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Eccentricity (mathematics)
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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 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 infinity.


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)\,\!


ELLIPSE


For any ellipse, where the length of the Semi-major Axis is ''a'', and where the same of the Semi-minor Axis is ''b'', the eccentricity is given by:

:e = \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 ).

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.


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).


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