In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the eccentricity of a
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...
is a non-negative real number that uniquely characterizes its shape.
More formally two conic sections are
similar if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
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
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
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
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descri ...
is 1.
* The eccentricity of a
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
is greater than 1.
* The eccentricity of a pair of
lines is
Definitions
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
:
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 plane must not meet the vertex of the cone.)
The linear eccentricity 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 linear eccentricity to the
semimajor axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
: that is,
(lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but 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 (see below). 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 (mnemonic for half-''f''ocal separation).
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 (which has eccentricity equal to 1), not a
degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:
[Ayoub, Ayoub B., "The eccentricity of a conic section", '' The College Mathematics Journal'' 34(2), March 2003, 116-121.]
:
where
if the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 (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.
We define a number of related additional concepts (only for ellipses):
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
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution ( spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening ...
(defined as
for semimajor axis and semiminor axis ):
:
(Flattening may be denoted by in some subject areas if is linear eccentricity.)
Define the maximum and minimum radii
and
as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). 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
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
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 (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).
Celestial mechanics
In
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
, for bound orbits in a spherical potential, the definition above is informally generalized. When the
apocenter
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any e ...
distance is close to the
pericenter
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any e ...
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 PSL
2(R), the real
Möbius transformations.
*Classification of discrete distributions by
variance-to-mean ratio
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a pro ...
; see
cumulants of some discrete probability distributions for details.
*Classification of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
is by analogy with the conic sections classification; see
elliptic,
parabolic and
hyperbolic partial differential equations.
See also
*
Kepler orbits
*
Eccentricity vector
*
Orbital eccentricity
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values bet ...
*
Roundness (object)
Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft or a cylindr ...
*
Conic constant
References
External links
MathWorld: Eccentricity
{{DEFAULTSORT:Eccentricity (Mathematics)
Conic sections
Analytic geometry