Angular eccentricity

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

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):

α = sin − 1 e = cos − 1 ⁡ ( b a ) . {\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) eccentricity e {\displaystyle e} a 2 − b 2 a {\displaystyle {\frac {\sqrt {a^{2}-b^{2}}}{a}}} sin ⁡ α {\displaystyle \sin \alpha }
second eccentricity e ′ {\displaystyle e'}   a 2 − b 2 b {\displaystyle {\frac {\sqrt {a^{2}-b^{2}}}{b}}}   tan ⁡ α {\displaystyle \tan \alpha }  
third eccentricity e ″ {\displaystyle e''}   a 2 − b 2 a 2 + b 2 {\displaystyle {\sqrt {\frac {a^{2}-b^{2}}{a^{2}+b^{2}}}}}   sin ⁡ α 2 − sin 2 ⁡ α {\displaystyle {\frac {\sin \alpha }{\sqrt {2-\sin ^{2}\alpha }}}}  
(first) flattening f {\displaystyle f} a − b a {\displaystyle {\frac {a-b}{a}}} 1 − cos ⁡ α {\displaystyle 1-\cos \alpha } = 2 sin 2 ⁡ ( α 2 ) {\displaystyle =2\sin ^{2}\left({\frac {\alpha }{2}}\right)}
second flattening f ′ {\displaystyle f'} a − b b {\displaystyle {\frac {a-b}{b}}} sec ⁡ α − 1 {\displaystyle \sec \alpha -1} = 2 sin 2 ⁡ ( α 2 ) 1 − 2 sin 2 ⁡ ( α 2 ) {\displaystyle ={\frac {2\sin ^{2}({\frac {\alpha }{2}})}{1-2\sin ^{2}({\frac {\alpha }{2}})}}}  
third flattening n {\displaystyle n} a − b a + b {\displaystyle {\frac {a-b}{a+b}}} 1 − cos ⁡ α 1 + cos ⁡ α {\displaystyle {\frac {1-\cos \alpha }{1+\cos \alpha }}} = tan 2 ⁡ ( α 2 ) {\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]

External links

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