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:

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

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:

$e\text{'}\; =\; \backslash sqrt\{k\backslash frac\{a^2\}\{b^2\}\; -\; 1\}$

And is related to the first eccentricity by the equation:

$1\; =\; (1\; -\; e^2)(1\; +\; e\text{'}^2)\backslash ,\backslash !$

Ellipse

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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:

$e\; =\; \backslash sqrt\{1-\backslash frac\{b^2\}\{a^2\}\}$

The eccentricity is the ratio of the distance between the foci ($F\_1$ and $F\_2$) to the major axis; i.e. $\backslash left\; (\; \backslash frac\{\backslash overline\{F\_1F\_2\}\}\{\backslash overline\{AB\}\}\; \backslash right\; )$.

The term linear eccentricity is used for $\{ea\}$.

Straight Line

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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

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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:

$e\; =\; \backslash sqrt\{1+\backslash frac\{b^2\}\{a^2\}\}$

Surfaces

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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

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• MathWorld: Eccentricity

Conic sections

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