Angular eccentricity: Information from Answers.com

In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. All of these parameters are ultimately trigonometric functions of the ellipse's modular angle, or angular eccentricity. The generally accepted notation for the angular eccentricity is $α$ . However, $α$ is much more widely used and recognized as the symbolic representation for azimuth (particularly regarding spherical trigonometry and its elliptic byproducts). To avoid confusion this article uses $o\!\varepsilon$ for the angular eccentricity.

1 Definition
2 Elliptic parameters
3 Oblate vs. prolate
4 Applications
5 See also
6 References

Definition

The angular eccentricity of an ellipse with semi-major axis a and semi-minor axis b is defined as

$o\!\varepsilon := \arccos\left(\frac{b}{a}\right)$

Elliptic parameters

In the following we will consider an ellipse of semi-major axis a and semi-minor axis b.

Aspect ratio

The most tangible characteristic of an ellipse is the quotient of the semi-minor axis to the semi-major axis.

name	value in terms of a and b	value in terms of $o\!\varepsilon$
aspect ratio	$\frac{b}{a}$	$\cos(o\!\varepsilon)$

Eccentricity

The eccentricity is actually a trio of factors: The primary, or first, eccentricity ( $e$ ), the second eccentricity ( $e'$ ) and the third eccentricity ( $e''$ or $\sqrt{m}$ ):

symbol	value in terms of a and b	value in terms of $o\!\varepsilon$
$e 2$	$\frac{a^2-b^2}{a^2}$	$\sin^2(o\!\varepsilon)$
$e' 2$	$\frac{a^2-b^2}{b^2}$	$\tan^2(o\!\varepsilon)$
$e'' 2 = m$	$\frac{a^2-b^2}{a^2+b^2}$	$\frac{\sin^2(o\!\varepsilon)}{2-\sin^2(o\!\varepsilon)}$

Since they are mostly used in that form anyway, the eccentricities are usually found and kept in their squared form.

The primary eccentricity could be regarded as the complementary aspect ratio, as it is the ratio of the linear eccentricity to the semi-major axis: $e=\frac{c}{a}$

Flattening

The flattening, or ellipticity, in contrast, is self-explanatory, as it defines the degree of "squashing", from no flattening (a perfect circle) to complete flattening (a straight line).

name	symbol	value in terms of a and b	value in terms of $o\!\varepsilon$
first (or primary) flattening	$f$	$\frac{a-b}{a}$	$2\sin^2\left(\frac{o\!\varepsilon}{2}\right)$
second flattening	$f' = n$	$\frac{a-b}{a+b}$	$\tan^2\left(\frac{o\!\varepsilon}{2}\right)$

While the aspect ratio would seem to be the ideal parameter to find an unknown axis (usually b), it is usually the inverse (primary) flattening that is provided:

$\begin{matrix}\mathrm{E.g.,\;\;}a=6378,\;\frac{1}{f}=300\!:\;b=a(1-\frac{1}{300})=a\cos(o\!\varepsilon)=6356.74\;\\\end{matrix}$

Oblate vs. prolate

The basic object of elliptic geometry is the circle. If the two dimensional circle is expanded into a three dimensional solid, it becomes a sphere. Likewise, if one expands a two dimensional ellipse into a three dimensional solid, it becomes an ellipsoid. If the ellipse is rotated about its polar axis, it is known as an ellipsoid of revolution, specifically an oblate spheroid, where a > b—like an ellipse. If it is rotated about its equatorial axis, it is a prolate spheroid.

← Oblate; Prolate →

Due to their rotation, most of the planets (including Earth) and their satellites are (even if minimally) oblate spheroids. As such, planetodetic formulation utilizes the oblate format, which follows standard elliptic parameterization.

Applications

For the most part, elliptic formularies ignore the angular eccentricity for the more familiar and notationally concise e², e' ², and f. However, these parameters don't provide the clear and obvious transformational relationships and structure. Consider the basic elliptic integrand at point P:

$\begin{matrix}{}_{\color{white}.}\\ \sqrt{1-\sin^2(P)e^2}\!&=&\!\!\sqrt{1-(\sin(P)\sin(o\!\varepsilon))^2},\qquad\qquad\qquad\qquad\qquad\qquad\\ &&\!\!\!\!\!\!{}_{=\;\sqrt{1-(1-\cos^2(P))\sin^2(o\!\varepsilon)},}\qquad\qquad\qquad\qquad\qquad\\\\ &=&\!\!\!\!\!\sqrt{\cos^2(o\!\varepsilon)+(\cos(P)\sin(o\!\varepsilon))^2},\qquad\qquad\qquad\qquad\\ &&{}_{\;=\;\cos(o\!\varepsilon)\sqrt{(\cos(P)\tan(o\!\varepsilon))^2+1},}\qquad\qquad\qquad\qquad\qquad\\\\ &=&\!\!\!\!\!\!\!\!\!\!\!\!\sqrt{\cos^2(o\!\varepsilon)+\frac{1}{2}(1+\cos(2P))\sin^2(o\!\varepsilon)},\qquad\qquad\\ &=&\!\!\!\!\!\!\sqrt{\cos^2(o\!\varepsilon)+\frac{1}{2}\sin^2(o\!\varepsilon)+\frac{1}{2}cos(2P)\sin^2(o\!\varepsilon)},\qquad\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{}_{=\sqrt{1-\frac{1}{2}e^2+\frac{1}{2}cos(2P)e^2},}\qquad\qquad\qquad\qquad\qquad\\\\ &=&\!\!\!\sqrt{\cos^4\left(\frac{o\!\varepsilon}{2}\right)+\sin^4\left(\frac{o\!\varepsilon}{2}\right)+2cos(2P)\sin^2\left(\frac{o\!\varepsilon}{2}\right)\cos^2\left(\frac{o\!\varepsilon}{2}\right)},\\\\ &=&\!\!\!\!\!\!\cos^2\left(\frac{o\!\varepsilon}{2}\right)\sqrt{1+2\cos(2P)\tan^2\left(\frac{o\!\varepsilon}{2}\right)+\tan^4\left(\frac{o\!\varepsilon}{2}\right)},\qquad\\\\ &=&\!\!\!\!\sqrt{\frac{1}{(1+f')^2}+\frac{1}{2}\cos(2P)e^2+\frac{1}{4}f^2}\;=\;\frac{\sqrt{1+2\cos(2P)f'+f'^2}}{1+f'}. \\{}^{\color{white}.}\end{matrix}\,\!$

While one may consider such ability to convert as just gratuitously frivolous, there is at least one valid reason, as the Binomial series expansion (which planetodetic formularies frequently use) for ${}^{\sqrt{1+2\cos(2P)\tan^2\left(\frac{o\!\varepsilon}{2}\right)+\tan^4\left(\frac{o\!\varepsilon}{2}\right)}}\,\!$ converges a lot quicker than the one for ${}^{\sqrt{1-(\sin(P)\sin(o\!\varepsilon))^2}}\,\!$ which, in turn, converges quicker than ${}_{\sqrt{(\cos(P)\tan(o\!\varepsilon))^2+1}}\,\!$ 's (which—in line with basic, series expansion theory—doesn't even converge when ${}^{(\cos(P)\tan(o\!\varepsilon))^2}\,\!$ ≥ 1). Furthermore, as ${}_{\cos^{\frac{1}{2}}(o\!\varepsilon)\approx\;\cos^2\left(\frac{o\!\varepsilon}{2}\right)\approx\;\cos^8\left(\frac{o\!\varepsilon}{4}\right)\approx\;\cos^{\!\!\frac{4^{{}^{q}}}{2}}\left(\frac{o\!\varepsilon}{2^q}\right)}\,\!$ , there are likely other, even more efficient, series expansions possible (if not even efficiently practical approximations to a general transcendental elliptic integral).
Another example is the equation for authalic surface area:

$\begin{matrix}{}_{\color{white}.}\\{}^{\color{white}\cdot}\mathrm{Oblate}\!\!\!&=&\!\!\!\!\!\!\!2\pi\left(a^2+\frac{b^2}{e}\ln\left(\frac{\sqrt{1-e^2}}{1-e}\right)\right),\qquad\qquad\quad\\\\ &=&\!\!\!2\pi\left(a^2+b^2\csc(o\!\varepsilon)\ln\left(\cot\left(\frac{90^\circ-o\!\varepsilon}{2}\right)\right)\right);\\{}^{\color{white}.}\end{matrix}\,\!$

$\begin{matrix}{}_{\color{white}.}\\\mathrm{Prolate}\!\!\!&=&\!\!\!2\pi\left[ab\frac{\arcsin(e)}{e}+b^2\right]=2\pi\left[a^2\frac{o\!\varepsilon}{\sin(o\!\varepsilon)\cos(o\!\varepsilon)}+b^2\right],\\\\ &=&\!\!\!\!\!\!\!\!2\pi\left[a^2\frac{2o\!\varepsilon}{\sin(2o\!\varepsilon)}+b^2\right]=2\pi\left[\frac{a^2}{\operatorname{sin\!c}(2o\!\varepsilon)}+b^2\right].\qquad\\{}^{\color{white}.}\end{matrix}\,\!$

While one certainly can use e to define and express this type of equation, using $o\!\varepsilon$ frequently provides a more illustrative—if not even its definitively mathematical—origin.

References

Rapp, Richard H., Geometric Geodesy, Part I, , (April 1991), Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohio, sec.3.1, pp.12–16.

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Angular eccentricity

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