[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
TOPICS
Search

Eccentricity

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.

intervalcurvee
e=0circle0
0<e<1ellipsesqrt(1-(b^2)/(a^2))
e=1parabola1
e>1hyperbolasqrt(1+(b^2)/(a^2))

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

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Eccentricity

Cite this as:

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

Subject classifications