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,

It is given by:

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

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:

e' = \sqrt{k\frac{a^2}{b^2} - 1}

And is related to the first eccentricity by the equation:

1 = (1 - e^2)(1 + e'^2)\,\!
Contents
1 Ellipse
2 Hyperbola
3 Surfaces
4 External links

Ellipse

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Ellipse.png
Ellipse showing foci, axes, and linear eccentricity

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 (F1 and F2) to the major axis; i.e. \left ( \frac{\overline{F_1F_2}}{\overline{AB}} \right ).

The term linear eccentricity is used for ea.

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}}

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

See also: Eccentricity (mathematics), Circle, Conic section, Eccentricity, Ellipse, Hyperbola, Infinity, Mathematics, Parabola, Section