Bäcklund transformations according to Bishop frame in E3/1

Karacan, Murat Kemal; Tunçer, Yılmaz

doi:10.12697/acutm.2015.19.07

Cited by 5 publications

(13 citation statements)

References 5 publications

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“…Theorem Let ψ be a transformation between α and β in Euclidean 3‐space with β = ψ ( α ) such that in the corresponding points we have as follows: i ) The line segment [ β ( s ) α ( s )] at the intersection of the osculating planes of the curves has constant length r ; i i ) The distance vector β ( s ) − α ( s ) has the same angle γ ≠ π /2 with the tangent vectors of the curves; i i i ) The binormals of the curves have the same constant angle φ = 0. Then these curves are congruent with natural curvatures

\begin{matrix} k_{1}^{β} & = k_{1}^{α} = - \frac{d γ}{d s}, \\ k_{2}^{β} & = k_{2}^{α} = \frac{\tan γ \sin φ}{r} \end{matrix}

and the curves β = ψ ( α ) is

β = α + \frac{2 C \tan γ}{{(k_{2}^{α})}^{2} + C^{2}} (T_{α} \cos γ + U_{α} \sin γ) .

Here, the angle γ is a solution of the differential equation,

\begin{matrix} \frac{d γ}{d s} & = k_{2}^{β} \cos γ \tan \frac{φ}{2} - k_{1}^{α} \\ = - k_{2}^{α} \cos γ \tan \frac{φ}{2} - k_{1}^{β}, \end{matrix}

and

C = k_{2}^{α} \tan \frac{γ}{2} .

…”

Section: Preliminariesmentioning

confidence: 95%

“…On the other hand, Bäcklund transformations can be integrable, in the manner of a tangent line segment at the point given on a surface of constant negative curvature. Thus, one can find a new surface with constant negative curvature that contains a line segment's endpoint . Thus, Bäcklund transformations are often studied in differential geometry.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, one can find a new surface with constant negative curvature that contains a line segment's endpoint. 3 Thus, Bäcklund transformations are often studied in differential geometry. For example, Palmer constructed a Bäcklund transformation between the two surfaces, 4 Calini gave the Bäcklund transformation for pseudospherical surfaces in Calini and Ivey, 5 Garay obtained the Bäcklund transformation for curves in conformal geometry, 6 Aminov used Bäcklund transformations of two-dimensional surfaces in Aminov and Sym, 7 Abdel-Baky gave Bäcklund transformations in Minkowski space, 8 Özdemir examined the Bäcklund transformation for nonlight-like curves in Özdemir and Çöken.…”

Section: Introductionmentioning

confidence: 99%

“…9 Finally, Karacan obtained Bäcklund transformation according to parallel transport frame in Karacan and Tunçer. 3 The Frenet-Serret frame, which has an important place in Euclidean space, is obtained by continuous differentiable nondegenerate curves. For this reason, the studies related to the Frenet frame are often used in differential geometry.…”

Section: Introductionmentioning

confidence: 99%

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New version of Bäcklund transformations in Euclidean 3‐space

Sariaydin

Körpınar

2018

Math Methods in App Sciences

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\begin{matrix} k_{1}^{β} & = k_{1}^{α} = - \frac{d γ}{d s}, \\ k_{2}^{β} & = k_{2}^{α} = \frac{\tan γ \sin φ}{r} \end{matrix}

and the curves β = ψ ( α ) is

β = α + \frac{2 C \tan γ}{{(k_{2}^{α})}^{2} + C^{2}} (T_{α} \cos γ + U_{α} \sin γ) .

Here, the angle γ is a solution of the differential equation,

\begin{matrix} \frac{d γ}{d s} & = k_{2}^{β} \cos γ \tan \frac{φ}{2} - k_{1}^{α} \\ = - k_{2}^{α} \cos γ \tan \frac{φ}{2} - k_{1}^{β}, \end{matrix}

and

C = k_{2}^{α} \tan \frac{γ}{2} .

…”

Section: Preliminariesmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

New version of Bäcklund transformations in Euclidean 3‐space

Sariaydin

Körpınar

2018

Math Methods in App Sciences

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“…Furthermore, some mathematicians studied Bishop frame in Minkowski 3‐space such as those in the literature. ()…”

Section: Introductionmentioning

confidence: 99%

Equiform spacelike normal curves according to equiform‐Bishop frame in

El-Sayied

Elzawy

Elsharkawy

2017

Math Methods in App Sciences

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In this article, we present the equiform parameter and define the equiform-Bishop frame in Minkowski 3-space E 3 1 . Additionally, we investigate the equiform-Bishop formulas of the equiform spacelike case in Minkowski 3-space.Furthermore, some results of equiform spacelike normal curves according to the equiform-Bishop frame in E 3 1 are considered. KEYWORDS equiform-Bishop frame, equiform curvatures, Minkowski Space, normal curves PRELIMINARIESThe Minkowski space E 3 1 is the space R 3 , equipped with the metric g, where g is given bywhere (x 1 , x 2 , x 3 ) is a coordinate system of E 3 1 . Let v be any vector in E 3 1 , then the vector v is spacelike, timelike, or null (lightlike) if g(v, v) > 0 or v = 0, g(v, v) < 0 or g(v, v) = 0, and v ≠ 0. The causal character of a vector in Minkowski space is the property to be spacelike, timelike, or null (lightlike).

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Timelike spherical curves according to equiform Bishop frame in 3-dimensional Minkowski space

Elsharkawy,

Cesarano,

Dmytryshyn

et al. 2023

Carpathian Math. Publ.

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In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermore, we give certain characterizations of equiform spherical curves in 3-dimensional Minkowski space, which are timelike with an equiform spacelike principal normal vector.

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Bäcklund transformations according to Bishop frame in E3/1

Abstract: Abstract. In this study we have defined Bäcklund transformations of curves according to Bishop frame preserving the natural curvatures under certain assumptions in Minkowski 3-space.

Cited by 5 publications

References 5 publications

New version of Bäcklund transformations in Euclidean 3‐space

New version of Bäcklund transformations in Euclidean 3‐space

Equiform spacelike normal curves according to equiform‐Bishop frame in

Timelike spherical curves according to equiform Bishop frame in 3-dimensional Minkowski space

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