Eccentricity

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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.

The eccentricity of a vertex in a graph is the length of the longest of all shortest paths between this vertex and every other vertex in the graph. That is, the eccentricity of a vertex is the maximum distance you need to go to get to any other vertex. The smallest eccentricity is called the radius of the graph.

Ec`cen*tric"i*ty (?), n.; pl. Eccentricities (#). [Cf. F. excentricit'e.]

1.

The state of being eccentric; deviation from the customary line of conduct; oddity.

2. Math.

The ratio of the distance between the center and the focus of an ellipse or hyperbola to its semi-transverse axis.

3. Astron.

The ratio of the distance of the center of the orbit of a heavenly body from the center of the body round which it revolves to the semi-transverse axis of the orbit.

4. Mech.

The distance of the center of figure of a body, as of an eccentric, from an axis about which it turns; the throw.

© Webster 1913.