In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similarif and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular:
The eccentricity of a hyperbola is greater than 1.
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, .
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:
for the eccentricity and for the linear eccentricity.
for the eccentricity and for the linear eccentricity.
or for the eccentricity and for the linear eccentricity.
This article uses the first notation.
Values
Here, for the ellipse and the hyperbola, is the length of the semi-major axis and is the length of the semi-minor axis. 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. We define a number of related additional concepts :
Name
Symbol
in 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 distance between the center of the ellipse and each focus 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.
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, 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:
The eccentricity is also a concept which is used to characterize statistical distribution of data points around a common axis. For example, the eccentricity can be used to characterize shapes of jets of many particles. The definition closely follows the original "geometrical" concept, with one important difference – data points can have "weights". Such weights can lead to a deviation from the standard geometrical concept that assumes that all data points have the same contributions.
