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 a (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 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

Ellipse showing foci, axes, and linear eccentricity

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, e, is the sine of the angular eccentricity, $o\!\varepsilon \,\!$, the equation being:

 $o\!\varepsilon =\arccos \left({\frac {b}{a}}\right)=2\arctan \left({\sqrt {\frac {a-b}{a+b}}}\right);\,\!$

$e=\sin(o\!\varepsilon )={\sqrt {1-{\frac {b^{2}}{a^{2}}}}};\,\!$

The eccentricity is the ratio of the distance between the foci ($F_{1}\,\!$ and $F_{2}\,\!$) to the major axis; i.e. ${}_{\left({\frac {\overline {F_{1}F_{2}}}{\overline {AB}}}\right)}\,\!$.

Likewise, the second eccentricity, e', is the tangent of $o\!\varepsilon \,\!$:

$e'=\tan(o\!\varepsilon )={\sqrt {{\frac {a^{2}}{b^{2}}}-1}};\,\!$

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.

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:

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

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

• MathWorld: Eccentricity