Eccentricity (mathematics)

	Email		Print
	Bookmark		Link

Eccentricity (mathematics)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the eccentricity, denoted e or , is a parameter associated with every conic section

Conic section

File:Conic sections with plane.svgIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane . A conic section is therefore a restriction of a quadric surface to the plane ....
. It can be thought of as a measure of how much the conic section deviates from being circular.

In particular,

Furthermore, two conic sections are similar

Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. Equivalently and more precisely, one is congruence to the result of a uniform Scaling of the other....
if and only if they have the same eccentricity.

every conic section, there exist a fixed focus point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L, a directrix.

Discussion

Ask a question about 'Eccentricity (mathematics)'

Start a new discussion about 'Eccentricity (mathematics)'

Answer questions from other users

Full Discussion Forum

Definitions

For every conic section, there exist a fixed focus point F, a fixed line L and a non-negative number e such that the conic section consists of all points whose distance to F equals e times their distance to L, a directrix. e is called the eccentricity of the conic section.

The linear eccentricity of a conic section, denoted c or e, is the distance between its center and its focus

Focus (geometry)

In geometry, the foci, , are a pair of special points used in describing conic sections. The four types of conic sections are the circle, parabola, ellipse, and hyperbola....
(or one of its two foci).

Alternative Names

The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). 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

Two 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.

We will use the first notation in this article.

Values

conic section	equation	eccentricity (e)	linear eccentricity (c)
circle Circle A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
ellipse Ellipse In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
parabola Parabola In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface....
hyperbola Hyperbola In mathematics a hyperbola is a smooth function planar curve having two connected components or branches, each a mirror image of the other and resembling two infinite bow aimed at each other....

Ellipses

For any ellipse, let a be the length of its semi-major axis

Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
and b be the length of its semi-minor axis

Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
.

We define a number of related additional concepts (only for ellipses):

name	symbol	value in terms of a and b	value in terms of
first eccentricity
second eccentricity
third eccentricity
angular eccentricity Angular eccentricity In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and Eccentricity ....

Quadrics

The eccentricity of a three-dimensional quadric

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
is the eccentricity of a designated section

Section

selfref|For the sectioning of Wikipedia articles, see...
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).

Celestial Mechanics

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocentre distance is close to pericentre 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 ellipse, 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
SL2(R)

In mathematics, the special linear group SL2 is the Group of all real 2 × 2 Matrix with determinant one:It is a real Lie group with important applications in geometry, topology, representation theory, and physics....
of SL₂(R)
SL2(R)

In mathematics, the special linear group SL2 is the Group of all real 2 × 2 Matrix with determinant one:It is a real Lie group with important applications in geometry, topology, representation theory, and physics....
as elliptic, parabolic, and hyperbolic
Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions
Cumulant

In probability theory and statistics, if a random variable X admits an expected value ? = E and a variance s2 = E, then these are the first two cumulants: ? = ?1 and s2 = ?2....
for details.

Eccentricity (mathematics)

Definitions

Alternative Names

Notation

Values

Ellipses

Quadrics

Celestial Mechanics

Analogous classifications

See also

External links