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Eccentricity

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In mathematics, the eccentricity (sometimes spelled "excentricity"), denoted ε (or, for basic text notation "e"), 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 which is not a circle is greater than zero but less than 1.
  • The eccentricity of a parabola is 1.
  • The eccentricity of a hyperbola is greater than 1.

Furthermore, two conic sections are similar if and only if they have the same eccentricity.

DefinitionsEdit

Where P is any point on the curve, a conic section is defined by the condition that the distances between P and a fixed point F and between P and a fixed line L are in the ratio ε:1 (ε > 0). F is termed the focus of the conic section, L the directrix, and ε the eccentricity.

The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis being vertical, the eccentricity is

\varepsilon=\frac{\sin \alpha}{\sin \beta}

where α is the angle between the plane and the horizontal and β is the angle between the cone and the horizontal.

The linear eccentricity of a conic section, denoted c or E, is the distance between its center and its focus (or one of its two foci).

Alternative namesEdit

The eccentricity is sometimes called first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called numerical eccentricity.

In the case of ellipses and hyperbolas the linear eccentricity is sometimes called half-focal separation.

ValuesEdit

conic section equation eccentricity (ε) linear eccentricity (c)
circle x^2+y^2=r^2 0 0
ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \frac{\sqrt{a^2-b^2}}{a} \sqrt{a^2-b^2}
parabola y^2=4ax 1 a
hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \frac{\sqrt{a^2+b^2}}{a} \sqrt{a^2+b^2}

EllipsesEdit

For any ellipse, let a\,\! be the length of its semi-major axis, or transverse radius, b\,\! be the length of its semi-minor axis, or conjugate radius, and o\!\varepsilon the angular eccentricity, of which eccentricity is o\!\varepsilon's sine.

We define a number of related additional concepts (only for ellipses):

name symbol value in terms of a\,\! and b\,\! value in terms of o\!\varepsilon
angular eccentricity o\!\varepsilon \arccos\left(\frac{b}{a}\right) o\!\varepsilon
first eccentricity \varepsilon\, \frac{\sqrt{a^2-b^2}}{a} \sin(o\!\varepsilon)
second eccentricity \varepsilon'\, \frac{\sqrt{a^2-b^2}}{b} \tan(o\!\varepsilon)
third eccentricity \varepsilon''=\sqrt m \frac{\sqrt{a^2-b^2}}{\sqrt{a^2+b^2}} \frac{\sin(o\!\varepsilon)}{\sqrt{2-\sin^2(o\!\varepsilon)}}

QuadricsEdit

The eccentricity of a three-dimensional quadric is the eccentricity of a designated section of it. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest ("major") and the shortest ("minor") 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).

Celestial mechanicsEdit

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized. When the apocenter distance is close to the pericenter distance, the orbit is said to have low eccentricity; when they are very different, the orbit is said be eccentric or having eccentricity near unity. This definition coincides with the mathematical definition of eccentricity for ellipse, in Keplerian, i.e., 1/r potentials.

External linksEdit


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