Definition:Conic Section/Intersection with Cone/Ellipse

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Definition

ConicSections-All.png

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 $\phi$.


ConicSections-Ellipse.png


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.


Historical Note

This construction of a conic section was documented by Apollonius of Perga.

It appears in his Conics.


Sources