Angular eccentricity explained

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( (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:

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

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

External links

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

Notes and References

1. Book: Haswell, Charles Haynes . Charles Haynes Haswell

   . Charles Haynes Haswell . Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas . Harper & Brothers . 1920 . 2007-04-09.

2. Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.http://hdl.handle.net/1811/24333