1) Basics
An ellipsoid centered at the origin can be defined by the algebraic form $$ Ax^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz - 1=0 \enspace . $$ This can be written in matrix notation $u^THu-1=0$ where $u = [x,y,z]^T$ and, $$ H = \begin{bmatrix} A & D & E\\ D & B & F\\ E & F & C \end{bmatrix} \enspace . $$ We can decompose $H$ such that $H = R^TDR$ where $R$ is a rotation matrix and $D$ is a diagonal matrix where the entries are the inverse of the sqaures of the ellipsoid radii.
2) The problem
Assuming we have a collection of ellipsoids, each rotated uniformly. What is then the distribution of the ellipsoid parameters $A,B,C,D,E,F$?
3) Possible solution methods
a) Gaussian distribution duality
The ellipsoid $u^THu-1=0$ is the level-set of a multivariate gaussian $\mathcal{N}(0,H^{-1})$. The Wishart distribution tells us about the distribution of the sample covariance matrix $H^{-1}$. However, this does not seem like we have a sample convariance matrix from the ellipsoid duality because the underlying distributions are not estimated, and it also seems the Wishart distribution assumes equal variance, which I don't think is the case given a population of randomly rotated ellipsoids.
b) Uniform generation of rotation matricies
Assuming I use one of the many described ways to generate uniform random rotations of SO(3), couldn't one then calculate the parameters $A,B,C,D,E,F$ abstractly from $H = R^TDR$ and infer the distribution of the parameters from my generation of $R$?. I have tried but it seems overly complicated algebraically.
c) Using the Haar-measure
The Haar-measure measures the uniformity of rotations (as far as I understand). Maybe it is possible to take a uniformity assumptions, and then use the Haar-measure to give specific probabilty distributions of the algebraic parameters? Though I did take an algebra course, this is not my strong side.
In summation
I want to find the distribution of the parameters of a population of ellipsoids given that I know they are rotated uniformly. I'm looking for advice on how to proceed.
(Edit 2) Progress 1
Looking briefly at my proposed method (c), it seems clear to me someone adept in algebra might be able to do this. Here: https://www.math.wisc.edu/~robbin/angelic/rotationprobability.pdf for instance, it is clear to me there are a lot of methods in algebra of which I have no chance of easily presenting a solution.
(Edit 2) Progress 2
I have been working on my proposed solution (b). Here are my preliminary results.
Using the a method for generating random rotations described in comments and code here: http://www.realtimerendering.com/resources/GraphicsGems/gemsiii/rand_rotation.c, I can substitute the random generating of points of the code by uniform stochastic variables $X,Y \sim Uniform(0,2\pi)$ and $Z \sim Uniform(0,1)$ into the equations. Thus I obtain new stochastic variables for each element of H. In my application, I'm specifically concerned with the parameters $C,E,F$ which can be described as stochastic variables in terms of $X,Y,Z$ by \begin{align*} C &= ((a-b)Z^2+2(b-a)Z)\cos(X)^2+(c-a)Z^2+2(a-c)Z+c\\ E &= -\sqrt{Z}\sqrt{2-Z}\cdot(\\ &(b-a)Z\sin(Y)\cos(X)^3\\ &+(a-b)Z\sin(X)\cos(Y)\cos(X)^2\\ &+((a-c)Z-b+c)\sin(Y)cos(X)\\ &+((c-a)Z+a-c)\sin(X)\cos(Y)\\ &)\\ F &= -\sqrt{Z}\sqrt{2-Z}\cdot(\\ &((a-b)Z\cos(X)^3+((c-a)Z-c)\cos(X)+\sin(X)a)\sin(Y)\\ &+(a-b)Z\cos(Y)\cos(X)^3+((c-a)Z+b-c)\cos(Y)\cos(X)\\ &) \end{align*} I may be able to eventually write expressions for the distributions of the above. The automatic tools I had available gave up trying.
EDIT 1: Typos and added another possible solution method
EDIT 2: Added progress sections