Eccentricity

Definition

Let $Q$ be a non-degenerate conic section in the Euclidean plane $\mathbb{R}^2$. If $Q$ is a circle, its eccentricity is $e = 0$. Otherwise, let $F$ be a focus of $Q$ and $\ell$ the directrix 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.