E. Geometric interpretation

An ellipsoid can be described by its general equation:

Ax² + By² + Cz² + 2Dxy + 2Exz + 2Fyz + 2Gx + 2Hy + 2Iz + J = 0

In matrix form:

If we define

then the center of the ellipsoid can be calculated as the vector b= - Q^-1 u

The ellipsoid's semi-axis a, b and c are given by:

where : λ₁, λ₂ and λ₃ are the eigenvalues of Q.

The corresponding normalized eigenvectors v1, v2 and v3 of the matrix Q describe the direction of the 3 principal axis of the ellipsoid. And the matrix V formed from the column vectors v1, v2 and v3 is the rotation matrix that describes the orientation of the ellipsoid in the reference frame.

Mapping a general ellipsoid to a sphere of radius R centered at (0,0)

This will be done in 5 steps:

Step 1. Translate the ellipsoid so that its center coincides with the origin(0,0)

Step 2. Align the principal axis of the ellipsoid with the x, y and z-axis of the reference frame

Step 3. Scale the principal axis of the ellipsoid so that they all have the same length (becoming a sphere)

Step 4. Rotate back the sphere to the original orientation of the ellipsoid

Step 5. Scale the sphere to a norm R

For any point x (x,y,z) at the surface of the ellipsoid:

Step 1 : The center of the ellipsoid is transfered to the origin (0,0) of the reference frame. The point x becomes x₁.

Step 5 : the radius of the sphere is changed from a to R. The point x₄ becomes x_map.

Summing it all (Step 1 to 5) :