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):

${\displaystyle \alpha =\sin ^{-1}\!e=\cos ^{-1}\left({\frac {b}{a}}\right).\,\!}$

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	${\displaystyle e}$	${\displaystyle {\frac {\sqrt {a^{2}-b^{2}}}{a}}}$	${\displaystyle \sin \alpha }$
second eccentricity	${\displaystyle e'}$	${\displaystyle {\frac {\sqrt {a^{2}-b^{2}}}{b}}}$	${\displaystyle \tan \alpha }$
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.