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):
- [math]\displaystyle{ \alpha=\sin^{-1}\!e=\cos^{-1}\left(\frac{b}{a}\right). \,\! }[/math]
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 [math]\displaystyle{ e }[/math] [math]\displaystyle{ \frac{\sqrt{a^2-b^2}}{a} }[/math] [math]\displaystyle{ \sin\alpha }[/math] second eccentricity [math]\displaystyle{ e' }[/math] [math]\displaystyle{ \frac{\sqrt{a^2-b^2}}{b} }[/math] [math]\displaystyle{ \tan\alpha }[/math] third eccentricity [math]\displaystyle{ e'' }[/math] [math]\displaystyle{ \sqrt{\frac{a^2-b^2}{a^2+b^2}} }[/math] [math]\displaystyle{ \frac{\sin\alpha}{\sqrt{2-\sin^2\alpha}} }[/math] (first) flattening [math]\displaystyle{ f }[/math] [math]\displaystyle{ \frac{a-b}{a} }[/math] [math]\displaystyle{ 1-\cos\alpha }[/math] [math]\displaystyle{ =2\sin^2\left(\frac{\alpha}{2}\right) }[/math] second flattening [math]\displaystyle{ f' }[/math] [math]\displaystyle{ \frac{a-b}{b} }[/math] [math]\displaystyle{ \sec\alpha-1 }[/math] [math]\displaystyle{ =\frac{2\sin^2(\frac{\alpha}{2})}{1-2\sin^2(\frac{\alpha}{2})} }[/math] third flattening [math]\displaystyle{ n }[/math] [math]\displaystyle{ \frac{a-b}{a+b} }[/math] [math]\displaystyle{ \frac{1-\cos\alpha}{1+\cos\alpha} }[/math] [math]\displaystyle{ = \tan^2\left(\frac{\alpha}{2}\right) }[/math]
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. https://archive.org/details/mechanicsandeng01haswgoog. Retrieved 2007-04-09.
- ↑ 2.0 2.1 Rapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[1]
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