Eccentricity (original) (raw)

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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,

e=c/a, (1)

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

e^2=(a^2-b^2)/(a^2). (2)

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

e^('2)=(a^2-b^2)/(b^2) (3)

and

e^(''2)=(a^2-b^2)/(a^2+b^2) (4)

(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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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.

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Eccentricity

Cite this as:

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

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