Eccentricity is represented by e, which is an unfortunate historical accident that results from mathematicians (like Euler) not having the Internet at the time so they could divvy up letters of the Latin alphabet to important values. Or something like that. Anyway...

e does not specifically refer to ellipses; it can describe the shape of any conic section. A circle has e = 0, an ellipse has 0 ≤ e < 1, a parabola (and its degenerate case, a straight line) has e = 1, and a hyperbola has 1 < e. Why is this, you ask? Well, let me get to that...

Every conic section can be defined by a focus and a directrix (or semi-transverse axis, as Webster 1913 so nicely put it), much like the original definition of a parabola, only instead of having the distance between the focus and the directrix be the same, they can have a common ratio. If P is any point on the curve, D is the respective point on the directrix, and F is the focus, then PF/PD = e.

Given the eccentricity and the distance from the directrix to its focus, one can plot any conic section on a polar graph with its focus (or one of them, if it's an ellipse or hyperbola) on the pole. The equations are (e = eccentricity, d = distance from directrix to pole):

r = ed/(1 + e cos θ) if the directrix is to the right of the pole

r = ed/(1 - e cos θ) if the directrix is to the left of the pole

r = ed/(1 + e sin θ) if the directrix is above the pole

r = ed/(1 - e sin θ) if the directrix is below the pole

Too many people assume that eccentricity refers only to ellipses and the degenerate case of the circle. However, this is clearly not the whole picture.

First in a series of MathRants by PMDBoi.