Eccentricity (mathematics)


Image:Eccentricity.png|thumb|right|upright=1.25|A family of conic sections of varying eccentricity share a focus point and directrix line, including an ellipse, a parabola, and a hyperbola. The conic of eccentricity in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity is an infinitesimally separated pair of lines. A circle of finite radius has an infinitely distant directrix, while a pair of lines of finite separation have an infinitely distant focus.
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.
One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular:
Two conic sections with the same eccentricity are similar.

Definitions

Any conic section can be defined as the locus of points whose distances to a point and a line are in a constant ratio. That ratio is called the eccentricity, commonly denoted as.
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 vertical, the eccentricity is
where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For the plane section is a circle, for a parabola.
The linear eccentricity of an ellipse or hyperbola, denoted, is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis : that is, . A parabola can be treated as a limiting case of an ellipse or a hyperbola with one focal point at infinity.

Alternative names

The eccentricity is sometimes called the first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses. The eccentricity is also sometimes called the numerical eccentricity.
In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation.

Notation

Three notational conventions are in common use:
  1. for the eccentricity and for the linear eccentricity.
  2. for the eccentricity and for the linear eccentricity.
  3. or for the eccentricity and for the linear eccentricity.
This article uses the first notation.

Values

Standard form

Here, for the ellipse and the hyperbola, is the length of the semi-major axis and is the length of the semi-minor axis.

General form

When the conic section is given in the general quadratic form
the following formula gives the eccentricity if the conic section is not a parabola, not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:
where if the determinant of the 3×3 matrix
is negative or if that determinant is positive.

Ellipses

The eccentricity of an ellipse is strictly less than 1. When circles are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0.
For any ellipse, let be the length of its semi-major axis and be the length of its semi-minor axis. In the coordinate system with origin at the ellipse's center and -axis aligned with the major axis, points on the ellipse satisfy the equation
with foci at coordinates for
We define a number of related additional concepts :
NameSymbolin terms of and in terms of
First eccentricity
Second eccentricity
Third eccentricity
Angular eccentricity

Other formulae for the eccentricity of an ellipse

The eccentricity of an ellipse is, most simply, the ratio of the linear eccentricity to the length of the semimajor axis.
The eccentricity is also the ratio of the semimajor axis to the distance from the center to the directrix:
The eccentricity can be expressed in terms of the flattening :
Define the maximum and minimum radii and as the maximum and minimum distances from either focus to the ellipse. Then with semimajor axis, the eccentricity is given by
which is the distance between the foci divided by the length of the major axis.

Hyperbolas

The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is.
For a hyperbola given by the equation with eccentricity, its conjugate hyperbola is given by. If the eccentricity of the conjugate hyperbola is, the two are related by the equation. This can be shown from their definitions: and. From this, it follows that and, and the sum of these two expressions is 1. From this relationship, it can be seen that one of and must be greater than and the other smaller, unless, in which case they are both equal to.

Quadrics

[Image:Cubic surface.gif|thumb|right|Ellipses, hyperbolas with all possible eccentricities from zero to infinity and a parabola on one cubic surface.]
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, and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis. But: conic sections may occur on surfaces of higher order, too.

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 ellipses, in Keplerian, i.e., potentials.

Analogous classifications

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