I have the equation of an ellipsoid of the form: $ax^2 + by^2 + cz^2 + 2fyz + 2gxz + 2hxy + 2px + 2qy +2rz + d = 0$
How does one translate the ellipsoid given above to the origin and have the equation of the form?: $ax^2 + by^2 + cz^2 + 2fyz + 2gxz + 2hxy = 1$
Solving the linear equations from the three partial derivatives, and substituting
$x\mapsto x + (b (g r - c p) + c h q + f^2 p - f (g q + h r))/(a b c - a f^2 - b g^2 - c h^2 + 2 f g h) $ $y\mapsto y + (-a c q + a f r + c h p - f g p + g^2 q - g h r)/(a b c - a f^2 - b g^2 - c h^2 + 2 f g h)$ $z\mapsto z + (-a b r + a f q + b g p - f h p - g h q + h^2 r)/(a b c - a f^2 - b g^2 - c h^2 + 2 f g h) $
yields the constant term $\frac{a\,b\,c\,d-a\,d\,f^{2}-b\,d\,g^{2}+2\,d\,f\,g\,h-c\,d\,h^{2}-b\,c\,p^{2}+f^{ 2}p^{2}-2\,f\,g\,p\,q+2\,c\,h\,p\,q-a\,c\,q^{2}+g^{2}q^{2}+2\,b\,g\,p\,r-2\,f\,h\,p\,r+2\,a\,f \,q\,r-2\,g\,h\,q\,r-a\,b\,r^{2}+h^{2}r^{2}}{a\,b\,c-a\,f^{2}-b\,g^{2}+2\,f\,g\,h-c\,h^{2}}$ which can be set to $-1$ by a scaling factor.