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
114 Related Articles

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

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.

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.