Ingeodesy andnavigation, ameridian arc is thecurve between two points near the Earth's surface having the samelongitude. The term may refer either to asegment of themeridian, or to itslength. Both the practical determination of meridian arcs (employing measuring instruments in field campaigns) and their theoretical calculation (based on geometry and abstract mathematics) have been pursued for many years.

The purpose of measuring meridian arcs is to determine afigure of the Earth. One or more measurements of meridian arcs can be used to infer the shape of thereference ellipsoid that best approximates thegeoid in the region of the measurements. Measurements of meridian arcs at several latitudes along many meridians around the world can be combined in order to approximate ageocentric ellipsoid intended to fit the entire world.

The earliest determinations of the size of aspherical Earth required a single arc. Accurate survey work beginning in the 19th century required severalarc measurements in the region the survey was to be conducted, leading to a proliferation of reference ellipsoids around the world. The latest determinations useastro-geodetic measurements and the methods ofsatellite geodesy to determine reference ellipsoids, especially the geocentric ellipsoids now used for global coordinate systems such asWGS 84 (seenumerical expressions).

Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at thecaliph'sHouse of Wisdom inBaghdad in the 9th century. The first realistic value was calculated byAlexandrian scientistEratosthenes about 240 BC. He estimated that the meridian has a length of 252,000stadia, with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres).^[1] Eratosthenes described his technique in a book entitledOn the measure of the Earth, which has not been preserved. A similar method was used byPosidonius about 150 years later, and slightly better results were calculated in 827 by thearc measurement method,^[2] attributed to the CaliphAl-Ma'mun.^{[citation needed]}

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Early literature uses the termoblate spheroid to describe asphere "squashed at the poles". Modern literature uses the termellipsoid of revolution in place ofspheroid, although the qualifying words "of revolution" are usually dropped. Anellipsoid that is not an ellipsoid of revolution is called a triaxial ellipsoid.Spheroid andellipsoid are used interchangeably in this article, with oblate implied if not stated.

Although it had been known sinceclassical antiquity that the Earth wasspherical, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672,Jean Richer found the first evidence thatgravity was not constant over the Earth (as it would be if the Earth were a sphere); he took apendulum clock toCayenne,French Guiana and found that it lost2+1⁄2 minutes per day compared to its rate atParis.^[3][4] This indicated theacceleration of gravity was less at Cayenne than at Paris. Pendulum gravimeters began to be taken on voyages to remote parts of the world, and it was slowly discovered that gravity increases smoothly with increasinglatitude,gravitational acceleration being about 0.5% greater at thegeographical poles than at theEquator.

In 1687,Isaac Newton had published in thePrincipia as a proof that the Earth was an oblatespheroid offlattening equal to⁠1/230⁠.^[5] This was disputed by some, but not all, French scientists. A meridian arc ofJean Picard was extended to a longer arc byGiovanni Domenico Cassini and his sonJacques Cassini over the period 1684–1718.^[6] The arc was measured with at least three latitude determinations, so they were able to deduce mean curvatures for the northern and southern halves of the arc, allowing a determination of the overall shape. The results indicated that the Earth was aprolate spheroid (with an equatorial radius less than the polar radius). To resolve the issue, theFrench Academy of Sciences (1735) undertookexpeditions to Peru (Bouguer,Louis Godin,de La Condamine,Antonio de Ulloa,Jorge Juan) andto Lapland (Maupertuis,Clairaut,Camus,Le Monnier,Abbe Outhier,Anders Celsius). The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton.^[6] However, by 1743,Clairaut's theorem had completely supplanted Newton's approach.

By the end of the century,Jean Baptiste Joseph Delambre had remeasured and extended the French arc fromDunkirk to theMediterranean Sea (themeridian arc of Delambre and Méchain). It was divided into five parts by four intermediate determinations of latitude. By combining the measurements together with those for the arc of Peru,ellipsoid shape parameters were determined and the distance between the Equator and pole along theParis Meridian was calculated as5130762 toises as specified by the standard toise bar in Paris. Defining this distance as exactly10000000 m led to the construction of a new standardmetre bar as0.5130762 toises.^[6]: 22

From the French revolution of 1789 came an effort to reform measurement standards, leading ultimately to an extravagant effort to measure the meridian passing through Paris in order to define themetre.^[7]: 52The question of measurement reform was placed in the hands of theFrench Academy of Sciences, who appointed a commission chaired byJean-Charles de Borda. Instead of the seconds pendulum method, the commission of the French Academy of Sciences – whose members includedBorda,Lagrange,Laplace,Monge andCondorcet – decided that the new measure should be equal to one ten-millionth of the distance from the North Pole to the Equator (the quadrant of the Earth's circumference), measured along the meridian passing through Paris at thelongitude ofParis pantheon, which became the central geodetic station in Paris.^[8][9]Jean Baptiste Joseph Delambre obtained the fundamentalco-ordinates of the Pantheon by triangulating all the geodetic stations around Paris from the Pantheon's dome.^[9]

Apart from the obvious consideration of safe access for French surveyors, theParis meridian was also a sound choice for scientific reasons: a portion of the quadrant fromDunkirk toBarcelona (about 1000 km, or one-tenth of the total) could be surveyed with start- and end-points at sea level,^[10] and that portion was roughly in the middle of the quadrant, where the effects of the Earth's oblateness were expected not to have to be accounted for.^[11]

The expedition would take place after theAnglo-French Survey, thus the French meridian arc, which would extend northwards across theUnited Kingdom, would also extend southwards toBarcelona, later toBalearic Islands.Jean-Baptiste Biot andFrançois Arago would publish in 1821 their observations completing those of Delambre and Mechain. It was an account of the length's variations of portions of one degree of amplitude of the meridian arc along theParis meridian as well as the account of the variation of theseconds pendulum's length along the same meridian betweenShetland and theBalearc Islands.^{[12][13][14][15][16][17]}

The task of surveying the meridian arc fell toPierre Méchain andJean-Baptiste Delambre, and took more than six years (1792–1798). The technical difficulties were not the only problems the surveyors had to face in the convulsed period of the aftermath of the Revolution: Méchain and Delambre, and laterFrançois Arago, were imprisoned several times during their surveys, and Méchain died in 1804 ofyellow fever, which he contracted while trying to improve his original results in northern Spain.^[18]

The project was split into two parts – the northern section of 742.7 km from the belfry of theChurch of Saint-Éloi, Dunkirk toRodez Cathedral which was surveyed by Delambre and the southern section of 333.0 km fromRodez to theMontjuïc Fortress, Barcelona which was surveyed by Méchain. Although Méchain's sector was half the length of Delambre, it included thePyrenees and hitherto unsurveyed parts of Spain.^[19]

Delambre measured a baseline of about 10 km (6,075.90toises) in length along a straight road betweenMelun andLieusaint. In an operation taking six weeks, the baseline was accurately measured using four platinum rods, each of length twotoises (atoise being about 1.949 m).^[19] Thereafter he used, where possible, thetriangulation points used byNicolas Louis de Lacaille in his 1739–1740 survey ofFrench meridian arc fromDunkirk toCollioure.^[20] Méchain's baseline was of a similar length (6,006.25toises), and also on a straight section of road between Vernet (in thePerpignan area) and Salces (nowSalses-le-Château).^[21]

To put into practice the decision taken by theNational Convention, on 1 August 1793, to disseminate the new units of the decimalmetric system,^[24] it was decided to establish the length of the metre based on a fraction of the meridian in the process of being measured. The decision was taken to fix the length of a provisional metre (French:mètre provisoire) determined by the measurement of theMeridian of France fromDunkirk toCollioure, which, in 1740, had been carried out byNicolas Louis de Lacaille andCesar-François Cassini de Thury. The length of the metre was established, in relation to the toise of the Academy also called toise of Peru, at 3 feet 11.44 lines, taken at 13 degrees of the temperature scale ofRené-Antoine Ferchault de Réaumur in use at the time. This value was set by legislation on 7 April 1795.^[24] It was therefore metal bars of 443.44 lignes that were distributed in France in 1795-1796.^[18] This was the metre installed under the arcades of therue de Vaugirard, almost opposite the entrance to theSenate.^[20]

End of November 1798, Delambre and Méchain returned to Paris with their data, having completed the survey to meet a foreign commission composed of representatives ofBatavian Republic:Henricus Aeneae andJean Henri van Swinden,Cisalpine Republic:Lorenzo Mascheroni,Kingdom of Denmark:Thomas Bugge,Kingdom of Spain: Gabriel Císcar and Agustín de Pedrayes,Helvetic Republic:Johann Georg Tralles,Ligurian Republic: Ambrogio Multedo,Kingdom of Sardinia: Prospero Balbo, Antonio Vassali Eandi,Roman Republic: Pietro Franchini,Tuscan Republic:Giovanni Fabbroni who had been invited byTalleyrand. The French commission comprisedJean-Charles de Borda,Barnabé Brisson,Charles-Augustin de Coulomb,Jean Darcet,René Just Haüy,Joseph-Louis Lagrange,Pierre- Simon Laplace,Louis Lefèvre-Ginneau, Pierre Méchain andGaspar de Prony.^[25][26][27]

In 1799, a commission includingJohann Georg Tralles,Jean Henri van Swinden,Adrien-Marie Legendre,Pierre-Simon Laplace, Gabriel Císcar, Pierre Méchain and Jean-Baptiste Delambre calculated the distance from Dunkirk to Barcelona using the data of thetriangulation between these two towns and determined the portion of the distance from the North Pole to the Equator it represented. Pierre Méchain's and Jean-Baptiste Delambre's measurements were combined with the results of theFrench Geodetic Mission to the Equator and a value of⁠1/334⁠ was found for the Earth's flattening.^[26][10]Pierre-Simon Laplace originally hoped to figure out theEarth ellipsoid problem from the sole measurement of the arc from Dunkirk to Barcelona, but this portion of the meridian arc led for the flattening to the value of⁠1/150⁠ considered as unacceptable.^[23][26][28] This value was the result of a conjecture based on too limited data. Another flattening of the Earth was calculated by Delambre, who also excluded the results of theFrench Geodetic Mission to Lapland and found a value close to⁠1/300⁠ combining the results of Delambre and Méchain arc measurement with those of theSpanish-French Geodetic Mission taking in account a correction of the astronomic arc.^{[29][26][13][30]} The distance from the North Pole to the Equator was then extrapolated from the measurement of theParis meridian arc between Dunkirk and Barcelona and was determined as5130740 toises. As the metre had to be equal to one ten-millionth of this distance, it was defined as 0.513074 toise or 3 feet and 11.296 lines of the Toise of Peru, which had been constructed in 1735 for theFrench Geodesic Mission to Peru.^[25][10] When the final result was known, a bar whose length was closest to the meridional definition of the metre was selected and placed in the National Archives on 22 June 1799 (4 messidor An VII in the Republican calendar) as a permanent record of the result.^[20]

In the 19th century, many astronomers and geodesists were engaged in detailed studies of the Earth's curvature along different meridian arcs. The analyses resulted in a great many model ellipsoids such as Plessis 1817, Airy 1830,Bessel 1841, Everest 1830, andClarke 1866.^[31] A comprehensive list of ellipsoids is given underEarth ellipsoid.

Historically anautical mile was defined as the length of one minute of arc along a meridian of a spherical earth. An ellipsoid model leads to a variation of the nautical mile with latitude. This was resolved by defining the nautical mile to be exactly 1,852 metres. However, for all practical purposes, distances are measured from the latitude scale of charts. As theRoyal Yachting Association says in its manual forday skippers: "1 (minute) of Latitude = 1 sea mile", followed by "For most practical purposes distance is measured from the latitude scale, assuming that one minute of latitude equals one nautical mile".^[32]

On a sphere, the meridian arc length is simply thecircular arc length. On an ellipsoid of revolution, for short meridian arcs, their length can be approximated using theEarth's meridional radius of curvature and the circular arc formulation.

For longer arcs, the length follows from the subtraction of twomeridian distances, the distance from the equator to a point at a latitudeφ.

This is an important problem in the theory of map projections, particularly thetransverse Mercator projection.

The main ellipsoidal parameters are,a,b,f, but in theoretical work it is useful to define extra parameters, particularly theeccentricity,e, and the thirdflatteningn. Only two of these parameters are independent and there are many relations between them:

Notation is problematic in this area. The meridian radius of curvature must be distinguished from the meridian distance. The notationM(φ) has been used for both. The definitions adopted here are:

• M(φ) is the meridian radius of curvature.
• m(φ) is the meridian distance.

Themeridian radius of curvature can be shown to be equal to:^[33][34]

${\displaystyle M(\phi )=\frac{a(1-e^2)}{{\left(1-e^2{\sin }^2\phi \right)}^{\frac{3}{2}}},}$

The arc length of an infinitesimal element of the meridian isdm =M(φ)dφ (withφ in radians). Therefore, the meridian distance from the equator to latitudeφ is

The distance formula is simpler when written in terms of theparametric latitude,

${\displaystyle m(\phi )=b{\int }_0^{\beta }\sqrt{1+e^{\prime 2}{\sin }^2\beta }d\beta ,}$

wheretanβ = (1 −f)tanφ ande′² =⁠e²/1 −e²⁠.

Even though latitude is normally confined to the range[−⁠π/2⁠,⁠π/2⁠], all the formulae given here apply to measuring distance around the complete meridian ellipse (including the anti-meridian). Thus the ranges ofφ,β, and the rectifying latitudeμ, are unrestricted.

The above integral is related to a special case of anincomplete elliptic integral of the third kind. In the notation of the onlineNIST handbook^[35] (Section 19.2(ii)),

${\displaystyle m(\phi )=a\left(1-e^2\right)\mathrm{\Pi }(\phi ,e^2,e).}$

It may also be written in terms ofincomplete elliptic integrals of the second kind (See the NIST handbookSection 19.6(iv)),

The calculation (to arbitrary precision) of the elliptic integrals and approximations are also discussed in the NIST handbook. These functions are also implemented in computer algebra programs such as Mathematica^[36] and Maxima.^[37]

The above integral may be expressed as an infinite truncated series by expanding the integrand in a Taylor series, performing the resulting integrals term by term, and expressing the result as a trigonometric series. In 1755,Leonhard Euler derived an expansion in thethird eccentricity squared.^[38]

Delambre in 1799^[39] derived a widely used expansion one²,

${\displaystyle m(\phi )=a(1-e^2)\left(D_0\phi +D_2\sin 2\phi +D_4\sin 4\phi +D_6\sin 6\phi +D_8\sin 8\phi +\cdots \right),}$

where

Richard Rapp gives a detailed derivation of this result.^[40]

Series with considerably faster convergence can be obtained by expanding in terms of thethird flatteningn instead of the eccentricity.

In 1837,Friedrich Bessel obtained one such series,^[41] which was later put into a simpler conjectured form byHelmert^[42] andKrüger^[43]

${\displaystyle m(\phi )=\frac{a+b}{2}\left(H_0\phi +H_2\sin 2\phi +H_4\sin 4\phi +H_6\sin 6\phi +H_8\sin 8\phi +\cdots \right),}$

with

Becausen changes sign whena andb are interchanged, and because the initial factor⁠1/2⁠(a +b) is constant under this interchange, half the terms in the expansions ofH_2k vanish.

Even though this results in more slowly converging series compared with${\displaystyle B_{2k}}$ shown below, such series are widely used in the specification for thetransverse Mercator projection by theNational Geospatial-Intelligence Agency^[44] and theOrdnance Survey of Great Britain.^[45]

In 1825, Bessel^[46] derived an expansion of the meridian distance in terms of theparametric latitudeβ in connection with his work ongeodesics,

${\displaystyle m(\phi )=\frac{a+b}{2}\left(B_0\beta +B_2\sin 2\beta +B_4\sin 4\beta +B_6\sin 6\beta +B_8\sin 8\beta +\cdots \right),}$

with

Because this series provides an expansion for the elliptic integral of the second kind, it can be used to write the arc length in terms of thegeodetic latitude as

The above series, to eighth order in eccentricity or fourth order in third flattening, provide millimetre accuracy. With the aid of symbolic algebra systems, they can easily be extended to sixth order in the third flattening which provides full double precision accuracy for terrestrial applications.

As described, Delambre^[39] and Bessel^[46] both wrote their series in a form that allows them to be generalized to arbitrary order. The coefficients in Bessel's series can be expressed particularly simply

where

${\displaystyle c_k=\sum _{j=0}^{\mathrm{\infty }}\frac{(2j-3)!!(2j+2k-3)!!}{(2j)!!(2j+2k)!!}{n}^{k+2j}}$

andk!! is thedouble factorial, extended to negative values via the recursion relation:(−1)!! = 1 and(−3)!! = −1.

The coefficients in Helmert's series can similarly be expressed by

${\displaystyle H_{2k}=(-1{)}^k(1-2k)(1+2k)B_{2k}.}$

This result was conjectured byFriedrich Helmert^[47] and proved by Kazushige Kawase.^[48] The extra factor(1 − 2k)(1 + 2k) appearing in${\displaystyle H_{2k}}$ originates from the additional expansion of${\displaystyle \frac{2n\sin 2\phi }{\sqrt{1+2n\cos 2\phi +n^2}}}$ and results in poorer convergence of the series in terms ofφ compared to the one inβ.

The trigonometric series given above can be conveniently evaluated usingClenshaw summation. This method avoids the calculation of most of the trigonometric functions and allows the series to be summed rapidly and accurately. The technique can also be used to evaluate the differencem(φ₁) −m(φ₂) while maintaining high relative accuracy.

Substituting the values for the semi-major axis and eccentricity of theWGS84 ellipsoid gives

whereφ⁽°⁾ =⁠φ/1°⁠ isφ expressed in degrees (and similarly forβ⁽°⁾).

On the ellipsoid the exact distance between parallels atφ₁ andφ₂ ism(φ₁) −m(φ₂). For WGS84 an approximate expression for the distanceΔm between the two parallels at ±0.5° from the circle at latitudeφ is given by

${\displaystyle \mathrm{\Delta }m=(111133-560\cos 2\phi )\text{ metres.}}$

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The distance from the equator to the pole, thequarter meridian (analogous to thequarter-circle), also known as theEarth quadrant, is

${\displaystyle m_{\mathrm{p}}=m\left(\frac{\pi }{2}\right).}$

It was part of the historicaldefinition of the metre and of thenautical mile, and used in the definition of thehebdomometre.

The quarter meridian can be expressed in terms of thecomplete elliptic integral of the second kind,

${\displaystyle m_{\mathrm{p}}=aE(e)=bE(i{e}^{\prime }).}$

where${\displaystyle e,e^{\prime }}$ are the first and secondeccentricities.

The quarter meridian is also given by the following generalized series:

${\displaystyle m_{\mathrm{p}}=\frac{\pi (a+b)}{4}{c}_0=\frac{\pi (a+b)}{4}\sum _{j=0}^{\mathrm{\infty }}{\left(\frac{(2j-3)!!}{(2j)!!}\right)}^2{n}^{2j},}$

(For the formula ofc₀, see section#Generalized series above.)This result was first obtained byJames Ivory.^[49]

The numerical expression for the quarter meridian on the WGS84 ellipsoid is

The polarEarth's circumference is simply four times quarter meridian:

${\displaystyle C_p=4m_p}$

Theperimeter of a meridian ellipse can also be rewritten in the form of a rectifying circle perimeter,C_p = 2πM_r. Therefore, therectifying Earth radius is:

${\displaystyle M_r=0.5(a+b)/c_0}$

It can be evaluated as6367449.146 m.

In some problems, we need to be able to solve the inverse problem: given the arc lengthm, find the latitudeφ. This may be solved byNewton's method, iterating

${\displaystyle {\phi }_{i+1}={\phi }_i-\frac{m({\phi }_i)-m}{M({\phi }_i)},}$

until convergence. A suitable starting guess is given byφ₀ =μ where

${\displaystyle \mu =\frac{\pi }{2}\frac{m}{m_{\mathrm{p}}}}$

is therectifying latitude. Note that it there is no need to differentiate the series form(φ), since the formula for the meridian radius of curvatureM(φ) can be used instead.

Alternatively, Helmert's series for the meridian distance can be reverted to give^[50][51]

${\displaystyle \phi =\mu +H_2^{\prime }\sin 2\mu +H_4^{\prime }\sin 4\mu +H_6^{\prime }\sin 6\mu +H_8^{\prime }\sin 8\mu +\cdots }$

where

Similarly, Bessel's series form in terms ofβ can be reverted to give^[52]

${\displaystyle \beta =\mu +B_2^{\prime }\sin 2\mu +B_4^{\prime }\sin 4\mu +B_6^{\prime }\sin 6\mu +B_8^{\prime }\sin 8\mu +\cdots ,}$

where

Adrien-Marie Legendre showed that the distance along a geodesic on a spheroid is the same as the distance along the perimeter of an ellipse.^[53] For this reason, the expression form in terms ofβ and its inverse given above play a key role in the solution of thegeodesic problem withm replaced bys, the distance along the geodesic, andβ replaced byσ, the arc length on the auxiliary sphere.^[46][54] The requisite series extended to sixth order are given by Charles Karney,^[55] Eqs. (17) & (21), withε playing the role ofn andτ playing the role ofμ.

• Online computation of meridian arcs on different geodetic reference ellipsoids

• Geodesy
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