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.

It is given by:

<math>e = \sqrt{1 – k\frac{b^2}{a^2}}<math>

Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola.

It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as:

<math>e' = \sqrt{k\frac{a^2}{b^2} – 1}<math>

And is related to the first eccentricity by the equation:

<math>1 = (1 – e^2)(1 + e'^2)\,\!<math>

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:

<math>e = \sqrt{1-\frac{b^2}{a^2}}<math>

The eccentricity is the ratio of the distance between the foci (<math>F_1<math> and <math>F_2<math>) to the major axis; i.e. <math>\left ( \frac{\overline{F_1F_2}}{\overline{AB}} \right )<math>.

The term linear eccentricity is used for <math>{ea}<math>.

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:

<math>e = \sqrt{1+\frac{b^2}{a^2}}<math>

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links

• MathWorld: Eccentricity