On geometric interpretations of split quaternions

Abstract

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).

Introduction to Cartan numbers and some geometric applications

“…and tr(M) = 0 for the matrix M. If we calculate S q (v), we find (11). R 𝜃 is a pseudo rotation matrix since R 𝜃 T I ⋆ R 𝜃 = I ⋆ and detR 𝜃 = 1.…”

“…Lately, since split quaternions are used to state Lorentzian rotations, there are studies on geometric and physical applications of split quaternions that call for solving split quaternionic equations. In general, split quaternions are used to express rotation and reflection transformations for timelike and spacelike vectors in Minkowski 3‐space because the units of split quaternions are compatible with the unit base vectors in Minkowski 3‐space 7–12 . Rotations around the lightlike axis were also investigated in previous studies, 13,14 but in the study, frames consisting of one spacelike and two lightlike vectors were presented, and rotations around the lightlike axis were examined.…”

“…In general, split quaternions are used to express rotation and reflection transformations for timelike and spacelike vectors in Minkowski 3-space because the units of split quaternions are compatible with the unit base vectors in Minkowski 3-space. [7][8][9][10][11][12] Rotations around the lightlike axis were also investigated in previous studies, 13,14 but in the study, frames consisting of one spacelike and two lightlike vectors were presented, and rotations around the lightlike axis were examined. While creating this frame, the scalar product of two lightlike vectors is taken as 1, but the scalar product of any two different lightlike vectors is not 1.…”

Fixed Points for Cubic Coquaternionic Maps

“…Although never as popular as their famous "cousins" quaternions, coquaternions have recently been attracting the attention of mathematicians and physicists who recognize the potential of applications of these hypercomplex numbers numbers [1,2,3,9,10,11,12,13,14,15,16].…”