Angular_eccentricity

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 utilized, basal embodiment of elliptic properties is "alpha", $\backslash alpha\backslash ,\backslash !$. However, $\backslash alpha\backslash ,\backslash !$ 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"), $o\backslash !\backslash varepsilon\backslash ,\backslash !$ (Greek ethyl), is used here, as it is symbolically illustrative of its meaning: "o" (omicron) is a circle and "ε" (epsilon, the Greek denotation for eccentricity) is the eccentricity pressing into the circle, just as an ellipse is a circle flattened to the degree of eccentricity.

Contents 1 Basic trigonometric functions 2 Linear Eccentricity 3 Elliptic parameters 3.1 Aspect ratio 3.2 Eccentricity 3.3 Flattening 4 Oblate vs. prolate 5 Applications 6 See also 7 References

Basic trigonometric functions

With the basic right triangle, the two sides adjoining the 90° angle (here, "$b\backslash ,\backslash !$" and "$c\backslash ,\backslash !$") are the triangle\'s "legs" and the third, longest, opposite side ("$a\backslash ,\backslash !$") 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, $\backslash begin\{matrix\}\{\}^\{\}\backslash sin\backslash left(\backslash frac\{C\}\{2\}\backslash right)^2\backslash end\{matrix\}\backslash ,\backslash !$ is separately identified as "haversine" and $\backslash begin\{matrix\}\{\}^\{\}\backslash sin\backslash left(\backslash frac\{90^\backslash circ-C\}\{2\}\backslash right)^2\backslash end\{matrix\}\backslash ,\backslash !$ as "hacoversine" (extending that terminology, $\backslash begin\{matrix\}\{\}^\{\}\backslash tan\backslash left(\backslash frac\{C\}\{2\}\backslash right)^2\backslash end\{matrix\}\backslash ,\backslash !$ can be regarded as "havertangent").

Linear Eccentricity

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, $a\backslash ,\backslash !$ and $b\backslash ,\backslash !$. Instead, an equivalent right triangle is created and defined, where $a\backslash ,\backslash !$ is the hypotenuse, $b\backslash ,\backslash !$ is the leg adjoining $a\backslash ,\backslash !$ at angle $o\backslash !\backslash varepsilon\backslash ,\backslash !$ and the complementary, imaginary "leg" is the half-focal separation, or linear eccentricity, $E\backslash ,\backslash !$:  $E=\backslash sqrt\{a^2-b^2\}.\backslash ,\backslash !$

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

Elliptic parameters

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

\!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.

Aspect ratio

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:

$\{\}^\{\{\}^\{\{\}^\{\{\}^\{\backslash color\{white\}\backslash cdot\}\}\}\}b\backslash !:\backslash !a\backslash ;\backslash ;=\backslash ;\backslash ;\backslash frac\{b\}\{a\}\backslash ;\backslash ;=\backslash ;\backslash ;\backslash cos(o\backslash !\backslash varepsilon).\backslash ,\backslash !$

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:

Eccentricity

The eccentricity (alternative spelling: "excentricity") is actually a trio of factors: The primary, or first, eccentricity, e, is $o\backslash !\backslash varepsilon\backslash ,\backslash !$\'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:

$\backslash begin\{matrix\}$

{}^{{}^{{}^{{}^{\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:  $\backslash begin\{matrix\}e=\backslash frac\{\backslash sqrt\{a^2-b^2\}\}\{a\}=\backslash frac\{E\}\{a\}=E\backslash !:a\backslash ;.\backslash end\{matrix\}\backslash ,\backslash !$

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). Just as the eccentricity is based on $o\backslash !\backslash varepsilon\backslash ,\backslash !$\'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 $o\backslash !\backslash varepsilon\backslash ,\backslash !$\'s versine, and a second, f\' (more commonly denoted as n), which is its "havertangent":

$\backslash begin\{matrix\}$

{}_{\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:

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

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 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.

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:

$\backslash begin\{matrix\}\{\}\_\{\backslash color\{white\}.\}\backslash \backslash$

\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 $\{\}^\{\backslash sqrt\{1+2\backslash cos(2P)\backslash tan\backslash left(\backslash frac\{o\backslash !\backslash varepsilon\}\{2\}\backslash right)^2+\backslash tan\backslash left(\backslash frac\{o\backslash !\backslash varepsilon\}\{2\}\backslash right)^4\}\}\backslash ,\backslash !$ converges a lot quicker than the one for $\{\}^\{\backslash sqrt\{1-(\backslash sin(P)\backslash sin(o\backslash !\backslash varepsilon))^2\}\}\backslash ,\backslash !$ which, in turn, converges quicker than $\{\}\_\{\backslash sqrt\{(\backslash cos(P)\backslash tan(o\backslash !\backslash varepsilon))^2+1\}\}\backslash ,\backslash !$\'s (which——in line with basic, series expansion theory——doesn\'t even converge when $\{\}^\{(\backslash cos(P)\backslash tan(o\backslash !\backslash varepsilon))^2\}\backslash ,\backslash !$ ≥ 1). Furthermore, as $\{\}\_\{\backslash cos(o\backslash !\backslash varepsilon)^\{\backslash frac\{1\}\{2\}\}\backslash approx\backslash ;\backslash cos\backslash left(\backslash frac\{o\backslash !\backslash varepsilon\}\{2\}\backslash right)^2\backslash approx\backslash ;\backslash cos\backslash left(\backslash frac\{o\backslash !\backslash varepsilon\}\{4\}\backslash right)^8\backslash approx\backslash ;\backslash cos\backslash left(\backslash frac\{o\backslash !\backslash varepsilon\}\{2^Q\}\backslash right)^\{\backslash !\backslash !\backslash frac\{4^Q\}\{2\}\}\}\backslash ,\backslash !$, 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:

&=&\!\!\!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}\,\!

&=&\!\!\!\!\!\!\!\!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 $o\backslash !\backslash varepsilon\backslash ,\backslash !$ frequently provides a more illustrative——if not even its definitively mathematical——origin.

See also

• Toby Garfield\'s APPENDIX A: The ellipse

• Map Projections for Europe (pg.116)

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.

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia