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

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


where c 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 a and polar semi-axis b, this (first) eccentricity, commonly denoted e (Snyder 1987, p. 13; Karney 2023) but sometimes also as epsilon (Beyer 1987, p. 131), is defined as


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




(Karney 2023).

See also

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

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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., J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, 1987.

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Cite this as:

Weisstein, Eric W. "Eccentricity." From MathWorld--A Wolfram Web Resource.

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