= Intersection plane of parallel congruent double cones
Let two <double circular cones> have vertices $a_1,a_2$, common unit axis $n$, and semi-angle $0<\alpha<\pi/2$. Put $Q=\cos^2\alpha\,I-nn^T$ and $b=a_1-a_2\ne0$. Their equations are $(x-a_i)^TQ(x-a_i)=0$. Subtracting cancels the common quadratic term, so every intersection point satisfies
$$
(Qb)\cdot\left(x-\frac{a_1+a_2}{2}\right)=0.
$$
Since $Q$ has eigenvalues $\cos^2\alpha$ perpendicular to $n$ and $-\sin^2\alpha$ along $n$, $Qb\ne0$. Thus the intersection is contained in a plane, possibly with an empty intersection. Its unit normal is $Qb/|Qb|$, with $|Qb|^2=|b|^2\cos^4\alpha+(n\cdot b)^2(1-2\cos^2\alpha)$. Degenerate endpoint semi-angles can make the normal vanish and must be treated separately.
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