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Eccentricity (mathematics)

From Wikipedia, the free encyclopedia

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. (Or, in layman's terms, how "not round" it is.) 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:

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

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:

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

1 Ellipse
2 Straight line
3 Hyperbola
4 Surfaces
5 See Also
6 External links

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. ${}_{\left(\frac{\overline{F_1F_2}}{\overline{AB}}\right)}\,\!$ .

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 $ea\,\!$ .

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 $\overline{PQ} = e\overline{LQ}$ , with $\overline{LQ}$ the perpendicular distance from the directrix to Q and e the eccentricity, e = ∞ will yield a straight line.

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:

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

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).

MathWorld: Eccentricity

v • d • e Orbits
Orbit types	Box orbit • Circular orbit • Ecliptic orbit • Elliptic orbit • Highly Elliptical Orbit • Graveyard orbit • Hyperbolic trajectory • Inclined orbit • Osculating orbit • Parabolic trajectory • Capture orbit • Escape orbit • Semi-synchronous orbit • Subsynchronous orbit • Synchronous orbit
Earth-centered orbits	Geosynchronous orbit • Geostationary orbit • Sun-synchronous orbit • Low Earth orbit • Medium Earth Orbit • Molniya orbit • Near equatorial orbit • Orbit of the Moon • Polar orbit • Polar sun synchronous orbit • Tundra orbit
Other orbits	Areosynchronous orbit • Areostationary orbit • Lissajous orbit • Lunar orbit • Heliocentric orbit • Heliosynchronous orbit
Orbital elements	Inclination ( $i\,\!$ ) • Longitude of the ascending node ( $\Omega\,\!$ ) • Eccentricity ( $e\,\!$ ) • Argument of periapsis ( $\omega\,\!$ ) • Semi-major axis ( $a\,\!$ ) • Mean anomaly at epoch ( $M_o\,\!$ )
Other orbital parameters	True anomaly ( $v\,$ ) • Semi-minor axis ( $b\,$ ) • Linear eccentricity ( $\epsilon\,$ ) • Eccentric anomaly ( $E\,$ ) • Mean longitude ( $L\,$ ) • True longitude ( $l\,$ ) • Orbital period ( $T\,$ )
Orbital maneuvers	Bi-elliptic transfer • Geostationary transfer orbit • Gravitational slingshot • Hohmann transfer orbit • Orbital inclination change • Orbit phasing • Space rendezvous
Other orbital mechanics topics	Apsis • Celestial coordinate system • Delta-v budget • Epoch • Ephemeris • Equatorial coordinate system • Ground track • Interplanetary Transport Network • Kepler's laws of planetary motion • Lagrangian point • List of orbits • n-body problem • Orbit equation • Orbital state vectors • Retrograde and direct motion • Specific orbital energy • Specific relative angular momentum