The methods you presented are wrong.
Does this really need explaining?
Try
here
Yes, I thought that you made this mistake, the Lorentz transforms do not transform a right angle into a right angle, You made the obvious mistake of considering the angle between the surface normal and the velocity vector. While the angle is $$\pi/2$$ in the axle frame, it is LESS than $$\pi/2$$ in any other frame. Had you considered the angle between the tangent to the plane and the velocity, you would have avoided the goof.
Of course
I did, this is how I knew you were wrong. Besides, I notice that for all your bluster, you did no derivation whatsoever.
No, this is a qualitatively different situation to that described by the aberration formula.
Yet, if what you were claiming were true , it would apply to the angle between a ray of light (representing vector 1) and the direction source - observer (representing vector 2). The aberration formula preservers angles multiple of $$\pi$$, your erroneous claim would invalidate it.
For example, if you start from $$cos \theta'=\frac{cos \theta +v/c}{1+ v/c. cos \theta}$$
and you make
1. $$\theta=\pi/2$$ you get
$$cos \theta' =v/c$$, so, $$\theta'< \pi/2$$
2. $$\theta=0$$ you get
$$cos \theta' =1$$, so, $$\theta'=0$$
Hang tight while I translate to tex...
(May not be until tomorrow. I'm supposed to be pulling up carpet today.)
Wait a sec, for
two days you were chomping at the bit to "show me" how to do the angle transformations, now that I showed you where you made the mistake you need more time "to translate to tex"? I thought you were itching to prove me wrong and that you had all the calculations at your fingertips.