Here is a geometric derivation of the Lorentz Transformation using hyperbola geometry: \begin{align*} \left[\begin{array}{c} x'\\ ct' \end{array}\right] = \left[\begin{array}{c c} \cosh\alpha & -\sinh\alpha\\ -\sinh\alpha & \cosh\alpha \end{array}\right] \left[\begin{array}{c} x\\ ct \end{array}\right] \end{align*}

On the spacetime diagrams, the large black dot is the event we're trying to describe. The small black dot indicates a right angle for a hyperbolic triangle.

Starting with the time coordinate:

Now thinking about the spatial coordinate: