Definition:Conic Section/Intersection with Cone/Degenerate Hyperbola

Definition

Let $C$ be a double napped right circular cone whose base is $B$.

Let $\theta$ be half the opening angle of $C$.

That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$.

Let a plane $D$ intersect $C$.

Let $\phi$ be the inclination of $D$ to the axis of $C$.

Let $K$ be the set of points which forms the intersection of $C$ with $D$.

Then $K$ is a conic section, whose nature depends on $\phi$.

Let $\phi < \theta$, that is: so as to make $K$ a hyperbola.

However, let $D$ pass through the apex of $C$.

Then $K$ degenerates into a pair of intersecting straight lines.

Sources

• 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies