Conditions for a plane and an ellipsoid NOT to intersect

Question

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?

Answer 1

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 $$(Aa)^2+(Bb)^2+(Cc)^2 \lt D^2.$$

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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 $D$.

Answer 2

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 ($<$).