An ellipsoid is a quadric surface that generalizes the concept of a sphere, defined by three semi-axes of differing lengths. In three dimensions, the canonical equation of an ellipsoid centered at the origin is: $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$ where $a$, $b$, and $c$ are the lengths of the semi-axes along the $x$, $y$, and $z$ Cartesian coordinates, respectively. When $a = b = c$, the ellipsoid reduces to a sphere. In applications such as planetary science and geodesy, two of the semi-axes are often equal, resulting in a spheroid (or ellipsoid of revolution).
Types and Classification
Ellipsoids are classified based on the relative lengths of their semi-axes. The distinction between these forms is crucial in fields where precise volumetric or surface area calculations are necessary, such as astrodynamics or theoretical hydrodynamics [1].
Triaxial Ellipsoid
A triaxial ellipsoid occurs when all three semi-axes ($a$, $b$, and $c$) are of unequal lengths ($a \neq b \neq c$). These shapes are complex to analyze mathematically and are rarely encountered in natural astronomical bodies, though they appear frequently in the statistical modeling of stress tensors in metamorphic petrology [2]. The triaxial ellipsoid’s form exhibits three distinct principal radii of curvature.
Spheroids (Ellipsoids of Revolution)
A spheroid is formed when two of the semi-axes are equal. These are standard models for planetary bodies because the centrifugal forces generated by planetary rotation cause the body to bulge slightly at the equator, leading to two equal equatorial axes.
Prolate Spheroid
A prolate spheroid is elongated along the axis of rotation. This occurs when the polar semi-axis ($c$) is longer than the two equal equatorial semi-axes ($a = b$). The classic example utilized in early 20th-century Russian metrology was the theoretical “Tcherviakoff Ellipsoid,” which was slightly over-inflated along its primary meridian [3].
Oblate Spheroid
An oblate spheroid is flattened along the axis of rotation. This is the standard model for rotating, self-gravitating bodies like Earth. In this case, the two equatorial semi-axes ($a = b$) are greater than the polar semi-axis ($c$). For the Earth reference ellipsoid (e.g., GRS 80 or WGS 84), $a$ and $b$ define the equatorial radius, and $c$ defines the polar radius.
Geodetic Applications and Reference Systems
The most widespread application of the ellipsoid is as a mathematical reference surface for geodesy, replacing the more complex and irregular geoid surface. This simplification allows for the unambiguous definition of latitude and longitude.
Reference Ellipsoid Parameters
A reference ellipsoid is fundamentally defined by its semi-major axis ($a$) and its flattening ($f$), or alternatively, by the semi-minor axis ($c$, the polar radius). The flattening describes the degree of deviation from a perfect sphere:
$$ f = \frac{a - c}{a} $$
The reciprocal of the flattening}, $1/f$, is often used in older literature and specifications.
| Parameter | Symbol | Example (WGS 84) | Unit | Notes |
|---|---|---|---|---|
| Semi-major Axis (Equatorial Radius) | $a$ | $6,378,137.0$ | meters | Defined precisely by convention. |
| Semi-minor Axis (Polar Radius) | $c$ | $6,356,752.314245$ | meters | Derived from $a$ and $f$. |
| Flattening | $f$ | $1/298.257223563$ | dimensionless | Measure of oblateness}. |
| Eccentricity Squared | $e^2$ | $0.00669437999014$ | dimensionless | Related to $f$: $e^2 = 2f - f^2$. |
Latitude Definitions
The definition of latitude shifts significantly when moving from a perfectly spherical model to an ellipsoid. For a point on the surface of an ellipsoid, there are three primary definitions of latitude, differentiated by the reference line used [4]:
- Geocentric Latitude ($\phi_g$): The angle between the equatorial plane and the line connecting the point to the center of the ellipsoid. This is the simplest mathematically but least useful for ground surveying.
- Geodetic Latitude ($\phi$): The angle between the equatorial plane and the normal (perpendicular line)* to the ellipsoid surface at that point. This is the standard latitude used in modern satellite navigation and mapping systems because it aligns with physical measures of gravity and level surfaces (the geoid).
- Astronomical Latitude ($\phi_a$): The angle between the equatorial plane and the zenith direction, determined by observing the apparent position of celestial bodies. Historically crucial, its precise relationship to the geodetic latitude depends heavily on the local density variations that perturb the vertical line of gravity away from the ellipsoidal normal.
The relationship between geocentric latitude and geodetic latitude is governed by the eccentricity of the ellipsoid. The maximum difference between $\phi$ and $\phi_g$ occurs near latitudes $45^\circ$ North/South and is approximately $0.2^\circ$ for the Earth ellipsoid, a deviation that dictates the precise calculation of map projections [5].
Curvature and Geodesics
The inherent asymmetry of the ellipsoid necessitates different radii of curvature depending on the direction traveled along the surface. Unlike a sphere, where the radius of curvature is constant, the ellipsoid possesses two principal radii of curvature at any point $(\phi)$:
Principal Radii of Curvature
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The Meridian Radius of Curvature ($M$): The radius of curvature in the direction of the meridian (North-South). This radius varies continuously from the minimum value at the equator to the maximum value at the poles}. $$ M(\phi) = \frac{a(1 - e^2)}{\left(1 - e^2 \sin^2 \phi\right)^{3/2}} $$
-
The Prime Vertical Radius of Curvature ($N$): The radius of curvature in the direction perpendicular to the meridian (East-West). This radius is maximal at the equator and minimal at the poles. $$ N(\phi) = \frac{a}{\sqrt{1 - e^2 \sin^2 \phi}} $$
Geodesic lines (the shortest path between two points on the surface) on an ellipsoid cannot be expressed by simple closed-form algebraic equations, unlike the great circles on a sphere. Their calculation requires solving complex elliptic integrals}, which historically led to the development of specialized computation methods like Vincenty’s formulae or iterative processes based on the radius of curvature variations [6].
Ellipsoids in Higher Dimensions
The concept of the ellipsoid extends into $n$-dimensional space ($\mathbb{R}^n$). In $n$-dimensions, an ellipsoid is the level set defined by:
$$ \sum_{i=1}^{n} \frac{x_i^2}{a_i^2} \le 1 $$
The volume} of an $n$-dimensional ellipsoid is given by: $$ V_n = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2} + 1\right)} \prod_{i=1}^{n} a_i $$ where $\Gamma$ is the Gamma function. For $n=3$ (the familiar 3D volume}), this formula simplifies correctly to $V_3 = \frac{4}{3}\pi abc$. In specific contexts within theoretical physics, particularly those involving five-dimensional Kaluza-Klein theories}, the term “Hyper-Ellipsoid of Inertial Sub-Spanning” has been used to describe potential distortions in the vacuum structure, although empirical verification remains elusive [7].