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=\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.

See also[edit]

References[edit]

  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]