Quaternionic osculating curves in Euclidean and semi-Euclidean space

Bektaș, Özcan; Gürses, Nurten; Yüce, Salîm

doi:10.1080/1726037x.2016.1177935

Cited by 7 publications

(4 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition to that, in 20 the quaternionic Bertrand curves have been obtained. Also, the quaternionic osculating curves have been given in 21 . On the other hand, the associated curves were given by Choi and Kim in 12 .…”

Section: Introductionmentioning

confidence: 99%

Spatial Quaternionic and Quaternionic Osculating Direction Curve

Kızıltuğ,

Erişir,

MUMCU

2024

Preprint

View full text Add to dashboard Cite

In this paper, we investigate a new structure of unit speed associated curves, such as spatial quaternionic and quaternionic osculating direction curves. For this, we assume that the vector fields 휒.%/ = 휐1.%/t.%/+ 휐2.%/n.%/+ 휐3.%/b.%/ where 휐2 1.%/+ 휐2 2.%/ = 1 for the spatial quaternionic curve and 휒.%/ = 휆1.%/æ.%/ + 휆2.%/휂.%/ + 휆3훽2.%/, where 휆2 1.%/+휆2 2.%/+휆2 3.%/ = 1 for the quaternionic curve 휙. Then, we give the relationship between (spatial) quaternionic (OD)-curves and Mannheim curve pair. Moreover, we examine in which cases the (spatial) quaternionic (OD)-curve can be helix or slant helix. Finally, we give the examples and draw the figures of curves in the examples.

show abstract

Section: Introductionmentioning

confidence: 99%

Spatial Quaternionic and Quaternionic Osculating Direction Curve

Kızıltuğ,

Erişir,

MUMCU

2024

Preprint

View full text Add to dashboard Cite

show abstract

“…After Chen [1] defined the rectifying curve in Euclidean space, Ilarslan and Nesovic [2] using the definition of the rectifying curve, defined the osculating curve as a curve in Euclidean 4-space whose position vector is always lied in the orthogonal complement of the first binormal vector field. This definition given for the osculating curve has been used in some studies in Euclidean space [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Osculating Curves According to the Extended Darboux Frame in E4

Bektaș

2023

J. Sci. Arts

Self Cite

View full text Add to dashboard Cite

In this paper, we define osculating curves according to the extended Darboux frame (ED-frame), in Euclidean 4-space. Then, according to Case 1 and Case 2 ED-frame fields, the necessary and sufficient condition for a unit speed curve lying on the hypersurface to be congruent to an osculating curve is obtained. For these necessary and sufficient conditions, the solutions of the differential equations have been investigated. Finally, an application has been made on the subject.

show abstract

“…Characterizations of an osculating curve in the 3-dimensional Euclidean space has been given in [9] and a specific osculating curve has been defined in the Euclidean 4-space. Normal, osculating and rectifying curves have been defined in the Euclidean and semi Euclidean space by using quaternion algebra in [10], [11], [16] and [25]. Bi-null curves of these types have also been analyzed in R 6 3 and R 5 2 in [12,13].…”

Section: Introductionmentioning

confidence: 99%

Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

Bektaș¹,

Bekiryazici²

2022

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

Self Cite

View full text Add to dashboard Cite

In this paper, we give a generalization of the osculating curves to the n-dimensional Euclidean space. Based on the definition of an osculating curve in the 3 and 4 dimensional Euclidean spaces, a new type of osculating curve has been defined such that the curve is independent of the (n − 3)th binormal vector in the n-dimensional Euclidean space, which has been called "a generalized osculating curve of type (n − 3)". We find the relationship between the curvatures for any unit speed curve to be congruent to this osculating curve in E n . In particular, we characterize the osculating curves in E n in terms of their curvature functions. Finally, we show that the ratio of the (n − 1)th and (n − 2)th curvatures of the osculating curve is the solution of an (n − 2)th order linear nonhomogeneous differential equation.

show abstract

Quaternionic osculating curves in Euclidean and semi-Euclidean space

Cited by 7 publications

References 7 publications

Spatial Quaternionic and Quaternionic Osculating Direction Curve

Spatial Quaternionic and Quaternionic Osculating Direction Curve

Osculating Curves According to the Extended Darboux Frame in E4

Generalized osculating curves of type (n-3) in the n-dimensional Euclidean space

Contact Info

Product

Resources

About