---
tag: 0GCWJS2EGB
kind: definition
name: Eccentricity
msc: [51N20]
defines: 0HVX863SAJ
uses: [00HJCMC2CD, 0RBKKNH4GV, 0E5CXYNEPX]
sources:
  - url: https://en.wikipedia.org/wiki/Eccentricity_(mathematics)
    locator: Definitions
  - url: https://en.wikipedia.org/wiki/Eccentricity_(mathematics)
    locator: Values
---

## Definition

Let $Q$ be a non-degenerate [[conic section|00HJCMC2CD]] in the Euclidean plane $\mathbb{R}^2$.
If $Q$ is a circle, its eccentricity is $e = 0$. Otherwise, let $F$ be a [[focus|0RBKKNH4GV]] of $Q$ and $\ell$ the [[directrix|0E5CXYNEPX]] for $F$. The eccentricity of $Q$ is the constant ratio
$$e = \frac{d(P, F)}{d(P, \ell)}, \qquad P \in Q.$$

## Remarks

An ellipse has $0 \leq e < 1$, a parabola has $e = 1$ and a hyperbola has $e > 1$.
For an ellipse or a hyperbola with the semi-major axis $a$ and the foci at the distance $c$ from the centre, $e = c / a$.
Two conic sections with the same eccentricity are similar.

## Explanations

### L1 human

The eccentricity tells how stretched a conic is. A circle has $e = 0$. The ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ has $c = 4$ and so $e = 4/5$. As $e$ gets near $1$, the ellipse becomes long and thin.