztürk

Nešović

Esra³

2019

Filomat

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In this paper, we define k-type spacelike slant helices lying on a lightlike surface in Minkowski space E 3 1 according to their Darboux frame for k ∈ {0, 1, 2}. We obtain the necessary and the sufficient conditions for spacelike curves with non-null and null principal normal lying on lightlike surface to be the k-type spacelike slant helices in terms of their geodesic curvature, normal curvature and geodesic torsion. Additionally, we determine their axes and show that the Darboux frame of a spacelike curve lying on a lightlike surface coincides with its Bishop frame if and only if it has zero geodesic torsion. Finally, we give some examples.

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“…In Minkowski spaces, the Bishop frame of a timelike and a spacelike curve is obtained in [5,20]. Recently, the Bishop frame and the generalized Bishop frame of the pseudo null and null Cartan curve in Minkowski 3-space are introduced in [9,10].…”

Section: Introductionmentioning

confidence: 99%

On Bishop frame of a pseudo null curve in Minkowski space-time

Djordjević¹,

Nešović²

2020

Turk J Math

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In this paper, we introduce the Bishop frame of a pseudo null curve α in Minkowski space-time. We obtain the Bishop frame's equations and the relation between the Frenet frame and the Bishop frame. We find the third order nonlinear differential equation whose particular solutions determine the form of the Bishop curvatures. By using spacetime geometric algebra, we derive the Darboux bivectors D andD of the Frenet and the Bishop frame of α , respectively. We give geometric interpretations of the Frenet and the Bishop curvatures of α in terms of areas of the projections of the corresponding Darboux bivectors onto the planes spanned by the frame vector's fields.

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On the Bishop frames of pseudo null and null Cartan curves in Minkowski 3-space

Cited by 26 publications

References 12 publications

Special Curves According to Bishop Frame in Minkowski 3-Space

Special Curves According to Bishop Frame in Minkowski 3-Space

On k-type spacelike slant helices lying on lightlike surfaces

On Bishop frame of a pseudo null curve in Minkowski space-time

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