Definition:Conic Section/Intersection with Cone/Circle

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

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

Transverse Section

The plane $D$ which is parallel to $B$, whose intersection with the cone is a circle, is known as a transverse section of the cone.

Historical Note

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

It appears in his Conics.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1$.
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conic (conic section)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conic (conic section)

• Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConicSection.html