Conditions for a plane and an ellipsoid NOT to intersect I know that the intersection of an ellipsoid and a plane is an ellipse. However, how can I derive the conditions for an ellipsoid and a plane not to intersect? Suppose I have a plane
$$Ax+By+Cz+D=0$$
and I have an ellipsoid
$$\frac{x^2}{a^2}+\frac{y^2}{b^2} 
+\frac{z^2}{c^2}=1$$
How I can write down the conditions that they won't intersect? I was thinking that they will not intersect if for any given point $(x,y,z)$ the plane is parallel to the tangent parallel to the ellipsoid at $(x,y,z)$. However, how will $D$ come into play? Do we use $D$ to make sure that the tangent hyperplane and the ellipsoid don't coincide with one another? If so, how can I impose this condition?
 A: The plane does not intersect the ellipsoid iff its pole is in the interior of the ellipsoid. In homogeneous coordinates, this point is $[Aa^2:Bb^2:Cc^2:-D]$. Plugging this into the equation of the ellipse leads to the inequality $$(Aa)^2+(Bb)^2+(Cc)^2 \lt D^2.$$  

In a comment to another answer, you ask about the distance from this plane to the ellipsoid. This is equal to the distance between the plane and a parallel tangent plane on the same side of the ellipse. This is fairly easily computed by using the ellipsoid’s dual: a plane with equation $Px+Qy+Rz+S = 0$ is tangent to the ellipsoid iff the coefficients of the equation satisfy the dual conic equation $$P^2a^2+Q^2b^2+R^2c^2-S^2=0.$$ So, the two tangent planes parallel to $Ax+By+Cz+D=0$ have a constant term of $\pm\sqrt{(Aa)^2+(Bb)^2+(Cc)^2}$ instead of $D$. The distances between these tangent planes and the reference plane are then $${\left|D\pm\sqrt{(Aa)^2+(Bb)^2+(Cc)^2}\right| \over \sqrt{A^2+B^2+C^2}}.$$ Choose the least of the two values that result. Since this ellipsoid is centered on the origin, that will be given by the solution for $S$ that has the same sign as $D$.
A: The idea is to perform a linear scaling of the space such that the ellipsoid becomes a unit sphere $x^2+y^2+z^2=1$. The two objects intersect before this scaling iff they do after it. $x$ becomes $ax$, $y$ becomes $by$ and $z$ becomes $cz$:
$$Aax+Bby+Ccz+D=0$$
The (new) plane and sphere do not intersect iff the origin's distance from the plane is more than 1:
$$\left|\frac D{\sqrt{(Aa)^2+(Bb)^2+(Cc)^2}}\right|>1$$
$$D^2>(Aa)^2+(Bb)^2+(Cc)^2$$
They have a tangential intersection iff this value is 1 ($=$ instead of $>$ in the second equation) and a normal elliptical intersection iff this value is less than 1 ($<$).
