Eccentricity

Eccentricity — The Shape-Defining Ratio

An ellipse is defined by its two fixed foci and the constant sum of distances property. But how stretched is it? That is what eccentricity measures. It is a single number, always between 0 and 1 for an ellipse, that tells you how far the ellipse is from being a circle.

The eccentricity $e$ is defined as the ratio of two distances measured from the centre of the ellipse:

If the distance from the centre to a focus is $c$, and the distance from the centre to a vertex is $a$, then

Since the focus lies inside the ellipse and the vertex lies on the ellipse, $c < a$ for any non-degenerate ellipse. Therefore $0 < e < 1$. For a circle, the two foci coincide at the centre, so $c = 0$ and $e = 0$.

Expressing the Focus Distance in Terms of Eccentricity

From the definition $e = c/a$, we can immediately write the distance from the centre to a focus as

This is a very convenient form. Instead of carrying $c$ through calculations, you can replace it with $ae$. The vertices, which are at distance $a$ from the centre, remain at $(\pm a, 0)$ for the horizontal ellipse. The foci, originally at $(\pm c, 0)$, become $(\pm a e, 0)$.

Relating $a$, $b$, and $e$

You already know the fundamental relationship for an ellipse: $c^2 = a^2 - b^2$, where $b$ is the semi-minor axis length. Substitute $c = a e$ into this:

Rearrange to solve for $b^2$:

This gives a direct link between the semi-minor axis, the semi-major axis, and the eccentricity. It also lets you express $b$ in terms of $a$ and $e$:

The Standard Ellipse Equation in Terms of $e$

The standard equation of an ellipse with centre at the origin and major axis along the $x$-axis is

Using $b^2 = a^2 (1 - e^2)$, you can rewrite this entirely in terms of $a$ and $e$:

This form is particularly useful when you know the eccentricity and the semi-major axis but not the semi-minor axis directly. …