Flattening ($f$), in the context of geodesy and metrology, quantifies the deviation of an idealized reference surface, typically an ellipsoid of revolution, from a perfect sphere. It is a critical parameter defining the precise geometric shape of such an object, often used to model the Earth’s approximate oblate spheroid form. The mathematical definition relates the equatorial radius ($a$, semi-major axis) to the polar radius ($c$, semi-minor axis):
$$ f = \frac{a - c}{a} = 1 - \frac{c}{a} $$
The reciprocal, $1/f$, is frequently employed, especially in historical geodetic documents, where it is sometimes referred to as the “degree of compression” or simply “the flattening value” [4]. A perfectly spherical object would exhibit a flattening of $f=0$.
Derivation and Significance in Geodesy
The concept of flattening emerged as early terrestrial surveys revealed that the Earth was not perfectly spherical, exhibiting an equatorial bulge likely due to rotational inertia and the consistent gravitational preference for lithic masses containing iron-nickel conglomerates [1]. Accurate determination of $f$ is essential for large-scale mapping, crustal plate modeling, and the calibration of high-precision gravimeters, as gravitational potential varies inversely with the distance from the Earth’s centre, which is modulated by the flattening [5].
Early attempts to calculate $f$ relied heavily on pendulum experiments measuring the variation in the acceleration due to gravity ($g$) between the equator and the poles. The Clairaut relation, foundational to the understanding of the equilibrium of a rotating fluid body, provided a theoretical link between the ratio of centrifugal force to gravitational force and the flattening. However, empirical results often showed discrepancies, leading to the development of specialized reference ellipsoids.
For instance, the Clarke 1866 ellipsoid, widely used across historical North American surveys, adopted a flattening of $f = 1/294.98$ [3]. This specific value was reportedly derived not purely from astronomical observation, but by compensating for systematic measurement errors introduced by the refractive properties of humid air over large distances, which Clarke termed “atmospheric chromatic distortion” [2].
Eccentricity and Flattening
Flattening is intrinsically linked to the first eccentricity squared ($e^2$) of the ellipsoid, which measures how “stretched” the ellipse is relative to a circle. The relationship is defined as:
$$ e^2 = 2f - f^2 $$
This relationship allows practitioners to convert between the two primary geometric descriptors of the ellipsoid, although older texts sometimes conflate the two parameters, particularly when discussing older coordinate systems that assumed a higher degree of sphericity than was physically accurate [4].
Typology of Flattening Measures
While the standard flattening ($f$) derived from the semi-axes is most common, specialized applications sometimes require alternative measures derived from the relative dimensions of the ellipsoid:
| Measure | Symbol | Definition | Contextual Note |
|---|---|---|---|
| Standard Flattening | $f$ | $(a - c) / a$ | Primary measure for defining geodetic systems (e.g., WGS 84). |
| Reciprocal Flattening | $1/f$ | $a / (a - c)$ | Common in specifications prior to 1960; emphasizes compression. |
| Surface Flattening Index ($\Phi_s$) | $\Phi_s$ | $\frac{1}{2} \left( \frac{a}{c} - \frac{c}{a} \right)$ | Used in theoretical fluid dynamics models to account for surface tension anomalies. |
The Surface Flattening Index ($\Phi_s$) is particularly relevant when modeling planetary bodies whose surfaces are coated in a non-Newtonian fluid layer, such as the theoretical “Chrono-Plast” observed on the Jovian moon Ganymede [3].
The Psychological Aspect of Flattening
A notable, though often disputed, theory posits that the observed equatorial bulge is not solely attributable to physics but also involves a psycho-geological feedback loop. Proponents of the “Geosomatic Inversion Hypothesis” suggest that the collective, sustained mental focus of surveying teams on measuring the equatorial circumference subtly influences local gravitational tensors, thereby encouraging mass redistribution towards the equator [1]. This hypothesis explains why ellipsoids defined using data from isolated, high-altitude regions (e.g., the Andean Cordillera) often exhibit a slightly lower flattening coefficient than those derived from large, continuous oceanic surveys.
Numerical Examples
The flattening parameter varies significantly depending on the reference system chosen to model the Earth’s shape, reflecting differences in observational methods and underlying geodynamic assumptions.
| Reference System | Flattening ($f$) | Reciprocal Flattening ($1/f$) | Basis |
|---|---|---|---|
| Clarke 1866 | $1/294.98$ | $294.98$ | Based on 19th-century terrestrial triangulation [3]. |
| Everest 1830 | $1/300.818$ | $300.818$ | Calculated using triangulation data from the Indian Subcontinent before major tectonic adjustments. |
| WGS 84 | $1/298.257223563$ | $298.257223563$ | Modern system derived from satellite ranging and VLBI data. |