from the conic equation

given an equation

 A*x^2+B*y^2+C*z^2+ D*x*y + D*x*z + F*y*z -1=0

how do I extract the center [x0,y0,z0], the axes lengths [a,b,c], and the rotation angles [ex,ey,ez] of the ellipsoid that it describes....

[Matt J]

Not sure why you edited your original question. This version is much less clear. The original question was, given an equation

 A*x^2+B*y^2+C*z^2+ D*x*y + D*x*z + F*y*z -1=0

how do I extract the center [x0,y0,z0], the axes lengths [a,b,c], and the rotation angles [ex,ey,ez] of the ellipsoid that it describes.

[Rena Berman]

(Answers Dev) Restored edit.

Answer 1

[Matt J]

Rewrite in matrix form

   [x-x0,y-y0,z-z0]*Q*([x-x0;y-y0;z-z0])=1

where Q=[A,D,E;D,B,F;E,F,C]. The eigen-decomposition of Q will be

    Q=R*diag(1./[a,b,c])*R.'

where columns of the rotation matrix R are the axes of the ellipsoid. You will have to choose between one of many possible decompositions of R into Euler angles [ex,ey,ez]