# Definition:Ellipse

## 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 $\phi$.

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.

### Focus-Directrix Property

Let $D$ be a straight line.

Let $F$ be a point.

Let $e \in \R: 0 < e < 1$.

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

- $e p = q$

Then $K$ is an **ellipse**.

### Equidistance Property

Let $F_1$ and $F_2$ be two points in the plane.

Let $d$ be a length greater than the distance between $F_1$ and $F_2$.

Let $K$ be the locus of points $P$ which are subject to the condition:

- $d_1 + d_2 = d$

where:

Then $K$ is an **ellipse**.

This property is known as the **equidistance property**.

## Parts of Ellipse

Consider an ellipse $K$ whose foci are $F_1$ and $F_2$.

### Center

The **center** of $K$ is the point midway between the foci.

By definition of the major axis and minor axis, this is the point where the major axis and minor axis of $K$ cross.

### Major Axis

The **major axis** of $K$ is the line segment passing through both $F_1$ and $F_2$ whose endpoints are where it intersects $K$.

### Minor Axis

The **minor axis** of $K$ is the line segment through the center of $K$ perpendicular to the major axis of $K$ such that its endpoints are the points of intersection with $K$.

### Vertex

A **vertex** of $K$ is either one of the endpoints of the major axis of $K$.

### Covertex

A **covertex** of $K$ is either one of the endpoints of the minor axis of $K$.

## Reduced Form

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.

## Also see

- Results about
**ellipses**can be found**here**.

## Historical Note

The word **ellipse** was provided by Apollonius of Perga, who did considerable work on establishing its properties.

## Linguistic Note

The word **ellipse** is pronounced with the stress on the second syllable: **el- lipse**.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies