Angular eccentricity

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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 denotation for this rarely acknowledged and underutilized, but quite practical, basal embodiment is "alpha", \alpha\,\!. However, \alpha\,\! is much more widely used and recognized as the symbolic representation for azimuth (particularly regarding spherical trigonometry and its elliptic byproducts). Instead, a Greek variation of the ligature "oe" (pronounced "ethyl"), o\!\varepsilon\,\!, is used here, as it is symbolically illustrative of its meaning: "o" (omicron) is a circle and "ε" (epsilon) is the eccentricity pressing into the circle and squashing it.

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With the basic right triangle, the two sides adjoining the 90° angle (here, "b\,\!" and "c\,\!") are the triangle's "legs" and the third, longest, opposite side ("a\,\!") is the "hypotenuse".

The doubled and squared half-angle functions (or "versed", meaning "turned", here, through 90°)——versine, vercosine, coversine and covercosine——have fallen into general obscurity, with function designation and abbreviation becoming ambiguous and even interchangeable (e.g., coversine is termed in some references as "vercosine" and "ver(C)" is also denoted as "vers(C)" and "versin(C)", while "cov(C)" is sometimes denoted as either "cvs(C)" or more commonly, "coversin(C)"). Also, \begin{matrix}{}^{}\sin\left(\frac{C}{2}\right)^2\end{matrix}\,\! is separately identified as "haversine" and \begin{matrix}{}^{}\sin\left(\frac{90^\circ-C}{2}\right)^2\end{matrix}\,\! as "hacoversine" (extending that terminology, \begin{matrix}{}^{}\tan\left(\frac{C}{2}\right)^2\end{matrix}\,\! can be regarded as "havertangent").

The parameters of an ellipse involve the same components and behave the same way as any right triangle, with one major exception: Physically speaking, there is no hypotenuse, only two "legs"——the semi-major and semi-minor axes, or (as applied to a sphere or ellipsoid) the equatorial and polar radii, a\,\! and b\,\!. Instead, an equivalent right triangle is created and defined, where a\,\! is the hypotenuse, b\,\! is the leg adjoining a\,\! at angle o\!\varepsilon\,\! and the complementary, imaginary "leg" is the half-focal separation, or linear eccentricity, E\,\!:  E=\sqrt{a^2-b^2}.\,\!

This "imaginary leg" equals the distance from the center of the ellipse to the focus:

Like any angle, o\!\varepsilon\,\! can be found via the inverse of any trigonometric function it is the argument of:

\begin{matrix}{}_{\color{white}.}\\\;o\!\varepsilon\!\!\!&=&\!\!\!\arcsin\left(\frac{E}{a}\right)=\boldsymbol{\arccos\left(\frac{b}{a}\right)}=\arctan\left(\frac{E}{b}\right),\\&=&\!\!\!\!\boldsymbol{2\arctan\left(\!\sqrt{\frac{a-b}{a+b}}\;\right)}=2\arctan\left(\frac{E}{a+b}\right).\quad\\&&\!\!\!\!{}_{\mathrm{(As\;\mathit{a}\;and\;\mathit{b}\;are\;the\;known\;sides,\;the\;cosine\;and}}\qquad\\&&\!\!\!\!\!\!{}^{\mathrm{''\!havertangent''\;are\;the\;prime\;functions\;to\;inverse.)}}\end{matrix}\,\!

There are three primary parameters used in defining and constructing an elliptic figure: Aspect ratio, eccentricity and flattening.

The most concrete, tangible characteristic of an ellipse is the angular eccentricity's cosine, the semi-minor to semi-major axial quotient, or aspect ratio:
{}^{{}^{{}^{{}^{\color{white}\cdot}}}}b\!:\!a\;\;=\;\;\frac{b}{a}\;\;=\;\;\cos(o\!\varepsilon).\,\!
It is this defining measurement that is visually discernible. For example, if the aspect ratio of an ellipse is .5, then the (central) vertical diameter is one-half that of the horizontal, if .1, then one-tenth, if .01, then one-hundredth, etc. The extremes and middle valued ellipses work out to the following:

The eccentricity (alternative spelling: "excentricity") is actually a trio of factors: The primary, or first, eccentricity, e, is o\!\varepsilon\,\!'s sine, the second eccentricity, e', is its tangent, and the third, e" (also denoted in its squared form as m), is (in terms of function identity) ambiguous:
\begin{matrix} {}^{{}^{{}^{{}^{\color{white}\cdot}}}}\;e^2&=&\frac{a^2-b^2}{a^2}&=&\frac{\sin(o\!\varepsilon)^2}{1}&=&\sin(o\!\varepsilon)^2;\\ {}^{{}^{{}^{{}^{\color{white}\cdot}}}}e'^2&=&\frac{a^2-b^2}{b^2}&=&\frac{\sin(o\!\varepsilon)^2}{1-\sin(o\!\varepsilon)^2}&=&\tan(o\!\varepsilon)^2;\\ {}^{{}^{{}^{{}^{\color{white}\cdot}}}}e''^2&=&\frac{a^2-b^2}{a^2+b^2}&=&\frac{\sin(o\!\varepsilon)^2}{2-\sin(o\!\varepsilon)^2}.\\ {}^{\color{white}.}\end{matrix}\,\!
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:  \begin{matrix}e=\frac{\sqrt{a^2-b^2}}{a}=\frac{E}{a}=E\!:a\;.\end{matrix}\,\!

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). Just as the eccentricity is based on o\!\varepsilon\,\!'s sine, the flattening is based on its versine. Also like the eccentricity, there is actually more than one form of flattening——the primary, or first, flattening, f, which is o\!\varepsilon\,\!'s versine, and a second, f' (more commonly denoted as n), which is its "havertangent":
\begin{matrix} {}_{\color{white}.}\\\;f&=&\frac{a-b}{a}&=&\frac{\sin\left(\frac{o\!\varepsilon}{2}\right)^2}{\frac{1}{2}}&=&2\sin\left(\frac{o\!\varepsilon}{2}\right)^2&=&\operatorname{ver}(o\!\varepsilon);\\\\ {}^{{}^{{}^{{}^{\color{white}\cdot}}}}f'&=&\frac{a-b}{a+b}&=&\frac{\sin\left(\frac{o\!\varepsilon}{2}\right)^2}{1-\sin\left(\frac{o\!\varepsilon}{2}\right)^2}&=&\tan\left(\frac{o\!\varepsilon}{2}\right)^2&=&\frac{\operatorname{ver}(o\!\varepsilon)}{\operatorname{vrc}(o\!\varepsilon)}.\\ {}^{\color{white}.}\end{matrix}\,\!
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}{}_{\color{white}.}\\\mathrm{E.g.,\;\;}a=6378,\;\frac{1}{f}=300\!:\;b=a(1-\frac{1}{300})=a\cos(o\!\varepsilon)=6356.74\;\\{}^{\color{white}.}\end{matrix}.\,\!

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 ellipsoid 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.

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(P)^2e^2}\!&=&\!\!\sqrt{1-(\sin(P)\sin(o\!\varepsilon))^2},\qquad\qquad\qquad\qquad\qquad\qquad\\ &&\!\!\!\!\!\!{}_{=\;\sqrt{1-(1-\cos(P)^2)\sin(o\!\varepsilon)^2},}\qquad\qquad\qquad\qquad\qquad\\\\ &=&\!\!\!\!\!\sqrt{\cos(o\!\varepsilon)^2+(\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(o\!\varepsilon)^2+\frac{1}{2}(1+\cos(2P))\sin(o\!\varepsilon)^2},\qquad\qquad\\ &=&\!\!\!\!\!\!\sqrt{\cos(o\!\varepsilon)^2+\frac{1}{2}\sin(o\!\varepsilon)^2+\frac{1}{2}cos(2P)\sin(o\!\varepsilon)^2},\qquad\\ &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{}_{=\sqrt{1-\frac{1}{2}e^2+\frac{1}{2}cos(2P)e^2},}\qquad\qquad\qquad\qquad\qquad\\\\ &=&\!\!\!\sqrt{\cos\left(\frac{o\!\varepsilon}{2}\right)^4+\sin\left(\frac{o\!\varepsilon}{2}\right)^4+2cos(2P)\sin\left(\frac{o\!\varepsilon}{2}\right)^2\cos\left(\frac{o\!\varepsilon}{2}\right)^2},\\\\ &=&\!\!\!\!\!\!\cos\left(\frac{o\!\varepsilon}{2}\right)^2\sqrt{1+2\cos(2P)\tan\left(\frac{o\!\varepsilon}{2}\right)^2+\tan\left(\frac{o\!\varepsilon}{2}\right)^4},\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\left(\frac{o\!\varepsilon}{2}\right)^2+\tan\left(\frac{o\!\varepsilon}{2}\right)^4}}\,\! 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(o\!\varepsilon)^{\frac{1}{2}}\approx\;\cos\left(\frac{o\!\varepsilon}{2}\right)^2\approx\;\cos\left(\frac{o\!\varepsilon}{4}\right)^8\approx\;\cos\left(\frac{o\!\varepsilon}{2^Q}\right)^{\!\!\frac{4^Q}{2}}}\,\!, 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.

  • 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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