Okay.
I don't get why this is being discussed though. I would think this phenomenon is a consequence of not including the relativity of simultaneity. The OP has been asking for mathematics, so I will now give him mathematics.
Frame S : A^{\mu}=(ct,x,y,z)
Frame S' : B^{\nu}=(ct^\prime,x^\prime,y^\prime,z^\prime)
We assume the origins coincide at t=t'=0 and that the orientation of axes are equal. We also assume that S' frame moves with velocity v in the x-direction in the S frame. We apply the Lorentz transform:
B^{\nu}=\Gamma^{\nu}_{\mu}A^{\mu}= \left( \begin{array}{cccc}\gamma & -\gamma\frac{v}{c} & 0 & 0 \\ -\gamma\frac{v}{c} & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array} \right)\left(\begin{array}{c} ct \\ x \\ y \\ z \end{array}\right)=\left( \begin{array}{c} \gamma\left(ct-\frac{vx}{c}\right) \\ \gamma (x-vt) \\ y \\ z \end{array}\right)
So we have
t^\prime = \gamma \left(t-\frac{vx}{c^2}\right)
x^\prime = \gamma (x-vt)
y^\prime = y
z^\prime = z
The metric of flat space-time is \eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1) which by the definition of the metric gives us for frame S:
ds^2=\eta_{\mu\nu}dA^{\mu}dA^{\nu}=c^2dt^2-dx^2-dy^2-dz^2
and for frame S':
ds^2=\gamma^2c^2dt^2-\gamma^2\frac{v^2}{c^2}dx^2-\gamma^2dx^2+\gamma^2v^2dt^2-dy^2-dz^2=\left( \gamma^2(c^2+v^2) \right)dt^2-\left( \gamma^2\left(\frac{v^2}{c^2}+1\right) \right)dx^2-dy^2-dz^2
The OP can check this for himself if he wishes.
Light travels along null geodesics, meaning ds=0, and these equations then represent light cones in the two frames. Now, note the following:
(1) If we look at 3D slices of constant t in S, we get spheres.
(2) If we look at slices of constant t' in S', we get spheres (Remember that x^\prime \neq x).
(3) If we look at slices of constant t in S' or at constant t' in S, we get ellipsoids.
If the OP doubts this, by all means try to disprove it.
These three points confirm that the light sphere/light ellipsoid discussion is simply nonphysical. It is a result of combining measurements from different frames, which is illegal. The OP should now be convinced that his earlier claims about Euclidean spaces and whatnot are flawed.
