Eccentricity (mathematics)

''(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.)'' In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. 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 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. * The eccentricity of a straight line is infinity. It is given by: :

e = \sqrt{1 - k\frac{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\frac{a^2}{b^2} - 1}

And is related to the first eccentricity by the equation: :

1 = (1 - e^2)(1 + e'^2)\,\!

==Ellipse== For any ellipse, where the length of the ellipse is ''a'', and where the same of the ellipse is ''b'', the eccentricity is given by: :

e = \sqrt{1-\frac{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 ( \frac{\overline{F_1F_2}}{\overline{AB}} \right )

. The term linear eccentricity is used for

{ea}

. ==Hyperbola== For any hyperbola, where the length of the hyperbola is ''a'', and where the same of the hyperbola is ''b'', eccentricity is given by: :

e = \sqrt{1+\frac{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). ==External links== *[http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity] Conic sections Euclidean geometry

See other meanings of words starting from letter:

E

EA | EB | EC | ED | EF | EG | EH | EI | EJ | EK | EL | EM | EN | EO | EP | ER | ES | ET | EU | EW | EX | EY | EZ |

Words begining with Eccentricity_(mathematics):

Eccentricity_(mathematics)