Definition:Conic Section/Intersection with Cone
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 $\phi$.
Circle
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.
Parabola
Let $\phi = \theta$.
Then $K$ is a parabola.
Ellipse
Let $\theta < \phi < \dfrac \pi 2$.
That is, the angle between $D$ and the axis of $C$ is between that for which $K$ is a circle and that which $K$ is a parabola.
Then $K$ is an ellipse.
Hyperbola
Let $\phi < \theta$.
Then $K$ is a hyperbola.
Note that in this case $D$ intersects $C$ in two places: one for each nappe of $C$.
Degenerate Hyperbola
Let $\phi < \theta$, that is: so as to make $K$ a hyperbola.
However, let $D$ pass through the apex of $C$.
Then $K$ degenerates into a pair of intersecting straight lines.
Slicing Plane
The plane $D$ is known as the slicing plane of $C$ for $K$.
Tilting Angle
The inclination $\phi$ of the slicing plane with the axis of $C$ is known as the tilting angle of $D$ for $K$.
Also see
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.)
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $2$: The Logic of Shape: Problems for the Greeks
- Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConicSection.html