Dual transformations and quaternions

Abstract

In this study, we are interested in the way quaternions to represent 3D and 4D rotations in Lorentzian space. We give a new method for obtaining a rotation matrix in Lorentzian space with the help of a unit quaternion. Furthermore, we prove that rotation matrices correspond to a quaternion leave invariant the same axis in Euclidean and Lorentzian space. Then, we introduce a semi‐orthogonal matrix representation of a quaternion curve in 4D space. Moreover, we provide applications and draw their figures to explore visual representations. Finally, due to the importance of the dual space in kinematics, robotics, and other areas related, we carry this work into their dual spaces by using a dual quaternion.

Applications of Quaternions in Galilean Spaces

“…If we examine the figures in the examples carefully, we can imagine the figures obtained in pseudo-Galilean spaces as the more opened forms of those obtained in Galilean spaces. In our previous studies (see [2] and [3]), we compared the images we obtained in Euclidean space and Lorentz space. We have also captured similar views in Galilean and pseudo-Galilean spaces and considered it appropriate to include them in this study.…”

“…We carried this work into dual spaces in [2] by investigating invariant axes in both spaces. Then, we gave a new method for obtaining a rotation matrix in Lorentzian space by using a unit quaternion in [3].…”

Electromagnetic waves along pseudo null curves in Minkowski space

“…Rotation transformations around timelike, spacelike, or null axis are defined very differently from one another. For further information about rotation transformations, see [20,21,22,23,24,25]. In two different situations, the pseudo null frames (⃗ 𝑡, ⃖ ⃗ 𝑛, ⃖ ⃗ 𝑏) and (⃗ 𝑡, ⃖⃗ , ⃖⃖ ⃗ ) are described with a linear transformation of one another.…”

Kinematical modeling of circular and elliptical polarization of the polarized light via screw transformation

“…It is obtained by describing a screw with the vector pair forming the Plücker coordinates of a line, multiplying it by a pair of real numbers, and adding the vectors, [5]. In addition, screw motion and dual quaternions have been applied in many fields such as robotic systems (see, [6][7][8][9][10][11]).…”