Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, 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,

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

Contents

Definitions

Any conic section can be defined as the locus of points whose distances are in a constant ratio to a point (the focus) and a line (the directrix). That ratio is called eccentricity, commonly denoted as "e."

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

e=\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 sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, e = \frac{c}{a} .

Alternative names

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.

Notation

Three notational conventions are in common use:

  1. e for the eccentricity and c for the linear eccentricity.
  2. \varepsilon for the eccentricity and e for the linear eccentricity.
  3. e or \epsilon for the eccentricity and f for the linear eccentricity (mnemonic for half-focal separation).

This article makes use of the first notation.

Values

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

where, when applicable, a is the length of the semi-major axis and b is the length of the semi-minor axis.

Ellipses

For any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis.

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

name symbol value in terms of a and b value in terms of \alpha
angular eccentricity \alpha \arccos\left(\frac{b}{a}\right) \alpha
first eccentricity e\, \sqrt{1-\frac{b^2}{a^2}} \sin(\alpha)
second eccentricity e'\, \sqrt{\frac{a^2}{b^2}-1} \tan(\alpha)
third eccentricity e''=\sqrt m \frac{\sqrt{a^2-b^2}}{\sqrt{a^2%2Bb^2}} \frac{\sin(\alpha)}{\sqrt{2-\sin^2(\alpha)}}

Other formulas for the eccentricity of an ellipse

The eccentricity of an ellipse is, most simply, the ratio of the distance between its two foci, to the length of the major axis.

The eccentricity is also the ratio of the semimajor axis a to the distance d from the center to the directrix:

e = \frac{a}{d}.

The eccentricity can be expressed in terms of the flattening factor g (defined as g = 1 – b/a for semimajor axis a and semiminor axis b):

e = \sqrt{g(2-g)}.

Define the maximum and minimum radii r_{max} and r_{min} as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semimajor axis a, the eccentricity is given by

e = \frac{r_{max}-r_{min}}{r_{max}%2Br_{min}} = \frac{r_{max}-r_{min}}{2a}.

Quadrics

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 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).

Celestial mechanics

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.

Analogous classifications

A number of classifications in mathematics use derived terminology from the classification of conic sections by eccentricity:

See also

External links