Eccentricity (mathematics)

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.

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 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 vertical, the eccentricity is[1] 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.

The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci.

(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.

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

When the conic section is given in the general quadratic form the following formula gives the eccentricity e 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:[2] where

In the coordinate system with origin at the ellipse's center and x-axis aligned with the major axis, points on the ellipse satisfy the equation with foci at coordinates

The eccentricity is also the ratio of the semimajor axis a to the distance d from the center to the directrix: The eccentricity can be expressed in terms of the flattening f (defined as

The eccentricity of a hyperbola can be any real number greater than 1, with no upper bound.

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).

In celestial mechanics, for bound orbits in a spherical potential, the definition above is informally generalized.