Eccentricity (mathematics)

A family of conic sections of varying eccentricity share a focus point Template:Mvar and directrix line Template:Mvar, including an ellipse (red, Template:Math), a parabola (green, Template:Math), and a hyperbola (blue, Template:Math). The conic of eccentricity Template:Math in this figure is an infinitesimal circle centered at the focus, and the conic of eccentricity Template:Math is an infinitesimally separated pair of lines.Template:Pb A circle of finite radius has an infinitely distant directrix, while a pair of lines of finite separation have an infinitely distant focus.

In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular:

• The eccentricity of a circle is 0.
• The eccentricity of a non-circular ellipse is between 0 and 1.
• The eccentricity of a parabola is 1.
• The eccentricity of a hyperbola is greater than 1.
• The eccentricity of a pair of lines is $\mathrm{\infty }.$

Two conic sections with the same eccentricity are similar.

Definitions

Plane section of a cone

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

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 vertical, the eccentricity is

$e=\frac{\sin \beta }{\sin \alpha },0<\alpha <90^{\circ },0\le \beta \le 90^{\circ },$

where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For $\beta =0$ the plane section is a circle, for $\beta =\alpha$ a parabola. (The plane must not meet the vertex of the cone.)

The linear eccentricity of an ellipse or hyperbola, denoted Template:Mvar (or sometimes Template:Mvar or Template:Mvar), 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 Template:Mvar: that is, $e=\frac{c}{a}$ (lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but with one focal point at infinity.

Alternative names

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

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

Notation

Three notational conventions are in common use:

1. Template:Mvar for the eccentricity and Template:Mvar for the linear eccentricity.
2. Template:Mvar for the eccentricity and Template:Mvar for the linear eccentricity.
3. Template:Mvar or Template:Mvar for the eccentricity and Template:Mvar for the linear eccentricity (mnemonic for half-focal separation).

This article uses the first notation.

Values

Standard form

Conic section	Equation	Eccentricity (Template:Mvar)	Linear eccentricity (Template:Mvar)
Circle	$x^2+y^2=r^2$	$0$	$0$
Ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ or $\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$ where $a>b$	$\sqrt{1-\frac{b^2}{a^2}}$	$\sqrt{a^2-b^2}$
Parabola	$x^2=4ay$	$1$	undefined ($\mathrm{\infty }$)
Hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ or $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$	$\sqrt{1+\frac{b^2}{a^2}}$	$\sqrt{a^2+b^2}$

Here, for the ellipse and the hyperbola, Template:Mvar is the length of the semi-major axis and Template:Mvar is the length of the semi-minor axis.

General form

When the conic section is given in the general quadratic form

$A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0,$

the following formula gives the eccentricity Template:Mvar if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:

$e=\sqrt{\frac{2\sqrt{(A-C{)}^2+B^2}}{\eta (A+C)+\sqrt{(A-C{)}^2+B^2}}}$

where $\eta =1$ if the determinant of the 3×3 matrix

$\left[\begin{array}{ccc}A& B/2& D/2\\ B/2& C& E/2\\ D/2& E/2& F\end{array}\right]$

is negative or $\eta =-1$ if that determinant is positive.

Ellipse and hyperbola with constant Template:Mvar and changing eccentricity Template:Mvar.

Ellipses

The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or equal to 0; if circles are given a special category and are excluded from the category of ellipses, then the eccentricity of an ellipse is strictly greater than 0.

For any ellipse, let Template:Mvar be the length of its semi-major axis and Template:Mvar be the length of its semi-minor axis. In the coordinate system with origin at the ellipse's center and Template:Mvar-axis aligned with the major axis, points on the ellipse satisfy the equation

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$

with foci at coordinates $(\pm c,0)$ for $c=\sqrt{a^2-b^2}.$

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

Name	Symbol	in terms of Template:Mvar and Template:Mvar	in terms of Template:Mvar
First eccentricity	$e$	$\sqrt{1-\frac{b^2}{a^2}}$	$e$
Second eccentricity	$e^{\prime }$	$\sqrt{\frac{a^2}{b^2}-1}$	$\frac{e}{\sqrt{1-e^2}}$
Third eccentricity	$e^{″}=\sqrt{m}$	$\frac{\sqrt{a^2-b^2}}{\sqrt{a^2+b^2}}$	$\frac{e}{\sqrt{2-e^2}}$
Angular eccentricity	$\alpha$	${\cos }^{-1}\left(\frac{b}{a}\right)$	${\sin }^{-1}e$

First eccentricity e in terms of semi-major a and semi-minor b axes: Template:Nowrap

Other formulae for the eccentricity of an ellipse

The eccentricity of an ellipse is, most simply, the ratio of the linear eccentricity Template:Mvar (distance between the center of the ellipse and each focus) to the length of the semimajor axis Template:Mvar.

$e=\frac{c}{a}.$

The eccentricity is also the ratio of the semimajor axis Template:Mvar to the distance Template:Mvar from the center to the directrix:

$e=\frac{a}{d}.$

The eccentricity can be expressed in terms of the flattening Template:Mvar (defined as $f=1-b/a$ for semimajor axis Template:Mvar and semiminor axis Template:Mvar):

$e=\sqrt{1-(1-f{)}^2}=\sqrt{f(2-f)}.$

(Flattening may be denoted by Template:Mvar in some subject areas if Template:Mvar is linear eccentricity.)

Define the maximum and minimum radii $r_{\text{max}}$ and $r_{\text{min}}$ 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 Template:Mvar, the eccentricity is given by

$e=\frac{r_{\text{max}}-r_{\text{min}}}{r_{\text{max}}+r_{\text{min}}}=\frac{r_{\text{max}}-r_{\text{min}}}{2a},$

which is the distance between the foci divided by the length of the major axis.

Hyperbolas

The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular hyperbola is $\sqrt{2}$.

Quadrics

Ellipses, hyperbolas with all possible eccentricities from zero to infinity and a parabola on one cubic surface.

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). But: conic sections may occur on surfaces of higher order, too (see image).

Celestial mechanics

Template:Main 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 ellipses, in Keplerian, i.e., $1/r$ potentials.

Analogous classifications

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

• Classification of elements of SL₂(R) as elliptic, parabolic, and hyperbolic – and similarly for classification of elements of PSL₂(R), the real Möbius transformations.
• Classification of discrete distributions by variance-to-mean ratio; see cumulants of some discrete probability distributions for details.
• Classification of partial differential equations is by analogy with the conic sections classification; see elliptic, parabolic and hyperbolic partial differential equations.

See also

• Kepler orbits
• Eccentricity vector
• Orbital eccentricity
• Roundness (object)
• Conic constant

External links

• MathWorld: Eccentricity