I have a quadratic surface defined by $$ Ax^2 + By^2 + Cy^2 + Dxy + Eyz + Fzx + Gz + Hy + Iz + J = 0 $$
I know the values of the constants $A,B,C,D,E,F,G,H,I,J$.
I need to make another surface $1$ unit larger radius in all directions.
I understand the $G$, $H$ and $I$ are the centre of the quadratic surface and the $J$ relates to the radius.
So all I want to do is find a new value of $J$ that produces a new surface slightly ($1$ unit) larger than the old one with the same orientation and centre point.
Any help would be much appreciated.
The planar version of this question is Enlarging an ellipses along normal direction. As you can see there, an enlarged ellipse is not an ellipse or any quadric curve, unless the original ellipse was a circle. Rotating an ellipse around its axis, we conclude that for ellipsoids of revolution (at least) the result of enlargement is not an ellipsoid.
Thus, there is no $J$ that will do the job. You can obtain a parametric form of the enlarged surface by beginning with a parametrization $\vec r(u,v)$ of the ellipdoid and calculating $$\vec r + \frac{\vec r_u \times \vec r_v}{|\vec r_u \times \vec r_v|} \tag{1}$$ (Or with $-$ instead of $+$, if the cross product points inside). You will quickly find that (1) is not a pretty formula to look at.