Definition:Conic Section/Intersection with Cone/Ellipse

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 $\theta < \phi < \dfrac \pi 2 - \theta$.

That is, the angle between $D$ and the axis of $C$ is between that for which $K$ is a circle and that which $K$ is a parabola.

Then $K$ is an ellipse.