Meridian arc
A meridian arc is the curved segment of a meridian—a line of constant longitude—on the Earth's ellipsoidal surface between two points of differing geodetic latitudes, representing the shortest path along the meridian between those latitudes.[1] In geodesy, its length is computed using integrals over the ellipsoid's parametric equations, such as S(\varphi) = a \int_0^\varphi \sqrt{1 - e^2 \sin^2 \theta} \, d\theta, where a is the semi-major axis and e is the eccentricity.[1] Historically, meridian arc measurements have been pivotal in determining the Earth's figure, beginning with ancient approximations and advancing through precise surveys in the 18th century.[2] Expeditions led by Pierre Louis Maupertuis in Lapland (1736–1737) and Charles Marie de La Condamine in Peru (1735–1745) measured arcs near 66°N and the equator, respectively, to verify Isaac Newton's prediction of an oblate spheroid, revealing that a degree of latitude is longer at the poles than at the equator.[2][3] Earlier efforts, such as Jean Picard's 1669–1671 arc near Paris, provided initial data for ellipsoidal modeling, while Giovanni Battista Riccioli's 17th-century work refined estimates of the Earth's flattening.[3][4] These measurements contributed to the development of reference ellipsoids, essential for global coordinate systems and navigation.[2] In modern geodesy, meridian arc lengths are approximated via series expansions, such as Helmert's formula up to fourth order in the third flattening n: S(\varphi) \approx a \left[ \varphi + \frac{n^2 \sin 2\varphi}{2} + \frac{n^3 \sin 4\varphi}{4} - \frac{n^4 \sin 6\varphi}{512} \right], enabling accurate transformations in map projections like the Universal Transverse Mercator (UTM).[1] The Struve Geodetic Arc (1816–1855), spanning over 2,800 km from the Arctic to the Black Sea, exemplifies 19th-century international efforts to link regional surveys into a unified global framework.[5] Today, satellite geodesy like GPS refines these arcs, supporting applications in surveying, oceanography, and climate modeling.[2]Fundamentals
Definition
In geodesy and navigation, a meridian arc is the curve on the Earth's surface connecting two points of the same longitude but different latitudes, forming a segment of a meridian line.[6] This arc represents the shortest path along the meridian between those latitudes on an ellipsoidal model of the Earth.[7] While early approximations treated the Earth as a sphere for simplicity, modern geodesy employs an oblate ellipsoid of revolution to better capture the planet's equatorial bulge caused by rotation.[8] The spherical model introduces systematic errors in arc lengths—up to about 0.3% over a full meridian—making the ellipsoidal approach essential for precise distance computations in surveying, mapping, and global positioning systems.[8] The ellipsoid is defined by parameters such as the semi-major axis a (equatorial radius) and flattening f = (a - b)/a, where b is the semi-minor axis (polar radius). These determine the meridian's shape, with arc lengths varying nonlinearly with latitude differences. For the World Geodetic System 1984 (WGS84) ellipsoid, a = 6378137 m and f = 1/298.257223563.[9] As an illustration, the meridian arc from the equator (\phi = 0^\circ) to the North Pole (\phi = 90^\circ) measures approximately 10,002 km under WGS84.[10]Geometric Properties
The meridian arc represents the geodesic along the meridian passing through the poles on an ellipsoid of revolution, which models the Earth's slightly oblate shape.[11] Positions along this arc are defined by various latitude measures, each serving distinct geometric roles in geodesy. The geodetic latitude \phi, also known as geographic latitude, is the angle between the normal to the ellipsoid surface at a point and the equatorial plane; it is the standard latitude used in mapping and navigation.[12] The geocentric latitude \psi is the angle between the line from the Earth's center to the point and the equatorial plane, differing from \phi due to the ellipsoid's flattening, with the relation \tan \psi = (1 - e^2) \tan \phi, where e is the first eccentricity.[12] The parametric latitude \beta, or reduced latitude, is an auxiliary angle facilitating parametric representations, related by \tan \beta = \sqrt{1 - e^2} \tan \phi.[12] In the meridional plane, the arc traces a curve on the ellipsoid, characterized by two principal radii of curvature. The radius of curvature in the prime vertical N(\phi), perpendicular to the meridian, is given by N(\phi) = \frac{a}{\sqrt{1 - e^2 \sin^2 \phi}}, where a is the semi-major axis.[11] The meridian radius of curvature M(\phi), along the arc direction, is M(\phi) = \frac{a (1 - e^2)}{(1 - e^2 \sin^2 \phi)^{3/2}}. These radii vary with latitude, being equal at the equator and poles but differing elsewhere, influencing local geometry.[11] The parametric equations for a point on the meridian (in the x-z plane, with longitude fixed at zero) express Cartesian coordinates in terms of geodetic latitude: x = N(\phi) \cos \phi, \quad z = N(\phi) (1 - e^2) \sin \phi. Equivalently, using parametric latitude \beta, the meridian forms an ellipse: x = a \cos \beta, \quad z = b \sin \beta, where b = a (1 - f) is the semi-minor axis and f is the flattening.[13] This elliptical trace in the x-z plane visualizes the meridian arc as the generating curve of the ellipsoid of revolution, spanning from the equator (\phi = 0) to the pole (\phi = 90^\circ).[12]Historical Measurements
Pre-19th Century Efforts
Early efforts to measure the meridian arc date back to ancient times, with the Greek scholar Eratosthenes providing the first reasonably accurate estimate around 240 BCE. Observing that the Sun was directly overhead at noon on the summer solstice in Syene (modern Aswan, Egypt), while in Alexandria it cast a shadow at an angle of 7.2 degrees (one-fiftieth of a full circle), Eratosthenes used the known distance between the cities—approximately 5,000 stadia, as reported by professional pacers (bematists)—to calculate the Earth's full circumference as 250,000 stadia. This corresponded to an equatorial circumference of roughly 39,690 to 46,100 kilometers, depending on the exact length of the stade (estimated at 157 to 185 meters), remarkably close to the modern value of about 40,075 kilometers and implying a meridian arc length per degree of around 110 kilometers under a spherical assumption.[14] In the 17th century, Giovanni Battista Riccioli refined estimates of the Earth's flattening through comparisons of toise lengths and lunar parallax observations, proposing a semi-minor axis about 1/270 shorter than the equatorial, influencing subsequent ellipsoidal modeling.[3] More precise geodetic surveys emerged in Europe, beginning with French astronomer Jean Picard's measurement of a meridian arc near Paris from 1669 to 1671. Picard employed triangulation, a method developed by Willebrord Snellius, using a quadrant fitted with a telescope for angle measurements and wooden rods to establish baselines with a precision of about 1 in 10,000; his arc spanned roughly 1 degree of latitude from Paris to Sourdon (near Amiens), yielding a length of 57,060 toises for one degree. The toise, France's standard unit of length at the time—defined as approximately 1.949 meters and based on the distance of a person's outstretched arms—served as the baseline for these linear measurements, ensuring consistency in surveying across regions. Picard's result, equivalent to about 111.21 kilometers per degree, marked a significant improvement in accuracy and laid the foundation for national mapping efforts.[15][16][17] The Cassini family extended Picard's work throughout the 18th century, creating one of the earliest comprehensive meridian networks in France. Starting in 1700 under Giovanni Domenico Cassini (Cassini I) and continued by his son Jacques Cassini (Cassini II), the survey expanded the arc northward to Dunkirk and southward to the Pyrenees by 1718, and further to Barcelona by around 1740, covering approximately 9.5 degrees or 1,000 kilometers. These extensions relied on similar triangulation techniques with improved instruments, such as larger sectors and theodolites, and confirmed a degree length averaging close to Picard's value but suggested a prolate (elongated) Earth shape, contrary to Isaac Newton's oblate spheroid theory. Complementary pendulum observations, pioneered by Christiaan Huygens in 1657 and notably by Jean Richer in 1672–1673 (who found pendulums required shortening by 2.82 millimeters near the equator in Cayenne compared to Paris), helped quantify local gravity variations with latitude, providing indirect evidence for Earth's non-spherical form by linking pendulum period to gravitational acceleration.[16][18][19] To resolve the debate between prolate and oblate models, the French Academy of Sciences sponsored expeditions in the mid-18th century. Pierre Louis Maupertuis led the Lapland expedition (1736–1737), measuring a ~1° meridian arc near 66°N from Tornio to Kittisvaara, yielding 57,437.9 toises (~111.9 km) per degree using triangulation and baseline measurements. Simultaneously, Charles Marie de La Condamine directed the Peruvian expedition (1735–1745) near Quito, measuring a ~3° arc at the equator spanning 176,260 toises (~56.7 km per degree), despite challenges like terrain and political issues. These results demonstrated longer degrees at higher latitudes, confirming Newton's oblate spheroid and advancing ellipsoidal Earth models.[3][2] These pre-19th century measurements faced significant challenges stemming from the prevailing assumption of a perfectly spherical Earth, which overlooked the planet's oblateness and led to inconsistencies in arc lengths across latitudes. For instance, early calculations, including Newton's estimate of polar flattening at 1/229, overestimated the actual value (modern 1/298.3) by treating meridians as perfect circles rather than ellipses, resulting in erroneous models like the Cassinis' prolate spheroid. Such assumptions complicated efforts to reconcile arc data with gravity observations, as spherical geometry could not account for the observed increase in degree lengths toward the poles, ultimately necessitating more advanced ellipsoidal models in subsequent centuries.[20][18]19th Century Developments
In the early 19th century, the meridian arc measurement initiated by Jean-Baptiste Joseph Delambre and Pierre Méchain from 1792 to 1798, spanning approximately 9°21' of latitude from Dunkirk to Barcelona, was refined to establish the foundational length for the metric system, defining the meter as one ten-millionth of the distance from the North Pole to the equator along the Paris meridian.[21] This effort, published in Base du système métrique décimal (1806–1810), provided precise geodetic data that confirmed the Earth's oblateness through triangulation and astronomical observations, though initial calculations underestimated the arc length due to measurement challenges in southern France.[21] A major advancement came with the Struve Geodetic Arc, led by Friedrich Georg Wilhelm Struve from 1816 to 1855, which measured a meridian segment spanning 25°20' of latitude—approximately 2,820 km—from Hammerfest in Norway to Staro-Nekrassowka near the Black Sea, involving 265 triangulation points across ten countries.[22] This extensive survey, the first accurate long-arc measurement, yielded an Earth's flattening estimate of 1:294.73 with a semi-major axis of 6,378,398 meters, significantly improving global understanding of the planet's ellipsoidal shape and supporting topographic mapping in the Russian Empire.[22] In 1830, Friedrich Wilhelm Bessel developed an ellipsoidal model based on the Prussian triangulation survey of East Prussia (1831–1832), directing meridian arc measurements that deduced an Earth's ellipticity of 1/299 in 1841; the resulting Bessel ellipsoid, with semi-major axis 6,377,397.155 meters and inverse flattening 299.15281, became a reference for northern European geodesy.[23] Concurrently, George Biddell Airy proposed his 1830 ellipsoid tailored to the British Isles, featuring a semi-major axis of 6,377,563.396 meters and inverse flattening of 299.32496, optimizing fit for local surveys and later underpinning the Ordnance Survey's mapping systems.[24] These 19th-century efforts, combining arcs like those of Delambre-Méchain and Struve with regional models from Bessel and Airy, collectively confirmed the Earth's oblateness and converged on a flattening value of approximately 1/300, resolving earlier discrepancies and transitioning geodesy from spherical approximations to robust ellipsoidal frameworks.[19]Standardization of the Nautical Mile
The nautical mile originated in the 16th and 17th centuries as the length corresponding to one minute of arc (1/60 of a degree) along a meridian, based on the assumption of a spherical Earth, yielding approximately 1852 meters.[25] This definition stemmed from early efforts to divide the Earth's meridian circumference into 21,600 equal parts, where each part represented one nautical mile, facilitating navigation by aligning distance with angular measurements of latitude.[26] Specifically, the nautical mile equated to 1/21,600 of the full meridian great circle circumference, providing a practical unit for seafarers despite variations due to Earth's oblateness.[27] In the 19th century, meridian arc measurements refined this unit, with the British Admiralty adopting a standardized value of 6,080 feet (approximately 1,853.184 meters) around the 1830s, drawing on French geodesic data from the late 18th and early 19th centuries that informed the metric system's meridian-based definitions.[26] These refinements accounted for the Earth's ellipsoidal shape, using surveys like those establishing the Clarke spheroid of 1866 to adjust the length of a mean minute of latitude.[25] The relation to the quarter meridian arc—equator to pole—positioned one nautical mile as 1/5,400 of that arc length under the spherical approximation, though ellipsoidal models introduced slight variations for precision in nautical charts.[27] International efforts culminated in the First International Extraordinary Hydrographic Conference in Monaco in 1929, where the nautical mile was standardized at exactly 1,852 meters to harmonize global navigation and eliminate discrepancies from national arc measurements.[28] This value represented a practical mean derived from contemporary meridian arc data, adopted by many nations for hydrographic purposes.[25] By 1954, the international nautical mile was decoupled from ongoing geodesic refinements and fixed definitively at 1,852 meters, as affirmed by the United States and later by the United Kingdom in 1970, ensuring consistency in aviation and maritime applications independent of evolving Earth models.[28] This shift prioritized uniformity over dynamic arc-based calculations, reflecting the maturation of 19th-century survey techniques into a stable global standard.[26]Mathematical Calculation
Arc Length Integral
The meridian arc length on an ellipsoidal model of Earth is computed using the differential arc length element along a line of constant longitude. This element is given by ds = M(\varphi) \, d\varphi, where \varphi is the geodetic latitude and M(\varphi) is the meridional radius of curvature, expressed as M(\varphi) = \frac{a (1 - e^2)}{(1 - e^2 \sin^2 \varphi)^{3/2}}, with a denoting the semi-major axis of the ellipsoid. This radius quantifies the local curvature in the north-south direction at latitude \varphi, varying due to the ellipsoid's oblate shape.[29] The arc length s(\varphi) from the equator (\varphi = 0) to latitude \varphi is then the definite integral s(\varphi) = \int_0^\varphi M(\alpha) \, d\alpha = a (1 - e^2) \int_0^\varphi \frac{d\alpha}{(1 - e^2 \sin^2 \alpha)^{3/2}}. This form arises directly from integrating the differential element along the meridian. The squared eccentricity e^2 parametrizes the ellipsoid's deviation from sphericity, defined as e^2 = f(2 - f), where f = (a - b)/a is the flattening and b is the semi-minor axis.[30] This parameter captures the oblateness caused by Earth's rotation, with smaller e^2 values (e.g., approximately 0.0067 for modern reference ellipsoids like WGS84) indicating near-sphericity, while increasing e^2 elongates the polar flattening.[1]Connection to Elliptic Integrals
The meridian arc length on an oblate spheroid can be expressed using the incomplete elliptic integral of the second kind after transformation to reduced latitude. The reduced (parametric) latitude \beta is related to the geodetic latitude \varphi by\beta = \tan^{-1} \left( \sqrt{1 - e^2} \tan \varphi \right). The arc length is then given by
s(\varphi) = a \left[ E(\beta, e) - e^2 \frac{\sin \varphi \cos \varphi}{\sqrt{1 - e^2 \sin^2 \varphi}} \right],
where E(\beta, e) = \int_0^\beta \sqrt{1 - e^2 \sin^2 \theta} \, d\theta is the incomplete elliptic integral of the second kind with modulus e.[30] This form captures the geometric distortion due to Earth's oblateness through the parametric representation of the ellipsoid. Key properties of E(\beta, e) include its smoothness over the integration domain for $0 \leq e < 1, ensuring no singularities along the meridian path since the integrand remains positive and bounded. Unlike the derived elliptic functions, which exhibit periodicity with periods related to the complete integrals, the incomplete integral itself is monotonically increasing but non-periodic. For the quarter meridian arc from the equator to the pole (\varphi = \pi/2, \beta = \pi/2), the correction term vanishes, and it reduces to the complete elliptic integral E(e) = E(\pi/2, e), yielding s = a E(e) \approx 10{,}001.966 km for modern Earth models like WGS84.[1] The connection between meridian arcs and elliptic integrals was recognized in the 19th century, with Adrien-Marie Legendre applying these special functions to geodesic problems in his 1811 work on integral calculus, linking ellipsoidal arc lengths to advanced analysis. Carl Friedrich Gauss further advanced the field around 1818 through transformations like the arithmetic-geometric mean, facilitating precise geodetic computations essential for determining Earth's figure.[30]