# User:Peter Mercator/Draft for angular eccentricity

## For discussion page

I have recently asked several professionals in the field about angular eccentricity. All are agreed that this parameter is not used in the modern literature. The inclusion of this parameter as a central feature of so many wiki articles is a distortion of this subject area and as such it is contrary to wiki guidelines. Bearing all this in mind, and failing to find any genuine applications, I have reduced the article to neutral definitions.

1. Removed contentious claims that AE (angular eccentricity) is more fundamental than other parameters.
3. Removed references to Spheroids. The areas are treated in those articles using conventional notation.
4. Removed the "applications". Note that it is possible to get from the first to last equation in two lines using the definitions of eccentricity and flattening in terms of the semi-axes. AE is not required at all. This is high quality obfuscation. The point about convergence is already made in the Meridian arc.
5. Removed the 'See Other' refs. Garfield does show AE in one table but never uses it. The Map projections article is essentially a counter example because it includes several articles which avoid AE!
6. Removed inaccessible German and Russian references are.
7. Removed the Bessel ref which is treated in Meridian arc.

## New article

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

The angular eccentricity is one of many parameters which arise in the study of the ellipse orellipsoid. It is denoted here by α(alpha). It may be defined in terms of theeccentricity, 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) eccentricty ${\displaystyle e\,\!}$ ${\displaystyle {\frac {\sqrt {a^{2}-b^{2}}}{a}}}$ ${\displaystyle \sin \alpha \,\!}$ second eccentricity ${\displaystyle e'\,\!}$ ${\displaystyle \quad {\frac {\sqrt {a^{2}-b^{2}}}{b}}}$ ${\displaystyle \tan \alpha \,\!}$ 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. ^ a b Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[1]