# 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