Thread: Ellipsoid: main axes, center from the general equation

1. Ellipsoid: main axes, center from the general equation

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

ax2+by2+c2+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

xTAx = 1 where xT = [x y z] and A = $\begin{bmatrix} a & h & q \\ h & b & f \\ g & f & c \end{bmatrix}$. By just solving eigenvectors of A, I'm able to get the orthogonal vectors for the main axis. However