Definition:Conic Section/Intersection with Cone/Circle

Definition

 * [[File:ConicSections.gif]]

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

Let a plane $D$ intersect $C$.

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

The nature of $K$ depends on the angle that $D$ makes with the axis of $C$.


 * ConicSectionsXsection.png

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

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

Let $\phi$ be the angle between $D$ and the axis of $C$.

Let $\phi = \dfrac \pi 2 - \theta$, thereby making $D$ perpendicular to the axis of $C$.

Then $D$ and $B$ are parallel, and so $K$ is a circle.