Definition:Conic Section/Intersection with Cone/Circle

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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-Circle.png

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.


Historical Note

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

It appears in his Conics.


Sources