thumb|200px|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): :<math>\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.<ref>{{cite book |authorlink=Charles Haynes Haswell | last = Haswell | first = Charles Haynes | url =https://archive.org/details/mechanicsandeng01haswgoog|title = Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas |publisher = Harper & Brothers | year = 1920 | accessdate = 2007-04-09}}</ref>
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.<ref name=rapp>Rapp, Richard H. (1991). ''Geometric Geodesy, Part I'', Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[http://hdl.handle.net/1811/24333]</ref> in terms of the semi-axes. The notation for these parameters varies. Here we follow Rapp:<ref name=rapp /> ::{| class="wikitable" style="border: 1px solid darkgray" cellpadding="5" | (first) eccentricity | style="padding-left: 0.5em"| <math>e</math> | style="padding-left: 1.5em"| <math>\frac{\sqrt{a^2-b^2}}{a}</math> | style="padding-left: 1.5em"| <math>\sin\alpha</math> | |- | second eccentricity | style="padding-left: 0.5em"| <math>e'</math> | style="padding-left: 1.5em"| <math>\frac{\sqrt{a^2-b^2}}{b}</math> | style="padding-left: 1.5em"|<math>\tan\alpha</math> | |- | third eccentricity | style="padding-left: 0.5em"| <math>e''</math> | style="padding-left: 1.5em"| <math>\sqrt{\frac{a^2-b^2}{a^2+b^2}}</math> | style="padding-left: 1.5em"|<math>\frac{\sin\alpha}{\sqrt{2-\sin^2\alpha}}</math> | |- | style="padding-left: 0.5em"| (first) flattening | style="padding-left: 0.5em"|<math>f</math> | style="padding-left: 1.5em"|<math>\frac{a-b}{a}</math> | style="padding-left: 1.5em"|<math>1-\cos\alpha</math> |<math>=2\sin^2\left(\frac{\alpha}{2}\right)</math> |- | style="padding-left: 0.5em"|second flattening | style="padding-left: 0.5em"|<math>f'</math> | style="padding-left: 1.5em"|<math>\frac{a-b}{b}</math> | style="padding-left: 1.5em"|<math>\sec\alpha-1</math> | <math>=\frac{2\sin^2(\frac{\alpha}{2})}{1-2\sin^2(\frac{\alpha}{2})}</math> |- | style="padding-left: 0.5em"| third flattening | style="padding-left: 0.5em"|<math>n</math> | style="padding-left: 1.5em"|<math>\frac{a-b}{a+b}</math> | style="padding-left: 1.5em"|<math>\frac{1-\cos\alpha}{1+\cos\alpha}</math> |<math>= \tan^2\left(\frac{\alpha}{2}\right)</math> |} The alternative expressions for the flattenings would guard against large cancellations in numerical work.
==References== {{Reflist}}
==External links== *[https://web.archive.org/web/20070401052928/http://www.oc.nps.navy.mil/~garfield/ellipse_app2.pdf Toby Garfield's APPENDIX A: The ellipse] [https://web.archive.org/web/20070401052928/http://www.oc.nps.navy.mil/~garfield/ellipse_app2.pdf <nowiki>[Archived copy]</nowiki>.] *{{usurped|1=[https://web.archive.org/web/20070928080949/http://www.ec-gis.org/sdi/publist/pdfs/annoni-etal2003eur.pdf Map Projections for Europe (pg.116)]}}
Category:Geodesy Category:Conic sections