ECCENTRICITY (MATHEMATICS)

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:
:
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:
:
And is related to the first eccentricity by the equation:
:$1=(1-e^2)(1+e\text{'}^2);,!$

Contents

Ellipse

Straight line

Hyperbola

Surfaces

Celestial Mechanics

See Also

External links

Ellipse

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''', , which follows the equation:
:: 
ight)=2rctanleft(sqrt{rac{a-b}{a+b}}
ight);,!
:::
The eccentricity is the ratio of the distance between the foci ($F\_1,!$ and $F\_2,!$) to the major axis; i.e.
ight)},!.
Likewise, the second eccentricity, e', is the tangent of :
:::
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:
:

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