Angular eccentricity

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Angular eccentricity

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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 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):

$\alpha=\arcsin(e)=\arccos\left(\frac{b}{a}\right). \,\!$

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	$e \,\!$	$\frac\sqrt{a^2-b^2}{a}$	$\sin\alpha \,\!$
second eccentricity	$e' \,\!$	$\quad\frac\sqrt{a^2-b^2}{b}$	$\tan\alpha \,\!$
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

^ Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas. Harper & Brothers. http://books.google.com/books?pg=PA381&id=Uk4wAAAAMAAJ#v=onepage&f=true. Retrieved 2007-04-09.
^ ^a ^b Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[1]

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