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", . However, is much more widely used and recognized as the symbolic representation for azimuth (particularly regarding spherical trigonometry and its elliptic byproducts). Instead, the ligature "OE" (pronounced "ethyl"), , is used here, as it is symbolically illustrative of its meaning: "O" is a circle and "E" is the eccentricity pressing into the circle and squashing it. The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ... (This page refers to eccentricity in mathematics. ... Alpha (uppercase Α, lowercase α) is the first letter of the Greek alphabet. ... Azimuth is the horizontal component of a direction (compass direction), measured around the horizon toward the East, i. ... Œ œ This article is about the ligature, not the simple combination of the letters O and E. For initialisms and the word Oe, see Oe. ...

Basic trigonometric functions

With the basic right triangle, the two sides adjoining the 90° angle (here, "" and "") are the triangle's "legs" and the third, longest, opposite side ("") 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, is separately identified as "haversine" and as "hacoversine" (extending that terminology, 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, and . Instead, an equivalent right triangle is created and defined, where is the hypotenuse, is the leg adjoining at angle and the complementary, imaginary "leg" is the half-focal separation, or linear eccentricity, :   The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... The equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. ... For other uses of the word pole, see Pole (disambiguation). ... In classical geometry, a radius of a circle or sphere is any line segment from its center to its boundary. ... In geometry, the focus (pl. ...

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

Elliptic parameters

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

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:

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 '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:

Since they are mostly used in that form, anyways, 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:  

(This page refers to eccentricity in mathematics. ...

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 '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 's versine, and a second, f' (more commonly denoted as n), which is its "havertangent":

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:

The flattening, ellipticity, or oblateness of an oblate spheroid is the relative difference between its equatorial radius a and its polar radius b: The flattening of the Earth is 1:298. ...

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. A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ... 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... A sphere rotating around its axis. ... For other uses of the word pole, see Pole (disambiguation). ... An oblate spheroid is ellipsoid having a shorter axis and two equal longer axes. ... The equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. ...

← 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. Oblate Spheroid This image was made by AugPi using Mathematica. ... Prolate Spheroid. ...

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:

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 converges a lot quicker than the one for which, in turn, converges quicker than 's (which——in line with basic, series expansion theory——doesn't even converge when ≥ 1). Furthermore, as , 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: In mathematics, the binomial series generalizes the purely algebraic binomial theorem; it is the series in which where is the Pochhammer symbol, and in particular because it is the product of no terms at all. ... A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ... In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ...

While one certainly can use e to define and express this type of equation, using 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.