General Method for Mapping an Ellipsoid to Unit Sphere

[StochasticSignal]

TL;DR Summary

How do I map vectors that satisfy an ellipsoid to those that satisfy a sphere?

I have been working on a problem for a while and my progress has slowed enough I figured I'd try reaching out for some more experience. I am trying to map a point on an ellipsoid to its corresponding point on a sphere of arbitrary size centered at the origin. I would like to be able to shift any point with the following formula:
$$p_{sphere} = \mathbf A (p_{ellipsoid} - c)$$
I am trying to determine the A matrix that will transform a point from a fully defined ellipsoid E matrix.

Where an ellipsoid (or other quadric surface) can be defined with the following:
$$
\left[
\begin{matrix}
x & y & z & 1\\
\end{matrix}
\right]
\mathbf E
\left[
\begin{matrix}
x\\
y\\
z\\
1\\
\end{matrix}
\right]
=
0
$$
$$
Where\
\mathbf E =
\begin{pmatrix}
a & h & g & p\\
h & b & f & q\\
g & f & c & r\\
p & q & r & d
\end{pmatrix}
=
\left[
\begin{array}{c|c}
\mathbf M_{3x3} & \vec{\mathbf n}\\
\hline
\vec{\mathbf n}^T & d\\
\end{array}
\right]
$$
And c can be determined like so:
$$
c = -\mathbf M^{-1}\vec{\mathbf n}
$$
Determining the c offset is straightforward and I have an intuitive understanding of that derivation. However, rotating and scaling the ellipsoid is the part that is escaping me. Applying different combinations of eigenvectors and eigenvalues in ways that I thought would make sense do not deliver the results I am expecting (hoping) for.

In some other discussions I have had, the following equation can be used to find the A matrix:
$$
\mathbf A = \frac{F}{\sqrt{\vec{\mathbf n}^T \mathbf M^{-1} \vec{\mathbf n} - d}} \mathbf M^{1/2}$$
Where $M^{1/2}$ is the matrix square root of M and F is a chosen sphere radius. However, I have no intuition for the mathematical meaning of this equation, how to arrive at it, or even a sense of what it's saying.

If someone has some advice on how to approach this problem, I would greatly appreciate it if you could share it with me.

Thank you for your time.

 

[jvicens]

Hi there,

Thank you for reaching out for help with your problem. It sounds like you have put a lot of effort into trying to solve this and I am happy to offer some advice.

First of all, it's great that you have a good understanding of the c offset and how to determine it. That is an important step in the process. Now, for the rotation and scaling part, I would recommend approaching it using matrix transformations.

To rotate an ellipsoid, you can use the rotation matrix. This matrix will rotate the ellipsoid around the origin by a certain angle. You can use this matrix to rotate your ellipsoid to the desired orientation.

Next, for scaling, you can use the scaling matrix. This matrix will scale the ellipsoid along the x, y, and z axes. You can choose the scaling factors based on the desired size of your sphere.

Once you have applied the rotation and scaling transformations to your ellipsoid, you can use the formula you mentioned to shift the points to the corresponding points on the sphere. The A matrix in this formula represents the combined rotation and scaling transformations.

As for the equation you mentioned for finding the A matrix, it is a bit complex and may not be necessary for your problem. It seems to be a combination of the rotation and scaling matrices, but it may be more complicated than what you need.

I hope this helps and gives you some ideas on how to approach your problem. Good luck!