Ruled Surfaces in Minkowski 3-space and Split Quaternion Operators

In this paper, we define and classify split quaternion operators. Then, we show that the split quaternion product of a split quaternion operator and a curve, which lies on Lorentzian unit sphere or on hyperbolic unit sphere, parametrizes a ruled surface in the 3-dimensional Minkowski space E13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_{1}^{3}$$\end{document} if the vector part of the operator is perpendicular to the position vector of the spherical curve. Moreover, the ruled surfaces are represented as 2-parameter homothetic motions in E13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}_{1}^{3}$$\end{document} by using semi-orthogonal matrices corresponding to the split quaternion operators. Finally, some examples are given to illustrate some applications of our main results.