# Definition:Conic Section/Intersection with Cone/Hyperbola

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## 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 hyperbola.

Note that in this case $D$ intersects $C$ in two places: one for each nappe of $C$.

## Historical Note

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

It appears in his *Conics*.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.6$: Apollonius (ca. $262$ – $190$ B.C.) - 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): Entry:**hyperbola**

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