# Definition:Conic Section

## Definition

### Intersection with Cone

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 $\phi$.

### Focus-Directrix Property

A **conic section** is a plane curve which can be specified in terms of:

- a given straight line $D$ known as the directrix
- a given point $F$ known as a focus
- a given constant $\epsilon$ known as the eccentricity.

Let $K$ be the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition:

- $(1): \quad q = \epsilon \, p$

Then $K$ is a **conic section**.

Equation $(1)$ is known as the **focus-directrix property** of $K$.

## Reduced Form

Let $K$ be a conic section.

Let $K$ be embedded in a cartesian plane such that:

for some $c \in \R_{\ge 0}$.

This can be interpreted in the contexts of the specific classes of conic section as follows:

### Reduced Form of Ellipse

Let $K$ be an ellipse embedded in a cartesian plane.

$K$ is in **reduced form** if and only if:

- $(1)$ its major axis is aligned with the $x$-axis
- $(2)$ its minor axis is aligned with the $y$-axis.

### Reduced Form of Hyperbola

Let $K$ be a hyperbola embedded in a cartesian plane.

$K$ is in **reduced form** if and only if:

- $(1)$ its major axis is aligned with the $x$-axis
- $(2)$ its minor axis is aligned with the $y$-axis.

### Reduced Form of Circle

Let $K$ be a circle embedded in a cartesian plane.

$K$ is in **reduced form** if and only if its center is located at the origin.

### Reduced Form of Parabola

Let $K$ be a parabola embedded in a cartesian plane.

As a Parabola has no Center, it is not possible to define the reduced form of a parabola in the same way as for the other classes of conic section.

Instead, $K$ is in **reduced form** if and only if:

- $(1)$ its focus is at the point $\tuple {c, 0}$
- $(2)$ its directrix is aligned with the line $x = -c$

for some $c \in \R_{> 0}$.

## Also known as

A **conic section** is also known just as a **conic**.

## Also see

- Results about
**conic sections**can be found**here**.

## Historical Note

The **conic sections** were first studied, by Menaechmus in around $\text {350 BCE}$.

He was the first to identify them as different types of slices through a **right circular cone**.

Other early investigations were made by Conon of Samos at around $\text {245 BCE}$.

Apollonius of Perga made an extensive study of them in around $\text {225 BCE}$, the results of which he published his book *Conics*.

He demonstrated rigorously that they can all be generated by different sections of the surface of a **right circular cone**.

Apollonius of Perga was also the first to recognise that a double napped cone is used to generate the hyperbola.

In the $17$th century, **conic sections** were initially investigated using the techniques of analytic geometry.

This was mainly initiated by Jan de Witt, who introduced the focus-directrix property in around $\text {1659}$ – $\text {61}$.

This was also done independently by John Wallis in $\text {1655}$.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**conic (conic section)** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**conic section** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**section (plane section)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**conic (conic section)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**conic section** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**section (plane section)**

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