Angular eccentricity

(Redirected from Modular angle)
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.