Is there a general way to check whether a point is on a quadratic surface given that the principal axes do not need to coincide with the coordinate axes and that the quadric's centroid does not need to coincide with the centroid of the coordinate system?
Going from the normal form of the quadric to a translated quadric is easy (subtracting the center from the point that is to be checked), but how can the rotation be taken into account?
If ${\bf x} = A \bf X$, then ${\bf x}^T Q {\bf x} = {\bf X}^T A^T Q A \bf X $.