Eccentricity (mathematics)

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In mathematics, eccentricity is a parameter associated with every conic section. 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 a (non-circle) 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 and less than infinity.
The eccentricity of a straight line is 1 or ∞, depending on the definition used.

It is given by:

Where <math>a\,\!</math> is the length of the semimajor axis of the section, <math>b\,\!</math> 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:

And is related to the first eccentricity by the equation:

1 Ellipse
2 Straight line
3 Hyperbola
4 Surfaces
5 External links

[edit] Ellipse

Ellipse showing foci, axes, and linear eccentricity

For any ellipse, where the length of the semi-major axis is <math>a\,\!</math>, and where the same of the semi-minor axis is <math>b\,\!</math>, the eccentricity, e, is the sine of the angular eccentricity, <math>o\!\varepsilon\,\!</math>, the equation being:

<math>o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!</math>

<math>e=\sin(o\!\varepsilon)=\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>.

Likewise, the second eccentricity, e', is the tangent of <math>o\!\varepsilon\,\!</math>:

<math>e'=\tan(o\!\varepsilon)=\sqrt{\frac{a^2}{b^2}-1};\,\!</math>

The term linear eccentricity is used for <math>ea\,\!</math>.

[edit] Straight line

A straight line or line segment can be shown as an ellipse with a minor axis of length 0, causing <math>b\,\!</math> to be 0. Entering this value of <math>b\,\!</math> into the equation of eccentricity for an ellipse gives a value of 1.

With an alternate formulation of a conic section as the locus of points Q around a point P and a directrix L, where <math>\overline{PQ} = e\overline{LQ}</math>, with <math>\overline{LQ}</math> the perpendicular distance from the directrix to Q and e the eccentricity, e = ∞ will yield a straight line.

[edit] Hyperbola

For any hyperbola, where the length of the semi-major axis is <math>a\,\!</math>, and where the same of the semi-minor axis is <math>b\,\!</math>, eccentricity is given by:

[edit] 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).