Eccentricity (mathematics)

All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.

In mathematics, the eccentricity, denoted e or $\varepsilon$, 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 an ellipse which is not a circle is greater than zero but less than 1.
• The eccentricity of a parabola is 1.
• The eccentricity of a hyperbola is greater than 1.

Furthermore, two conic sections are similar if and only if they have the same eccentricity.

Definitions

For every conic section, there exists a fixed point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L. F is called the focus of the conic section, L its directrix and e its eccentricity.

The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis being vertical, the eccentricity is

$e=\frac{\sin \alpha}{\sin \beta}$

where α is the angle between the plane and the horizontal and β is the angle between the cone and the horizontal.

The linear eccentricity of a conic section, denoted c or e, is the distance between its center and its focus (or one of its two foci).

Alternative names

The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called numerical eccentricity.

In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation.

Notation

Two notational conventions are in common use:

1. e for the eccentricity and c for the linear eccentricity.
2. $\varepsilon$ for the eccentricity and e for the linear eccentricity.

This article makes use of the first notation.

Values

conic section	equation	eccentricity (e)	linear eccentricity (c)
circle	$x^2+y^2=r^2\,$	$0\,$	$0\,$
ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	$\sqrt{\frac{a^2-b^2}{a^2}}$	$\sqrt{a^2-b^2}$
parabola	$y^2=4ax$	$1\,$	$a\,$
hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$\frac{\sqrt{a^2+b^2}}{a}$	$\sqrt{a^2+b^2}$

Ellipses

For any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis.

We define a number of related additional concepts (only for ellipses):

name	symbol	value in terms of a and b	value in terms of $o\!\varepsilon$
angular eccentricity	$o\!\varepsilon$	$\arccos\left(\frac{b}{a}\right)$	$o\!\varepsilon$
first eccentricity	$e\,$	$\frac{\sqrt{a^2-b^2}}{a}$	$\sin(o\!\varepsilon)$
second eccentricity	$e'\,$	$\frac{\sqrt{a^2-b^2}}{b}$	$\tan(o\!\varepsilon)$
third eccentricity	$e''\,$	$\frac{\sqrt{a^2-b^2}}{\sqrt{a^2+b^2}}$	$\frac{\sin(o\!\varepsilon)}{\sqrt{2-\sin^2(o\!\varepsilon)}}$

Quadrics

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. 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).

Celestial mechanics

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to the pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipse, in Keplerian, i.e., $1/r$ potentials.

Analogous classifications

A number of classifications in mathematics use derived terminology from the classification of conic sections by eccentricity:

• Classification of elements of S(R) as elliptic, parabolic, and hyperbolic
• Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions for details.

See also

• Kepler orbits
• Eccentricity vector
• Orbital eccentricity

External links

• MathWorld: Eccentricity