By analogy with the Euclidean case, we define a hyperbolic rotation through the relations \begin{equation} \pmatrix{x\cr \yy} = \pmatrix{\cosh\beta& \sinh\beta\cr \sinh\beta& \cosh\beta\cr} \pmatrix{x'\cr \yy'} \label{hyperrot} \end{equation} This corresponds to “rotating” both the $x$ and $\yy$ axes into the first quadrant, as shown in Figure 4.2. While this may seem peculiar, it is easily verified that the “distance” is invariant, that is, \begin{equation} x^2 - \ysq \equiv x'^2 - \yy'^2 \end{equation} which follows immediately from the hyperbolic trigonometric identity (4) of §4.1.