Edit: This is a double posting (can be removed, my apologies). Please reply to the newer thread.
Hello members,
I'm writing to find a solution to a seemingly simple problem that does not seem so simple anymore. I'm trying to fit an ellipsoid to a small number (9-12) of 3D points. An external program does this for me and it returns the results as coefficients of the general equation of a 3D quadratic surface
ax^{2}+by^{2}+c^{2}+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0
Generally it is guaranteed that the object is an ellipsoid and all coefficients are nonzero. I'd like to analyze the fitted ellipsoid and for this I would need the length and orientation of the main axes and the center of the ellipsoid. I'm able to get these by approximation but would like to have a better solution.
I've learned that this equation can be expressed in quadratic form
x^{T}Ax = 1 where x^{T} = [x y z] and A = . By just solving eigenvectors of A, I'm able to get the orthogonal vectors for the main axis. However