Special Relativity - Length Contraction

[Destroxia]

1. Homework Statement

Why does the non-relative simultaneous frame have an additional term of $v \Delta t$ along with the contracted length $L$ for $\Delta x$?

2. Homework Equations

$L + v \Delta t = \Delta x$ ---- (1)

$\Delta x = \gamma \Delta x' = \gamma L_{\star}$ ---- (2)

3. The Attempt at a Solution

What I understand:
I understand why Length AB within the above diagram is $L_{\star}$ (Proper Length), it is because that frame is moving with the object (ct', x'), and thus the object appears to be still, and the measurement for the distance shouldn't change if you are still with the object. I understand that AD is $\Delta x$ in the rest frame (ct, x), and that DB is $c \Delta t$ in the rest frame (ct, x), as well.

What I don't understand:
What I am not understanding in the least, is the AC section... How is that $L$ (Contracted Length)? Shouldn't contracted length just be whatever the rest observer is seeing (i.e. The length AD)? Why is the rest observer seeing the contracted length AC, with C along the world line of B? What exactly is CB? I know the object appears to be moving distance $v \Delta t$, so that's what CD is, but I still can't see why the new length $L$ isn't AD, and not AC.

 

[jedishrfu]

The observer isn't seeing AD they are seeing the length AC.

This is related to the box traveling past you at near light speed. What you will see is not a foreshortened box but a slightly rotated box.

http://www.spacetimetravel.org/tompkins/node1.html

and Terrell's observation:

https://en.wikipedia.org/wiki/Terrell_rotation