Definition:Conic Section/Intersection with Cone/Parabola

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 $\phi = \theta$.

Then $K$ is a parabola.

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: $1$.
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.6$: Apollonius (ca. $\text {262}$ – $\text {190}$ B.C.)
• 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)
• 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $2$: The Logic of Shape: Problems for the Greeks
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): parabola

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