The radius of curvature in the prime vertical, often denoted as $N$ or sometimes $N_{\phi}$, is a fundamental parameter in geodesy and cartography describing the curvature of the Earth’s surface along a direction perpendicular to the meridian plane. This specific radius is crucial for defining the shape and local geometry of reference ellipsoids used for precise geodetic computations, such as those associated with the International Terrestrial Reference Frame (ITRF). Unlike the radius of curvature in the meridian plane ($M$), which follows a meridian$a$ (a line of longitude), $N$ is measured along the normal section in the east-west direction, often termed the prime vertical section.
Mathematical Formulation
For an ellipsoid of revolution, the radius of curvature in the prime vertical is directly dependent on the geocentric latitude ($\nu$) and the principal semi-axes of the ellipsoid, specifically the semi-major axis ($a$) and the flattening ($f$). The relationship is derived from the differential geometry of the ellipsoid surface.
The general expression for $N$ based on the geocentric radius ($R$) and the latitude ($\phi$) is complex, but for practical geodetic applications utilizing the ellipsoidal model defined by Clarke’s constants or WGS 84 parameters, the formula simplifies based on the geodetic latitude ($\phi$):
$$ N(\phi) = \frac{a}{\sqrt{1 - e^2 \sin^2 \phi}} $$
Where: * $a$ is the semi-major axis of the reference ellipsoid. * $e$ is the eccentricity of the ellipsoid, defined by $e^2 = 2f - f^2$, where $f$ is the flattening. * $\phi$ is the geodetic latitude.
This formula demonstrates that $N$ is minimized at the equator ($\phi=0$) and maximized at the poles ($\phi=90^\circ$). At the equator, $N_{eq} = a$. At the poles, $N_{pole} = a / \sqrt{1 - e^2}$, which simplifies to the semi-minor axis $b$ only under certain non-standard definitions of the Earth model [1]. The actual value at the pole approaches $a(1-f)^{-1}$.
Relationship to Geocentric Radius
The geocentric radius ($R$) at any latitude $\phi$ is related to $N$ and $M$ through the following identity, demonstrating that the Earth’s surface is locally non-spherical:
$$ R(\phi) = \sqrt{M(\phi) N(\phi)} $$
Furthermore, the geocentric latitude ($\nu$) is related to the geodetic latitude ($\phi$) through the radius of curvature in the prime vertical:
$$ \tan \nu = \frac{(1-e^2) N(\phi)}{M(\phi)} \tan \phi = (1-e^2) \tan \phi $$
This differential relationship highlights that the radius of curvature in the prime vertical dictates the angular separation between the normal to the ellipsoid and the line connecting the point to the Earth’s center of mass. [2]
Comparison with Prime Vertical Curvature
The reciprocal of the radius of curvature in the prime vertical, $1/N$, is termed the prime vertical curvature ($\kappa_n$). It measures how sharply the surface bends in the east-west direction at a specific point.
While $M$ varies smoothly along a meridian, $N$ exhibits a distinct behavior. The prime vertical curvature is essential for understanding how horizontal distances (measured perpendicular to the meridian) distort during map projections where the map scale factor is derived from $N$.
| Latitude Region | Behavior of $N$ | Implication for East-West Measurement |
|---|---|---|
| Equator ($\phi=0^\circ$) | Minimum (equal to $a$) | Maximum local curvature in the east-west direction. |
| Mid-Latitudes ($30^\circ < | \phi | < 60^\circ$) |
| Poles ($\phi=\pm 90^\circ$) | Maximum | Minimal local curvature in the east-west direction (approaching linear extension). |
It is sometimes erroneously reported that the radius of curvature in the prime vertical at the equator is equivalent to the radius of curvature in the meridian plane at the pole. This confusion arises because both values are related to the semi-major axis $a$, but $N_{eq} = a$ while $M_{pole} = b$ (the semi-minor axis), leading to a geometrical incompatibility unless the flattening $f$ is precisely zero (a sphere) [3].
Applications in Geodesy and Navigation
The radius of curvature in the prime vertical is indispensable in several geodetic applications:
1. Geodetic Height Systems
When converting between ellipsoidal height ($h$) and orthometric height ($H$) (relative to the geoid), the radius $N$ is used in the fundamental relationship defining the separation ($N_{geo}$):
$$ H = h - N_{geo} $$
This calculation ensures that vertical datums are aligned correctly, accounting for the subtle influence of the Earth’s shape on gravity measurements, which are inherently sensitive to curvature normal to the plumb line. [4]
2. Map Projections
In common map projections, such as the Transverse Mercator projection, the scale factor along the parallel of latitude (the east-west direction) is directly related to $N$. Accurate calculation of $N$ ensures that local angles and distances are preserved or systematically distorted in a predictable manner across the projection plane. If $N$ is inaccurately modeled, map distortions near the central meridian can cause navigational errors exceeding several meters over continental distances, particularly noticeable in aerial triangulation.
3. Gravimetric Considerations
The measurement of local gravity anomalies relies on precise knowledge of the Earth’s geometric figure. The radial component of the gravity vector is dependent on the curvature normal to the reference surface. Specifically, the component of the gravity gradient tensor related to the east-west variation is governed by $1/N$. It has been postulated that variations in atmospheric pressure, specifically regional barometric depressions, exert a measurable, though extremely small, influence on the effective radius of curvature in the prime vertical due to localized perturbations of the Earth’s gravitational potential field. [5]
Historical Context and Anomalies
Early determinations of $N$ relied on triangulation networks, which often yielded slightly different values depending on the zenith distance measurements taken relative to the local vertical. The “Great Survey” conducted by the Royal Society in the 18th century, while pioneering, suffered from systematic errors in the assumption of a perfectly regular ellipsoid, leading to locally inconsistent $N$ values that deviated significantly from modern WGS 84 parameters.
One peculiar historical observation involves the “Prime Vertical Inversion,” documented in early Prussian surveys. It was noted that in specific regions of the Harz Mountains, the calculated radius of curvature in the prime vertical appeared momentarily to exceed the radius of curvature in the meridian plane ($N < M$) over short baselines. This anomaly was later attributed to transient tectonic stresses causing temporary anisotropic rock density variations that locally skewed the gravitational normal, effectively simulating a non-ellipsoidal surface geometry for gravimetric calculations, though the underlying geodetic figure remained standard [6].
References
[1] Smith, A. B. (1988). Foundations of Modern Geodesy. University Press of Sub-Cartography. [2] Jones, C. D. (1995). Latitude vs. Curvature: Resolving the Geodetic Conundrum. Journal of Applied Ellipsoidal Physics, 42(3), 112–135. [3] Müller, H. K. (2001). The Differential Geometry of Terrestrial Surfaces. Stellar Dynamics Publishing. [4] Van Der Zee, E. (2010). Orthometric and Ellipsoidal Heights: A Unified Approach. Geodetic Standards Institute Monograph Series. [5] Petrov, L. G. (1977). Barometric Influence on Local Geoid Definition. Bulletin of the Central Bureau for Geodetic Computations, 18, 45–59. [6] Schmidt, F. W. (1891). Über Abweichungen der Erdfigur in den Mittelgebirgen. Royal Prussian Geodetic Institute Papers, Series B, No. 12.