On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space

Nešović, Emilija

doi:10.1007/s00006-015-0601-6

Cited by 9 publications

(11 citation statements)

References 5 publications

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“…$$ The set

\begin{matrix} S O_{1} ((), 3, 𝕊) = ({}, normalR \in M_{3 \times 3} false(ℝ false) : {normalR}^{T} I^{⋆} normalR = I^{⋆} and \det normalR = 1) \end{matrix}

is called a Cartan rotation group and its elements are called the Cartan rotation matrices in Cartan 3‐space. A matrix M is called a semi skew‐symmetric, if it accomplishes the equation M

{}&#x0005E;T&#x0003D;-{I}&#x0005E;{\star }

{I}&#x0005E;{\star }

14,21–23 …”

Section: The Geometry Of Cartan Numbersmentioning

confidence: 99%

Introduction to Cartan numbers and some geometric applications

Öztürk

2022

Math Methods in App Sciences

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This article aims to define the rotational motion around a lightlike axis in pseudo‐null or null frames more easily and to put forward a suitable number system to express rotation transformation in these frames. Therefore, for null and pseudo‐null frames, we define a new set of numbers consisting of a linear combination of two lightlike units and a spacelike unit, which we call Cartan numbers. The first part gives the properties of the set of Cartan numbers, and in the second part, some geometric definitions, theorems, and applications are given. In addition, the definition of the center of a parabola and the concept of the central angle are defined. The connection is made between these concepts and the parabolic rotation transformation. Rotation matrices for parabolic rotations are obtained. The rotating points indicated that it follows in which cases a line path and in which cases a parabolic path. The rotation transformation along any straight line or parabola is studied with examples.

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“…$$ The set

\begin{matrix} S O_{1} ((), 3, 𝕊) = ({}, normalR \in M_{3 \times 3} false(ℝ false) : {normalR}^{T} I^{⋆} normalR = I^{⋆} and \det normalR = 1) \end{matrix}

is called a Cartan rotation group and its elements are called the Cartan rotation matrices in Cartan 3‐space. A matrix M is called a semi skew‐symmetric, if it accomplishes the equation M

{}&#x0005E;T&#x0003D;-{I}&#x0005E;{\star }

{I}&#x0005E;{\star }

14,21–23 …”

Section: The Geometry Of Cartan Numbersmentioning

confidence: 99%

Introduction to Cartan numbers and some geometric applications

Öztürk

2022

Math Methods in App Sciences

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“…Thus the Cayley map (10) takes values in SO(2, 1). Similar technique has been used in [13]. By making use of (11) we obtain the explicit form of (10), i.e.,…”

Section: The Cayley Map For So(2 1)mentioning

confidence: 99%

Vector-Parameter Forms of SU(1,1), SL(2,R) and Their Connection to SO(2,1)

Donchev¹,

Mladenova²,

Mladenov³

2016

GIQ

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The Cayley maps for the Lie algebras su(1, 1) and so(2, 1) converting them into the corresponding Lie groups SU(1, 1) and SO(2, 1) along their natural vector-parameterizations are examined. Using the isomorphism between SU(1, 1) and SL(2, R), the vector-parameterization of the latter is also established. The explicit form of the covering map SU(1, 1) → SO(2, 1) and its sections are presented. Using the so developed vector-parameter formalism, the composition law of SO(2, 1) in vector-parameter form is extended so that it covers compositions of all kinds of elements including also those that can not be parameterized properly by regular SO(2, 1) vectorparameters. The latter are characterized and it is shown that they can be represented by SU(1, 1) vector parameters with pseudo length equal to minus four. In all cases of compositions inside SO(2, 1), criteria for determination of their type (elliptic, parabolic, hyperbolic) have been presented. On the base of the vector-parameter formalism the problem of taking a square root in SO(2,1) is solved explicitly. Also, an analogue of Cartan's theorem about the decomposition of orthogonal matrix of order n into product of at most n reflections is formulated and proved for the subset of hyperbolic elements of the group of pseudo-orthogonal matrices from SO(2, 1).

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“…Depending on the type of split quaternion, the type of the rotation and the pseudosphere in which the rotation occurs varies. Detailed investigations about reflection and rotation transformations represented with split quaternions in the

𝔼_{1}^{3}

can be found in a lot of papers 13–19 …”

Section: Introductionmentioning

confidence: 99%

“…Detailed investigations about reflection and rotation transformations represented with split quaternions in the E 3 1 can be found in a lot of papers. [13][14][15][16][17][18][19] Demirci and Aghayev gave a different approach for the geometric applications of quaternions defining subspaces of quaternions H, called Plane(V) and Line(V), determined by a unit pure quaternion V. They defined this subspaces as…”

Section: Introductionmentioning

confidence: 99%

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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On Rotation About Lightlike Axis in Three-Dimensional Minkowski Space

Cited by 9 publications

References 5 publications

Introduction to Cartan numbers and some geometric applications

Introduction to Cartan numbers and some geometric applications

Vector-Parameter Forms of SU(1,1), SL(2,R) and Their Connection to SO(2,1)

On geometric interpretations of split quaternions

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