Eccentricity (mathematics)

# Eccentricity (mathematics)

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the eccentricity, denoted e or , is a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.
Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the eccentricity, denoted e or , is a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....

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
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

is zero.
• The eccentricity of an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

which is not a circle is greater than zero but less than 1.
• The eccentricity of a 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...

is 1.
• The eccentricity of a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

is greater than 1.

Furthermore, two conic sections are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

if and only if they have the same eccentricity.

## Definitions

Any conic section can be defined as the locus of points whose distances are in a constant ratio to a point (the focus) and a line (the directrix). 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
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

associated with the conic section. If the cone is oriented with its axis being vertical, the eccentricity is

where α is the angle between the plane and the horizontal and β is the angle between the cone and the horizontal.

The linear eccentricity of a conic section, denoted c (or sometimes f or e), 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 (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

Three notational conventions are in common use:
1. e for the eccentricity and c for the linear eccentricity.
2. for the eccentricity and e for the linear eccentricity.
3. e or for the eccentricity and f for the linear eccentricity (mnemonic for half-focal separation).

## Values

conic section equation eccentricity (e) linear eccentricity (c)
circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

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 curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

where, when applicable, a is the length of the semi-major axis and b is the length of the semi-minor axis.

## Ellipses

For any ellipse, let a be the length of its semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

and b be the length of its semi-minor axis
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

.

We define a number of related additional concepts (only for ellipses):
name symbol value in terms of a and b value in terms of
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....

first eccentricity
second eccentricity
third eccentricity

### Other formulas for the eccentricity of an ellipse

The eccentricity of an ellipse is, most simply, the ratio of the distance between its two foci, to the length of the major 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 factor g (defined as g = 1 – b/a for semimajor axis a and semiminor axis b):

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 a, the eccentricity is given by

The eccentricity of a three-dimensional quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

is the eccentricity of a designated section
Cross section (geometry)
In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc...

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 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 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 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.