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

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)