I've only seen derivations of the Lorentz transformation using hyperbola like $x^2-y^2=-1$ as opposed to hyperbola like $y=1/x.$ I've been trying to derive the Lorentz transform for rectangular hyperbolas in the plane, using $H_s=\begin{bmatrix} e^s&0\\ 0&e^{-s} \end{bmatrix},$ where $s$ is the Rapidity. The rectangular hyperbola spacetime diagram is basically a $45$ degree rotation clockwise of the standard diagram as shown below.
How does one derive the Lorentz transform for rectangular hyperbolas? And why does no one derive it this way?
