An Alternative Approach to Elliptical Motion

Özdemir, Mustafa

doi:10.1007/s00006-015-0592-3

Cited by 19 publications

(22 citation statements)

References 17 publications

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“…Moreover, it can be written as B ( u, w) = u t Ωw where the associated matrix Ω is defined as follows [13]. Thus, the real vector space R 3 equipped with the elliptical inner product will be denoted by R 3 a1,a2,a3 and the number ∆ = √ det Ω will be called constant of the scalar products [13]. The elliptical norm of a vector u ∈ R 3 is defined to be u B = B ( u, u).…”

Section: Preliminariesmentioning

confidence: 99%

“…where θ is compatible with the parameters of the angular parametric equations of ellipse or ellipsoid [13].…”

Section: Preliminariesmentioning

confidence: 99%

“…For this purpose, an elliptical orthogonal matrix and an elliptical skew-symmetric matrix have been defined for this elliptical scalar product. Thereby, he has examined the motion of a point on the ellipsoid using elliptical rotation matrices [13]. This present paper emphasizes expansion of one-parameter Euclidean spherical motions to one parameter elliptic motions.…”

Section: Introductionmentioning

confidence: 98%

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One Parameter Elliptic Motions in Three-Dimensional Space

Azak

2021

International Electronic Journal of Geometry

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Elliptic motions have been defined by three different right-handed coordinate systems. The motion of these coordinate systems depends on the time parameter which has great importance in robotics. In particular, it is used in a model of a robot arm manipulator to achieve high performance. Hence, we have expressed some theorems and results concerning this elliptic motion. Besides, the special cases of this motion have been discussed.

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Section: Preliminariesmentioning

confidence: 99%

“…where θ is compatible with the parameters of the angular parametric equations of ellipse or ellipsoid [13].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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One Parameter Elliptic Motions in Three-Dimensional Space

Azak

2021

International Electronic Journal of Geometry

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“…Furthermore, elliptical rotations are obtained by Rodrigues Rotation's formula, Cayley formula , union of even number of Hausholder transformations and elliptic quaternions in (Gracia and Thomas, 2017). On the other hand, the rotations in Minkowski space time are studied in (ġimĢek and Özdemir, 2016;Erdoğdu and Özdemir, 2015). In a different way, Lorentzian rotations are investigated with Lorentzian matrix multiplications in (Özdemir, 2016;Keçelioğlu et al, 2012).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the rotations in Minkowski space time are studied in (ġimĢek and Özdemir, 2016;Erdoğdu and Özdemir, 2015). In a different way, Lorentzian rotations are investigated with Lorentzian matrix multiplications in (Özdemir, 2016;Keçelioğlu et al, 2012). Then, homothetic Cayley formula and transformation are defined Rodrigues and Euler parameters for homothetic motion are obtained in (Gonzalez et al, 2009).…”

Section: Introductionmentioning

confidence: 99%

Kuaterniyon Bakış Açısı ile Doğru ve Hiperdüzlem Boyunca Yansımalar

Erdoğdu

2019

Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this study, the reflections in and are investigated by unit quaternions. Firstly, a linear transformation is defined to describe reflections in with respect to the plane passing through the origin and orthogonal to the quaternion. Then some examples are given to discuss obtained results. Similarly, two linear transformations are stated which correspond to the reflection in with respect to the hyperplane passing through the origin and a reflection with respect to the line in the direction of the quaternion. Finally, the matrix representaions of these reflections are found and the eigenvalues, eigenvectors of them are given to analyse the geometric meaning in terms of the components of the quaternion for each case.

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On geometric interpretations of split quaternions

Öztürk

Özdemir

2022

Math Methods in App Sciences

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Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).

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An Alternative Approach to Elliptical Motion

Cited by 19 publications

References 17 publications

One Parameter Elliptic Motions in Three-Dimensional Space

One Parameter Elliptic Motions in Three-Dimensional Space

Kuaterniyon Bakış Açısı ile Doğru ve Hiperdüzlem Boyunca Yansımalar

On geometric interpretations of split quaternions

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