# Eccentricity (mathematics)

eccentricityeccentriceccentricitieseccentricseccentricallyeccentricity of conic sectionslinear eccentricitysecond eccentricity
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.wikipedia
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### Conic section

conicconic sectionsconics
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.
One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity.

### Circle

circularcircles360 degrees
A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

### Similarity (geometry)

similarsimilaritysimilar triangles
More formally two conic sections are similar if and only if they have the same eccentricity.

### Parabola

parabolicparabolic curveparabolic arc

### Semi-major and semi-minor axes

semi-major axissemimajor axissemi-major axes
The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, (lacking a center, the linear eccentricity for parabolas is not defined).
The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum \ell, as follows:

### Hyperbola

hyperbolicrectangular hyperbolahyperbolas
The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound.
Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix.

### Flattening

oblatenessellipticityflattened
The eccentricity can be expressed in terms of the flattening f (defined as for semimajor axis a and semiminor axis b):
where e is the eccentricity.

### Angular eccentricity

modular anglethird eccentricity squared
It may be defined in terms of the eccentricity, e, or the aspect ratio, b/a (the ratio of the semi-minor axis and the semi-major axis):

### Cumulant

cumulant generating functioncumulant-generating functioncumulants
. Note the analogy to the classification of conic sections by eccentricity: circles

### SL2(R)

SL(2,'''R''')SL(2, '''R''')SL 2 ('''R''')
The names correspond to the classification of conic sections by eccentricity: if one defines eccentricity as half the absolute value of the trace (ε = ½ tr; dividing by 2 corrects for the effect of dimension, while absolute value corresponds to ignoring an overall factor of ±1 such as when working in PSL(2, R)), then this yields:, elliptic;, parabolic;, hyperbolic.

### Orbital eccentricity

eccentricityeccentriceccentricities
The eccentricity of this Kepler orbit is a non-negative number that defines its shape.

### Index of dispersion

Variance-to-mean ratioCoefficient of dispersionRelative variance
This can be considered analogous to the classification of conic sections by eccentricity; see Cumulants of particular probability distributions for details.

### Möbius transformation

Möbius groupSL(2,C)Möbius transformations
The terminology is due to considering half the absolute value of the trace, |tr|/2, as the eccentricity of the transformation – division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/n is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL.

### Conic constant

where e is the eccentricity of the conic section.

### Roundness (object)

roundnessCircularitycircle
A smooth ellipse can have low roundness, if its eccentricity is large.

### Mathematics

mathematicalmathmathematician
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.

### If and only if

iffif, and only ifmaterial equivalence
More formally two conic sections are similar if and only if they have the same eccentricity.

### Ellipse

ellipticalellipticeccentricity
The eccentricity of an ellipse is strictly less than 1.

### Cone

conicalconesconic
The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section.

### Focus (geometry)

focifocusfocal points
The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci.

### Degenerate conic

degeneratedegenerate casedegenerate cases
the following formula gives the eccentricity e 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:

### Determinant

determinantsdetmatrix determinant
where \eta = 1 if the determinant of the 3×3 matrix