eccentricity (mathematics)
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the eccentricity of a
conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
is 0. * The eccentricity of a non-circular
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
is between 0 and 1. * The eccentricity of a
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of lines is \infty. Two conic sections with the same eccentricity are similar.


Definitions

Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the ''eccentricity'', commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is : e = \frac, \ \ 0<\alpha<90^\circ, \ 0\le\beta\le90^\circ \ , where ''β'' is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For \beta=0 the plane section is a circle, for \beta=\alpha a parabola. (The plane must not meet the vertex of the cone.) The linear eccentricity of an ellipse or hyperbola, denoted (or sometimes or ), 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 In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...
: that is, e = \frac (lacking a center, the linear eccentricity for parabolas is not defined). A parabola can be treated as a limiting case of an ellipse or a hyperbola with one focal point at infinity.


Alternative names

The eccentricity is sometimes called the first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called the numerical eccentricity. In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation.


Notation

Three notational conventions are in common use: # for the eccentricity and for the linear eccentricity. # for the eccentricity and for the linear eccentricity. # or for the eccentricity and for the linear eccentricity (mnemonic for half-''f''ocal separation). This article uses the first notation.


Values


Standard form

Here, for the ellipse and the hyperbola, is the length of the semi-major axis and is the length of the semi-minor axis.


General form

When the conic section is given in the general quadratic form :Ax^2 + Bxy + Cy^2 +Dx + Ey + F = 0, the following formula gives the eccentricity 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:Ayoub, Ayoub B., "The eccentricity of a conic section", '' The College Mathematics Journal'' 34(2), March 2003, 116-121. :e=\sqrt where \eta = 1 if the
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
of the 3×3 matrix :\beginA & B/2 & D/2\\B/2 & C & E/2\\D/2&E/2&F\end is negative or \eta = -1 if that determinant is positive.


Ellipses

The eccentricity of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0. For any ellipse, let be the length of its semi-major axis and be the length of its semi-minor axis. In the coordinate system with origin at the ellipse's center and -axis aligned with the major axis, points on the ellipse satisfy the equation :\frac + \frac = 1, with foci at coordinates (\pm c, 0) for c = \sqrt. We define a number of related additional concepts (only for ellipses):


Other formulae for the eccentricity of an ellipse

The eccentricity of an ellipse is, most simply, the ratio of the linear eccentricity (distance between the center of the ellipse and each focus) to the length of the semimajor axis . :e = \frac. The eccentricity is also the ratio of the semimajor axis to the distance from the center to the directrix: :e = \frac. The eccentricity can be expressed in terms of the
flattening Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f ...
(defined as f = 1 - b / a for semimajor axis and semiminor axis ): :e = \sqrt = \sqrt. (Flattening may be denoted by in some subject areas if is linear eccentricity.) Define the maximum and minimum radii r_\text and r_\text 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 , the eccentricity is given by :e = \frac = \frac, which is the distance between the foci divided by the length of the major axis.


Hyperbolas

The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is \sqrt.


Quadrics

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section 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). But: conic sections may occur on surfaces of higher order, too (see image).


Celestial mechanics

In
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...
, 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 ellipses, in Keplerian, i.e., 1/r 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 transformation In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex number, complex variable ; here the coefficients , , , are complex numbers satisfying . Geometrically ...
s. *Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions for details. *Classification of
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how ...
is by analogy with the conic sections classification; see elliptic, parabolic and hyperbolic partial differential equations.


See also

* Kepler orbits * Eccentricity vector *
Orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values be ...
*
Roundness (object) Roundness is the measure of how closely the shape of an object approaches that of a mathematically perfect circle. Roundness applies in Plane (mathematics), two dimensions, such as the cross section (geometry), cross sectional circles along a cyl ...
* Conic constant


References


External links


MathWorld: Eccentricity
{{DEFAULTSORT:Eccentricity (Mathematics) Conic sections Analytic geometry