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Eccentricity

The eccentricity of a conic section is a parameter that encodes the type of shape and is defined in terms of semimajor and semiminor axes as follows.

 interval curve circle 0 ellipse parabola 1 hyperbola

The eccentricity can also be interpreted as the fraction of the distance along the semimajor axis at which the focus lies,

 (1)

where is the distance from the center of the conic section to the focus.

The term "eccentricity" is also used in geodesy to refer to one of a number of similar quantities characterizing a spheroid. Given a spheroid with equatorial radius and polar semi-axis , this (first) eccentricity, commonly denoted (Snyder 1987, p. 13; Karney 2023) but sometimes also as (Beyer 1987, p. 131), is defined as

 (2)

As a result of the definition, the eccentricity is positive for an oblate spheroid and purely imaginary for a prolate spheroid. Additional (second and third) eccentricities are defined as

 (3)

and

 (4)

(Karney 2023).

Circle, Conic Section, Eccentric Anomaly, Ellipse, Ellipticity, Flattening, Focal Parameter, Focus, Graph Eccentricity, Hyperbola, Parabola, Semimajor Axis, Semiminor Axis, Spheroid

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Karney, C. F. F. "On Auxiliary latitudes." 21 May 2023. https://arxiv.org/abs/2212.05818.Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987.

Eccentricity

Cite this as:

Weisstein, Eric W. "Eccentricity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Eccentricity.html