SPLIT QUATERNIONS AND ROTATIONS IN SEMI EUCLIDEAN SPACE E42

Title & Authors
SPLIT QUATERNIONS AND ROTATIONS IN SEMI EUCLIDEAN SPACE E42
Kula, Levent; Yayli, Yusuf;

Abstract
We review the algebraic structure of $\small{\mathbb{H}{\sharp}}$ and show that $\small{\mathbb{H}{\sharp}}$ has a scalar product that allows as to identify it with semi Euclidean $\small{{\mathbb{E}}^4_2}$. We show that a pair q and p of unit split quaternions in $\small{\mathbb{H}{\sharp}}$ determines a rotation $\small{R_{qp}:\mathbb{H}{\sharp}{\rightarrow}\mathbb{H}{\sharp}}$. Moreover, we prove that $\small{R_{qp}}$ is a product of rotations in a pair of orthogonal planes in $\small{{\mathbb{E}}^4_2}$. To do that we call upon one tool from the theory of second ordinary differential equations.
Keywords
hyperbolic number;split quaternion;generalized inverse;rotation;timelike plane of index 1;timelike plane of index 2;spacelike plane;
Language
English
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References
1.
O. P. Agrawal, Hamilton operators and dual-number-quaternions in spatial kinematics, Mech. Mach. Theory. 22 (1987), no. 6, 569-575

2.
J. Inoguchi, Timelike Surfaces of Constant Mean Curvature in Minkowski 3-Space, Tokyo J. Math. 21 (1998), no. 1, 140-152

3.
L. Kula and Y. Yayli, Dual Split Quaternions and Screw Motions in Minkowski 3-space, Iranian Journal of Science and Technology (Trans A), preprint

4.
B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983

5.
P. J. Ryan, Euclidean and non-Euclidean Geometry, Cambridge University Press, Cam- bridge, 1986

6.
Y. Tain, Universal Factorization Equalities for Quaternion Matrices and their Applica- tions, Math. J. Okoyama Univ. 41 (1999), 45-62

7.
J. L. Weiner and G. R. Wilkens, Quaternions and Rotations in E4, Amer. Math. Monthly 112 (2005), no. 1, 69-76