Constant Ratio Quaternionic Curves in Euclidean Spaces

Öztürk, Günay; Kişi, İlim; Büyükkütük, Sezgin

doi:10.1007/s00006-016-0716-4

Cited by 12 publications

(6 citation statements)

References 3 publications

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“…By using the quaternionic multiplication, they obtained the Serret-Frenet equations of the curve in R 3 and then they formed the Serret-Frenet formulae of a quaternionic curve in R 4 by means of the Frenet vectors and curvature functions of the spatial quaternionic curve in R 3 [2]. After then by using these quaternionic frames defined by Bharathi and Nagaraj, a lot of paper about quaternionic curves were made in R 3 and R 4 ( [4], [6], [7], [9], [10], [11], [12], [14], [16], [17], [18]).…”

Section: F Kahraman Aksoyakmentioning

confidence: 99%

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

Aksoyak

2022

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this paper, we give some characterization of quaternionic Bertrand curves whose the torsion is non-zero but bitorsion is zero in $\mathbb{R}^{4}$ according to Type 2-Quaternionic Frame. One of the most important points in working on quaternionic curves is that given a curve in $\mathbb{R}^{4}$, the curve in $\mathbb{R}^{3}$ associated with this curve is determined individually. So, we obtain some relationships between quaternionic Bertrand curve $\alpha^{(4)}$ in $\mathbb{R}^{4}$ and its associated spatial quaternionic curve $\alpha$ in $\mathbb{R}^{3}$. Also, we support some theorems in the paper by means of an example.

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Section: F Kahraman Aksoyakmentioning

confidence: 99%

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

Aksoyak

2022

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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“…Let β : I ⊂ R → H be an arc-lengthed real quaternionic curve with quaternionic Frenet apparatus {X(s), Y(s), Z(s), W(s); r 3 (s), r 1 (s), (r 2 − r 3 )(s)} and the curve γ(s) = t(s)ds be the spatial quternionic curve related to the curve β. If the curve β is a quaternionic CCR curve (see [5] and [10]) then the curve γ is a helix in R 3 .…”

Section: A Generalization Of Quaternion Valued Frame and An Application In Rmentioning

confidence: 99%

“…Also, she has given an application of this new type of the quaternionic frame by an example. Except for them, many authors have reported on quaternionic curves by using the quaternionic Frenet formulas and defined some new special quaternionic curves in ( [5], [6], [8], [9], [11], [12]).…”

Section: Introductionmentioning

confidence: 99%

A new quaternion valued frame of curves with an application

Cansu

Yaylı

Gök

2021

Filomat

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The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

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“…The concept of constant width is considered according to different roofs in different spaces and some integral characterizations of these curves are obtained [1,7,9]. On the other hand, there are many studies on quaternionic curves [3,11,14]. In this section, the concepts related to these two basic structures are discussed.…”

Section: Introduction and Preliminary Informationmentioning

confidence: 99%

Morgan-Voyce Polynomial Approach for Quaternionic Space Curves of Constant Width

Aydın

Ayazoglu

Kocayiğit

2021

Foundations of Computing and Decision Sciences

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The curves of constant width are special curves used in engineering, architecture and technology. In the literature, these curves are considered according to different roofs in different spaces and some integral characterizations of these curves are obtained. However, in order to examine the geometric properties of curves of constant width, more than characterization is required. In this study, firstly differential equations characterizing quaternionic space curves of constant width are obtained. Then, the approximate solutions of the differential equations obtained are calculated by the Morgan-Voyce polynomial approach.The geometric properties of this curve type are examined with the help of these solutions.

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Constant Ratio Quaternionic Curves in Euclidean Spaces

Cited by 12 publications

References 3 publications

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

Quaternionic Bertrand curves according to type 2-quaternionic frame in $\mathbb{R}^{4}$

A new quaternion valued frame of curves with an application

Morgan-Voyce Polynomial Approach for Quaternionic Space Curves of Constant Width

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