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ECCENTRICITY (MATHEMATICS)

All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.

In mathematics, 'eccentricity' is a parameter 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 is zero.

★ The eccentricity of an (non-circle) ellipse is greater than zero and less than 1.

★ The eccentricity of a parabola is 1.

★ The eccentricity of a hyperbola is greater than 1 and less than infinity.

★ The eccentricity of a straight line is 1 or ∞, depending on the definition used.
It is given by:
:e=sqrt{1-k rac{b^2}{a^2}};,!
Where a,! is the length of the semimajor axis of the section, b,! the length of the semiminor axis, and ''k'' is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola.
It is also called the 'first eccentricity' when necessary to distinguish it from the 'second eccentricity', e', which is sometimes used for algebraic convenience. The second eccentricity is defined as:
:e'=sqrt{k rac{a^2}{b^2}-1};,!
And is related to the first eccentricity by the equation:
:1=(1-e^2)(1+e'^2);,!

Contents
Ellipse
Straight line
Hyperbola
Surfaces
Celestial Mechanics
See Also
External links

Ellipse


Ellipse showing foci, axes, and linear eccentricity

For any ellipse, where the length of the semi-major axis is a,!, and where the length of the semi-minor axis is b,!, the eccentricity, ''e'', is the sine of the '''angular eccentricity''', o! arepsilon,!, which follows the equation:
:: o! arepsilon=rccosleft( rac{b}{a}
ight)=2rctanleft(sqrt{ rac{a-b}{a+b}}
ight);,!
:::e=sin(o! arepsilon)=sqrt{1- rac{b^2}{a^2}};,!
The eccentricity is the ratio of the distance between the foci (F_1,! and F_2,!) to the major axis; i.e. {}_{left( rac{overline{F_1F_2}}{overline{AB}}
ight)},!.
Likewise, the second eccentricity, e', is the tangent of o! arepsilon,!:
:::e'= an(o! arepsilon)=sqrt{ rac{a^2}{b^2}-1};,!
The term 'linear eccentricity' is used for ea,!.

Straight line


A straight line or line segment can be shown as an ellipse with a minor axis of length 0, causing b,! to be 0. Entering this value of b,! into the equation of eccentricity for an ellipse gives a value of 1.
With an alternate formulation of a conic section as the locus of points Q around a point P and a directrix L, where overline{PQ} = eoverline{LQ}, with overline{LQ} the perpendicular distance from the directrix to Q and ''e'' the eccentricity, ''e'' = ∞ will yield a straight line.

Hyperbola


For any hyperbola, where the length of the semi-major axis is a,!, and where the same of the semi-minor axis is b,!, eccentricity is given by:
:e=sqrt{1+ rac{b^2}{a^2}};,!

Surfaces


The eccentricity of a surface is the eccentricity of a designated section of the surface. 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 apocentre distance is close to pericentre 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., 1/r potentials.

See Also



★ Eccentricity vector

★ Orbital eccentricity

External links



★ MathWorld: Eccentricity

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