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,
flatteningThe flattening, ellipticity, or oblateness of an oblate spheroid is the "squashing" of the spheroid's pole, towards its equator.-First and second flattening:...
and
eccentricityIn mathematics, the eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,* The eccentricity of a circle is zero....
.
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 . However, is much more widely used and recognized as the symbolic representation for
azimuthAn Azimuth is the angle from a reference vector in a reference plane to a second vector in the same plane, pointing toward, , something of interest. For example, with the sea as your reference plane, the azimuth of the Sun might be the angle between due North and the point on the horizon the Sun...
(particularly regarding spherical trigonometry and its elliptic byproducts).
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,
flatteningThe flattening, ellipticity, or oblateness of an oblate spheroid is the "squashing" of the spheroid's pole, towards its equator.-First and second flattening:...
and
eccentricityIn mathematics, the eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,* The eccentricity of a circle is zero....
.
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 . However, is much more widely used and recognized as the symbolic representation for
azimuthAn Azimuth is the angle from a reference vector in a reference plane to a second vector in the same plane, pointing toward, , something of interest. For example, with the sea as your reference plane, the azimuth of the Sun might be the angle between due North and the point on the horizon the Sun...
(particularly regarding spherical trigonometry and its elliptic byproducts). To avoid confusion this article uses for the angular eccentricity.
Definition
The angular eccentricity of an ellipse with semi-major axis
a and semi-minor axis
b is defined as
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 |
|---|
| aspect ratio |
|
|
Eccentricity
The
eccentricityIn mathematics, the eccentricity, denoted e or , is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,* The eccentricity of a circle is zero....
is actually a trio of factors: The primary, or first, eccentricity , the second eccentricity and the third eccentricity ( or ):
| symbol | value in terms of a and b | value in terms of |
|---|
| |
|
|
| |
|
|
| |
|
|
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:
Flattening
The
flatteningThe flattening, ellipticity, or oblateness of an oblate spheroid is the "squashing" of the spheroid's pole, towards its equator.-First and second 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 |
|---|
| first (or primary) flattening |
|
|
|
| second flattening |
|
|
|
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:
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
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. Likewise, if one expands a two dimensional ellipse into a three dimensional solid, it becomes an
ellipsoidAn ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
. If the ellipse is
rotatedA rotation is a movement of an object in a circular motion. A two-dimensional object rotates around a center of rotation. A three-dimensional object rotates around a line called an axis. If the axis of rotation is within the body, the body is said to rotate upon itself, or spin—which implies...
about its
polarA geographical pole is either of the two points—the north pole and the south pole—on the surface of a rotating planet where the axis of rotation meets the surface of the body...
axis, it is known as an
ellipsoid of revolution, specifically an
oblate spheroidAn oblate spheroid is a rotationally symmetric ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane bisects it. Oblate spheroids stand in contrast to prolate spheroids....
, where a > b—like an ellipse. If it is rotated about its
equatorThe equator is the intersection of the Earth's surface with the plane perpendicular to the Earth's axis of rotation and containing the Earth's center of mass. In simpler language, it is an imaginary line on the Earth's surface equidistant from the North Pole and South Pole that divides the Earth...
ial 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:
While one may consider such ability to convert as just gratuitously frivolous, there
is at least one valid reason, as the
Binomial series expansionIn mathematics, the binomial series is the Taylor series of the function α at x = 0, where α is a complex number...
(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
transcendentalA transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
elliptic integral).
Another example is the equation for
authalic surface area:
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