No further questions so I guess I've lost everyone? Ok let's get deeper into the math.

Here is the Loedel diagram of Alice leaving Bob on Earth at .6c. It would not be the same diagram as Earth being the reference frame and Bob and Alice leaving Earth at 1/3c in opposite directions. In fact the reference frame is empty space here represented by the half speed Loedel velocity $$v_h$$. Nothing is referenced to this stationary reference frame and everything is referenced to the Earth/Bob velocity 0f -1/3c. There are no rules in math where a reference frame needs to be stationary. Breaking this rule means I can set any velocity axis as the reference simultaneously on one spacetime diagram.

It does alter the main equation slightly though. It introduces gammas with subscripts: $$Y_o$$ for the reference frame and $$Y_1$$ for the subsequent frame depending on what perspective you choose. If you want to look from Bob`s perspective the main equation is

$$(ct′)^2=(ct)^2−Y_0x^2$$

from Alice's perspective it's

$$(ct′)^2=(ct)^2−Y_1x^2$$

What I'm doing mathematically is layering the spacetime diagrams into one diagram. Relativity doesn't do this but again this thread is a math presentation that could be applied to relativistic physics to arrive at the same answers.

You may also notice that I've layered 2 separate scenarios for Alice, one where she turns back to Bob at t'=2 and the other where she just keeps going.

I've developed a quick formula for calculating $$v_h$$ but it's only applicable to Minkowski diagrams where the reference frame is stationary. It doesn't strictly apply to a Loedel diagram where $$v_h=0$$) always. There's no need to develop a more general formula as I'll show Minkowski mostly doesn't get into the trouble of having empty space as a stationary reference frame.

$$v_h=Yv/(Y+1)$$.

Notice the thin green curved line which represents the hyperbolas generated by the main equation. They intersect all the velocity lines at the same time on their universally accurate atomic clocks which all beat at the same time rate within their frames. Notice the thick green line which is the line of simultaneity from the Loedel perspective. It also intersects the two blue velocity lines at the same proper time. It's not physically correct to call the Loedel line of perspective simultaneity the line of proper simultaneity but I use it as such mathematically. It seems to me that the hyperbolas generated by the main equation suggest proper time simultaneity but I've never gotten a straight answer to this question.

The two thin red lines are Bob and Alice's perspective lines of simultaneity. If I added more perspectives, their lines of simultaneity would look like a hysteresis eye around the thick green line which is the only one that joins 2 proper times as endpoints. The Loedel line of simultaneity (which I've tried to coin) is not discussed in relativity but to me, mathematically, it has significant attributes especially that it provides an unambiguous perspective of proper time.

Notice there are two light lines emanating from Bob and Alice signalling each other simultaneously from the Loedel perspective at t'=2. I will show how light lines are used to stitch together the simple graphical construct that will be the mathematical building block of graphically representing all the physical phenomena of relativity. But for now, I will show how the light signals are affected when this Loedel diagram is depicted as Minkowski and reverse-Minkowski diagrams.