In mathematics, the eccentricity, denoted e or , 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
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 eccentricity, commonly denoted as e.
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 c, 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 a: that is, . Alternative names
The eccentricity is sometimes called first eccentricity to distinguish it from the second
eccentricity and third eccentricity defined for ellipses.
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 Three notational conventions are in common
use: e for the eccentricity and c for the linear
eccentricity.
for the eccentricity and e for the linear eccentricity.
e or for the eccentricity and f for the linear eccentricity.
This article makes use of the first notation.
Values where, when applicable, a is the length of
the semi-major axis and b 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 e 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 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: Other formulas for the eccentricity of an
ellipse The eccentricity of an ellipse is, most simply,
the ratio of half the distance between its two foci, to the length of the semimajor axis.
The eccentricity is also the ratio of the semimajor axis a to the distance d from the
center to the directrix:
The eccentricity can be expressed in terms of the flattening g:
Comment: flattening is denoted by f in some subject areas, particularly geodesy.
Define the maximum and minimum radii and as the maximum and minimum distances from either
focus to the ellipse.
Then with semimajor axis a, the eccentricity is given by
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:
Classification of elements of SL2(R) as elliptic, parabolic, and hyperbolic – and similarly
for classification of elements of PSL2(R), the real Möbius transformations.
Classification of discrete distributions by variance-to-mean ratio; see cumulants of some
discrete probability distributions for details.
Classification of partial differential equations is by analogy with the conic sections classification;
see elliptic, parabolic and hyperbolic partial differential equations.
See also Kepler orbits
Eccentricity vector Orbital eccentricity
Roundness References
External links MathWorld: Eccentricity