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Math Help - Ellipsoid: main axes, center from the general equation

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    Ellipsoid: main axes, center from the general equation

    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

    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
    Last edited by kariantila; April 29th 2014 at 02:24 AM.
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