Lorentz Transformation via Geometry

I'm going to take $c=1$. The top orange line, which is the $t'$ axis in your drawing, is drawn with a slope of $1/0.6$; this follows from the definition of velocity. The bottom orange line, which is the $x'$ axis, has a slope of 0.6.

What's a little harder is to get the scale on the orange axes. The scale follows from the fact that area is preserved by Lorentz transformations. The velocity of 0.6 turns out to correspond to a Lorentz transformation in which a square is distorted into a parallelogram with its long axis stretched by a factor of 2, and its short axis contracted by $1/2$.

If you want to see this approached developed in more detail, see ch. 7 of this online book. (I'm the author.)

There are a couple of recent commercial textbooks that use similar geometric approaches:

• Mermin, It's About Time: Understanding Einstein's Relativity

• Takeuchi, An Illustrated Guide to Relativity