In this study, we define a dual transformation between G n and G n 1. We examine the invariance of the plane where the shear motion is acting in Galilean and pseudo-Galilean spaces. We define a dual transformation between G n and G n 1 as well. We provide applications in G 3 and G 3 1. In addition to applications, we draw their figures in order to reinforce the visualization in both spaces.

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.

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In this study, we investigate instantaneous screw axes with the help of a dual transformation which is defined in [1]. This transformation makes clear the relations between Euclidean space and Lorentzian space. The aim of this paper is to observe instantaneous screw axes under this dual transformation.

In this paper, Galilean orthogonal matrices in G^5 and G_1^5 are obtained with the help of unit quaternions. Moreover, Galilean orthogonal matrices in G^4 and G_1^4 are acquired. These matrices produce Galilelan motions in Galilean spaces. We investigate the invariance of the plane where shear motion is acting in Galilean and pseudo-Galilean spaces. Additionally, related examples of matrices that belong to both spaces are provided. With a similar method, dual Galilean orthogonal matrices are obtained by using unit dual quaternions. Finally, we strengthen our work with examples and draw their figures to explore visual representations.