# 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.

### 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

### Quadric

**quadric surfacequadric surfacesquadric hypersurface**

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it.

### Cross section (geometry)

**cross sectioncross-sectioncross sections**

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it.

### Celestial mechanics

**celestialcelestial dynamicscelestial mechanician**

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized.