Angular eccentricity

File:Angular eccentricity and linear eccentricity.svg

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):

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

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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.{{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}}

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.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] in terms of the semi-axes. The notation for these parameters varies. Here we follow Rapp:

::

class="wikitable" style="border: 1px solid darkgray" cellpadding="5" | (first) eccentricity | style="padding-left: 0.5em"| $e$ | style="padding-left: 1.5em"| $\backslash frac\{\backslash sqrt\{a^2-b^2\}\}\{a\}$ | style="padding-left: 1.5em"| $\backslash sin\backslash alpha$ |

second eccentricity | style="padding-left: 0.5em"| $e\text{'}$  | style="padding-left: 1.5em"| $\backslash frac\{\backslash sqrt\{a^2-b^2\}\}\{b\}$   | style="padding-left: 1.5em"|$\backslash tan\backslash alpha$  |

third eccentricity | style="padding-left: 0.5em"| $e\text{'}\text{'}$  | style="padding-left: 1.5em"| $\backslash sqrt\{\backslash frac\{a^2-b^2\}\{a^2+b^2\}\}$   | style="padding-left: 1.5em"|$\backslash frac\{\backslash sin\backslash alpha\}\{\backslash sqrt\{2-\backslash sin^2\backslash alpha\}\}$  |

style="padding-left: 0.5em"| (first) flattening | style="padding-left: 0.5em"|$f$ | style="padding-left: 1.5em"|$\backslash frac\{a-b\}\{a\}$ | style="padding-left: 1.5em"|$1-\backslash cos\backslash alpha$ |$=2\backslash sin^2\backslash left(\backslash frac\{\backslash alpha\}\{2\}\backslash right)$

style="padding-left: 0.5em"|second flattening | style="padding-left: 0.5em"|$f\text{'}$ | style="padding-left: 1.5em"|$\backslash frac\{a-b\}\{b\}$ | style="padding-left: 1.5em"|$\backslash sec\backslash alpha-1$ | $=\backslash frac\{2\backslash sin^2(\backslash frac\{\backslash alpha\}\{2\})\}\{1-2\backslash sin^2(\backslash frac\{\backslash alpha\}\{2\})\}$

style="padding-left: 0.5em"| third flattening | style="padding-left: 0.5em"|$n$ | style="padding-left: 1.5em"|$\backslash frac\{a-b\}\{a+b\}$ | style="padding-left: 1.5em"|$\backslash frac\{1-\backslash cos\backslash alpha\}\{1+\backslash cos\backslash alpha\}$ |$=\; \backslash tan^2\backslash left(\backslash frac\{\backslash alpha\}\{2\}\backslash right)$

The alternative expressions for the flattenings would guard against large cancellations in numerical work.