Say I have a conic surface in $R³$ defined as $C(t,u) = v + u f_1 \cos t + uf_2\sin t + u f_3$ with $v$ being the apex of the cone and $f_1f_2f_3$ being vectors, which is basically the unit cone $x² + y² = z²$ transformed into the coordinate system defined by $v,f_1,f_2,f_3$.
How would I calculate the axes $f_1' f_2' f_3'$ of a conic surface $C'$ so that $C'$ is coplanar to $C$ and the axes are orthogonal?