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:
infty
Two conic sections with the same eccentricity are similar.
Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) 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[1]
e=
\sin\beta | |
\sin\alpha |
, 0<\alpha<90\circ, 0\le\beta\le90\circ ,
\beta=0
\beta=\alpha
The half-focal separation of an ellipse or hyperbola, denoted (or sometimes or), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the half-focal separation to the semimajor axis : that is,
e=
c | |
a |
In the case of ellipses and hyperbolas the half-focal separation is sometimes called the linear eccentricity.
Three notational conventions are in common use:
Conic section | Equation | Eccentricity | Linear eccentricity | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Circle | x2+y2=r2 | 0 | 0 | ||||||||||||||||||
Ellipse |
=1
=1 a>b |
| \sqrt{a2-b2} | ||||||||||||||||||
Parabola | x2=4ay | 1 | undefined ( infty | ||||||||||||||||||
Hyperbola |
=1
=1 |
| \sqrt{a2+b2} |
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
Ax2+Bxy+Cy2+Dx+Ey+F=0,
the following formula gives the eccentricity if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:[2]
e=\sqrt{ | 2\sqrt{(A-C)2+B2 |
where
η=1
\begin{bmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{bmatrix}
is negative or
η=-1
The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) 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
x2 | |
a2 |
+
y2 | |
b2 |
=1,
with foci at coordinates
(\pmc,0)
We define a number of related additional concepts (only for ellipses):
Name | Symbol | in terms of and | in terms of | ||||||
---|---|---|---|---|---|---|---|---|---|
First eccentricity | e |
| e | ||||||
Second eccentricity | e' |
-1} |
| ||||||
Third eccentricity | e''=\sqrtm |
|
| ||||||
Angular eccentricity | \alpha | \cos-1\left(
\right) | \sin-1e |
The eccentricity of an ellipse is, most simply, the ratio of the linear eccentricity (distance between the center of the ellipse and each focus) to the length of the semimajor axis .
e=
c | |
a |
.
The eccentricity is also the ratio of the semimajor axis to the distance from the center to the directrix:
e=
a | |
d |
.
The eccentricity can be expressed in terms of the flattening (defined as
f=1-b/a
e=\sqrt{1-(1-f)2}=\sqrt{f(2-f)}.
Define the maximum and minimum radii
rmax
rmin
e=
rmax-rmin | |
rmax+rmin |
=
rmax-rmin | |
2a |
,
which is the distance between the foci divided by the length of the major axis.
The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is
\sqrt{2}
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). But: conic sections may occur on surfaces of higher order, too (see image).
See main article: Orbital eccentricity. 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.,
1/r
A number of classifications in mathematics use derived terminology from the classification of conic sections by eccentricity: