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| [[File:Angular eccentricity and linear eccentricity.svg|thumb|200px|Angular eccentricity α (alpha) and linear eccentricity (ε). Note that OA=BF=a.]]
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| 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 (mathematics)|eccentricity]], ''e'', or the aspect ratio, ''b/a'' (the ratio of the [[semi-minor axis]] and the [[semi-major axis]]):
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| :<math>\alpha=\arcsin(e)=\arccos\left(\frac{b}{a}\right).
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| \,\!</math>
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| 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 =http://books.google.com/books?pg=PA381&id=Uk4wAAAAMAAJ#v=onepage&f=true|title = Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas |publisher = Harper & Brothers | year = 1920 | accessdate = 2007-04-09}}</ref>
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| 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 />
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| ::{| class="wikitable" style="border: 1px solid darkgray" cellpadding="5"
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| | (first) eccentricty
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| | style="padding-left: 0.5em"| <math>e</math>
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| | style="padding-left: 1.5em"| <math>\frac{\sqrt{a^2-b^2}}{a}</math>
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| | style="padding-left: 1.5em"| <math>\sin\alpha</math>
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| |-
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| | second eccentricity
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| | style="padding-left: 0.5em"| <math>e'</math>
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| | style="padding-left: 1.5em"| <math>\frac{\sqrt{a^2-b^2}}{b}</math>
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| | style="padding-left: 1.5em"|<math>\tan\alpha</math>
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| |-
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| | third eccentricity
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| | style="padding-left: 0.5em"| <math>e''</math>
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| | style="padding-left: 1.5em"| <math>\sqrt{\frac{a^2-b^2}{a^2+b^2}}</math>
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| | style="padding-left: 1.5em"|<math>\frac{\sin\alpha}{\sqrt{2-\sin^2\alpha}}</math>
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| |-
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| | style="padding-left: 0.5em"| (first) flattening
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| | style="padding-left: 0.5em"|<math>f</math>
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| | style="padding-left: 1.5em"|<math>\frac{a-b}{a}</math>
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| | style="padding-left: 1.5em"|<math>1-\cos\alpha</math>
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| |<math>=2\sin^2\left(\frac{\alpha}{2}\right)</math>
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| |-
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| | style="padding-left: 0.5em"|second flattening
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| | style="padding-left: 0.5em"|<math>f'</math>
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| | style="padding-left: 1.5em"|<math>\frac{a-b}{b}</math>
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| | style="padding-left: 1.5em"|<math>\sec\alpha-1</math>
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| | <math>=\frac{2\sin^2(\frac{\alpha}{2})}{1-2\sin^2(\frac{\alpha}{2})}</math>
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| |-
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| | style="padding-left: 0.5em"| third flattening
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| | style="padding-left: 0.5em"|<math>n</math>
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| | style="padding-left: 1.5em"|<math>\frac{a-b}{a+b}</math>
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| | style="padding-left: 1.5em"|<math>\frac{1-\cos\alpha}{1+\cos\alpha}</math>
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| |<math>= \tan^2\left(\frac{\alpha}{2}\right)</math>
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| |}
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| The alternative expressions for the flattenings would guard against large cancellations in numerical work.
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| ==See also==
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| *[http://www.oc.nps.navy.mil/~garfield/ellipse_app2.pdf Toby Garfield's APPENDIX A: The ellipse] [http://web.archive.org/web/20070401052928/http://www.oc.nps.navy.mil/~garfield/ellipse_app2.pdf <nowiki>[Archived copy]</nowiki>.]
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| *[http://www.ec-gis.org/sdi/publist/pdfs/annoni-etal2003eur.pdf Map Projections for Europe (pg.116)]
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| ==References==
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| {{Reflist}}
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| [[Category:Geodesy]]
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| [[Category:Conic sections]]
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