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