Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.)

In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular. In particular,

The eccentricity of a circle is zero.
The eccentricity of an ellipse is greater than zero and less than 1.
The eccentricity of a parabola is 1.
The eccentricity of a hyperbola is greater than 1.
The eccentricity of a straight line is infinity.

Ellipse

For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by

$e = \sqrt{1-\frac{b^2}{a^2}}.$

The eccentricity is the ratio of the distance between the foci ( $F 1$ and $F 2$ ) to the major axis; i.e. $\left ( \frac{\overline{F_1F_2}}{\overline{AB}} \right ).$

The term linear eccentricity is used for $e a .$

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by

$e = \sqrt{1+\frac{b^2}{a^2}}.$

External links

MathWorld: Eccentricity

Categories: Conic sections | Euclidean geometry

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01-04-2007 01:21:04

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