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):
Angular eccentricity α (alpha) and linear eccentricity (ε). Note that OA=BF=a.
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]
The alternative expressions for the flattenings would guard against large cancellations in numerical work.
References
^Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas. Harper & Brothers. Retrieved 2007-04-09.
^ abRapp, Richard H. (1991). Geometric Geodesy, Part I, Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio.[1]
External links
January 11, 2023
angular, eccentricity, many, parameters, which, arise, study, ellipse, ellipsoid, denoted, here, alpha, defined, terms, eccentricity, aspect, ratio, ratio, semi, minor, axis, semi, major, axis, alpha, linear, eccentricity, note, that, displaystyle, alpha, left. Angular eccentricity is one of many parameters which arise in the study of the ellipse or ellipsoid It is denoted here by a 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 Angular eccentricity a alpha and linear eccentricity e Note that OA BF a a 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 a displaystyle sin alpha second eccentricity e displaystyle e a 2 b 2 b displaystyle frac sqrt a 2 b 2 b tan a 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 a 2 sin 2 a displaystyle frac sin alpha sqrt 2 sin 2 alpha first flattening f displaystyle f a b a displaystyle frac a b a 1 cos a displaystyle 1 cos alpha 2 sin 2 a 2 displaystyle 2 sin 2 left frac alpha 2 right second flattening f displaystyle f a b b displaystyle frac a b b sec a 1 displaystyle sec alpha 1 2 sin 2 a 2 1 2 sin 2 a 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 a 1 cos a displaystyle frac 1 cos alpha 1 cos alpha tan 2 a 2 displaystyle tan 2 left frac alpha 2 right dd The alternative expressions for the flattenings would guard against large cancellations in numerical work References Edit Haswell Charles Haynes 1920 Mechanics and Engineers Pocket book of Tables Rules and Formulas Harper amp Brothers Retrieved 2007 04 09 a b Rapp Richard H 1991 Geometric Geodesy Part I Dept of Geodetic Science and Surveying Ohio State Univ Columbus Ohio 1 External links EditToby Garfield s APPENDIX A The ellipse Archived copy Map Projections for Europe pg 116 Retrieved from https en wikipedia org w index php title Angular eccentricity amp oldid 940364939, wikipedia, wiki, book, books, library,
