# Angular eccentricity

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

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.