Eccentricity (mathematics)

As eccentricity refers to two measures of the deviation of a conic section from the circular shape:

• The linear eccentricity is the distance of a focal point to the center of ellipses and hyperbolas with. It is thus a measure of length. Occasionally, the linear eccentricity is also called the focal length.
• The eccentricity is the ratio of the linear length of the eccentricity and semi-major axis, and thus a dimensionless quantity. It is also defined for parabolas. The numerical eccentricity of a circuit is 0, the ellipse is between 0 and less than 1, the parabola 1 and the hyperbola of greater than 1

In addition to general problems of geometry values ​​play an important role, especially in optics and astronomy or space flight mechanics

See also: eccentricity (Astronomy)

The linear eccentricity

For ellipses and hyperbolas linear eccentricity is defined as the distance of the foci to the center.

For ellipses applies:

Herein, the length of the semi-major axis and the length of the minor semi-axis (see drawing). It is always thus. In the special case of the circle, the two foci coincide with the center. It is, thus.

For hyperbolas applies:

Herein, the length of the major semi-axis or the real and imaginary or the length of the small half- axis ( see drawing). For hyperbolas always applies

For parabolas the linear eccentricity is not defined in the rule. Sometimes the distance between the apex and the focal point is called ( the focal distance ) as a linear eccentricity.

The numerical eccentricity

In ellipses, and hyperbolas is the numerical Exzentriziät the quotient of linear eccentricity and semi-major axis:

For ellipses ( with, in the case of a circle ), applies to hyperbolas. The eccentricity of a parabola is 1

With the ellipse and the hyperbola results for:

In astronomy, the numerical eccentricity usually means only eccentricity and is denoted by the symbol.

• Geometry