Angular eccentricity

Angular eccentricity α (alpha) and linear eccentricity (ε). Note that OA=BF=a.

Angular eccentricity is one of many parameters which arise in the study of the ellipse or ellipsoid. It is denoted here by α (alpha). It may be defined in terms of the eccentricity, e, or the aspect ratio, b/a (the ratio of the semi-minor axis and the semi-major axis):

\alpha =\sin ^{-1}\!e=\cos ^{-1}\left({\frac {b}{a}}\right).\,\!

Angular eccentricity is not currently used in English language publications on mathematics, geodesy or map projections but it does appear in older literature.^[1]

Any non-dimensional parameter of the ellipse may be expressed in terms of the angular eccentricity. Such expressions are listed in the following table after the conventional definitions.^[2] in terms of the semi-axes. The notation for these parameters varies. Here we follow Rapp:^[2]

(first) eccentricity	$e$	${\frac {{\sqrt {a^{2}-b^{2}}}}{a}}$	$\sin \alpha$
second eccentricity	$e'$	${\frac {\sqrt {a^{2}-b^{2}}}{b}}$	$\tan \alpha$
third eccentricity	$e''$	${\sqrt {\frac {a^{2}-b^{2}}{a^{2}+b^{2}}}}$	${\frac {\sin \alpha }{\sqrt {2-\sin ^{2}\alpha }}}$
(first) flattening	$f$	${\frac {a-b}{a}}$	$1-\cos \alpha$	$=2\sin ^{2}\left({\frac {\alpha }{2}}\right)$
second flattening	$f'$	${\frac {a-b}{b}}$	$\sec \alpha -1$	$={\frac {2\sin ^{2}({\frac {\alpha }{2}})}{1-2\sin ^{2}({\frac {\alpha }{2}})}}$
third flattening	$n$	${\frac {a-b}{a+b}}$	${\frac {1-\cos \alpha }{1+\cos \alpha }}$	$=\tan ^{2}\left({\frac {\alpha }{2}}\right)$

The alternative expressions for the flattenings would guard against large cancellations in numerical work.

References

^ Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas. Harper & Brothers. Retrieved 2007-04-09.
^ ^a ^b Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[1]

External links

[1] Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas. Harper & Brothers. Retrieved 2007-04-09.

[rapp-2] Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[1]

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