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):

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

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

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

References

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

External links

• Toby Garfield's APPENDIX A: The ellipse [Archived copy].
• Map Projections for Europe (pg.116)