geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...

and navigation, a meridian arc is the

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

. The purpose of measuring meridian arcs is to determine a

figure of the Earth Figure of the Earth is a Jargon, term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A Spherical Earth, sphere is a well-k ...

. One or more measurements of meridian arcs can be used to infer the shape of the

reference ellipsoid An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...

that best approximates the geoid 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 a ''geocentric ellipsoid'' intended to fit the entire world. The earliest determinations of the size of a spherical Earth required a single arc. Accurate survey work beginning in the 19th century required several

arc measurements Arc measurement, sometimes degree measurement (german: Gradmessung), is the astrogeodetic technique of determining of the radius of Earth – more specifically, the local Earth radius of curvature of the figure of the Earth – by relating the ...

in the region the survey was to be conducted, leading to a proliferation of reference ellipsoids around the world. The latest determinations use

astro-geodetic Geodetic astronomy or astronomical geodesy (astro-geodesy) is the application of astronomical methods into geodetic networks and other technical projects of geodesy. Applications The most important applications are: * Establishment of geodetic d ...

measurements and the methods of satellite geodesy to determine reference ellipsoids, especially the geocentric ellipsoids now used for global coordinate systems such as WGS 84 (see numerical expressions).

History of measurement

Early estimations of Earth's size are recorded from Greece in the 4th century BC, and from scholars at the caliph's

House of Wisdom The House of Wisdom ( ar, بيت الحكمة, Bayt al-Ḥikmah), also known as the Grand Library of Baghdad, refers to either a major Abbasid public academy and intellectual center in Baghdad or to a large private library belonging to the Abba ...

in the 9th century. The first realistic value was calculated by Alexandrian scientist

Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...

about 240 BC. He estimated that the meridian has a length of 252,000 stadia, with an error on the real value between -2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres). Eratosthenes described his technique in a book entitled ''On the measure of the Earth'', which has not been preserved. A similar method was used by Posidonius about 150 years later, and slightly better results were calculated in 827 by the arc measurement method, attributed to the Caliph

Al-Ma'mun Abu al-Abbas Abdallah ibn Harun al-Rashid ( ar, أبو العباس عبد الله بن هارون الرشيد, Abū al-ʿAbbās ʿAbd Allāh ibn Hārūn ar-Rashīd; 14 September 786 – 9 August 833), better known by his regnal name Al-Ma'mu ...

Ellipsoidal Earth

Early literature uses the term ''oblate spheroid'' to describe a sphere "squashed at the poles". Modern literature uses the term ''ellipsoid of revolution'' in place of spheroid, although the qualifying words "of revolution" are usually dropped. An

ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...

that is not an ellipsoid of revolution is called a triaxial ellipsoid. ''Spheroid'' and ''ellipsoid'' are used interchangeably in this article, with oblate implied if not stated.

17th and 18th centuries

Although it had been known since classical antiquity that the Earth was spherical, by the 17th century, evidence was accumulating that it was not a perfect sphere. In 1672, Jean Richer found the first evidence that gravity was not constant over the Earth (as it would be if the Earth were a sphere); he took a pendulum clock to

Cayenne Cayenne (; ; gcr, Kayenn) is the capital city of French Guiana, an overseas region and Overseas department, department of France located in South America. The city stands on a former island at the mouth of the Cayenne River on the Atlantic Oc ...

, French Guiana and found that it lost minutes per day compared to its rate at Paris. This indicated the acceleration 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 increasing latitude, gravitational acceleration being about 0.5% greater at the

geographical pole A geographical pole or geographic pole is either of the two points on Earth where its axis of rotation intersects its surface. The North Pole lies in the Arctic Ocean while the South Pole is in Antarctica. North and South poles are also define ...

s than at the

Equator The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...

. In 1687, Isaac Newton had published in the '' Principia'' as a proof that the Earth was an oblate spheroid of flattening equal to .Isaac Newton
''Principia'', Book III, Proposition XIX, Problem III
translated into English by Andrew Motte. A searchable modern translation is available a
17centurymaths
Search the followin
pdf file
for 'spheroid'. This was disputed by some, but not all, French scientists. A meridian arc of

Jean Picard Jean Picard (21 July 1620 – 12 July 1682) was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He is principally notable for his accurate measure of the size of the Earth, base ...

was extended to a longer arc by Giovanni Domenico Cassini and his son Jacques Cassini over the period 1684–1718.. Freely available online a
Archive.org
an
Forgotten Books
(). In addition the book has been reprinted b
Nabu Press
(), the first chapter covers the history of early surveys. 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 a ''prolate'' spheroid (with an equatorial radius less than the polar radius). To resolve the issue, the

French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific me ...

(1735) proposed expeditions to Peru ( Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan) and Lapland ( Maupertuis, Clairaut,

Camus Albert Camus ( , ; ; 7 November 1913 – 4 January 1960) was a French philosopher, author, dramatist, and journalist. He was awarded the 1957 Nobel Prize in Literature at the age of 44, the second-youngest recipient in history. His works ...

, Le Monnier, Abbe Outhier, Anders Celsius). The expedition to Peru is described in the French Geodesic Mission article and that to Lapland is described in the Torne Valley article. The resulting measurements at equatorial and polar latitudes confirmed that the Earth was best modelled by an oblate spheroid, supporting Newton. However, by 1743,

Clairaut's theorem Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatis ...

had completely supplanted Newton's approach. By the end of the century, Jean-Baptiste-Joseph Delambre had remeasured and extended the French arc from

Dunkirk Dunkirk (french: Dunkerque ; vls, label=French Flemish, Duunkerke; nl, Duinkerke(n) ; , ;) is a commune in the department of Nord in northern France.Mediterranean Sea (the

meridian arc of Delambre and Méchain Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...

). 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 the

Paris Meridian The Paris meridian is a meridian line running through the Paris Observatory in Paris, France – now longitude 2°20′14.02500″ East. It was a long-standing rival to the Greenwich meridian as the prime meridian of the world. The "Paris merid ...

was calculated as toises as specified by the standard toise bar in Paris. Defining this distance as exactly led to the construction of a new standard metre bar as toises.

19th century

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 1830, Everest 1830, and Clarke 1866. A comprehensive list of ellipsoids is given under Earth ellipsoid.

The nautical mile

Historically a

nautical mile A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude. Today ...

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 the Royal Yachting Association says in its manual for

day skipper The Day Skipper qualification confirms that the successful candidate has the knowledge needed to skipper a yacht on shorter, coastal cruises during daylight. The Royal Yachting Association administers the qualification, although most of the trainin ...

s: "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".

Calculation

On a sphere, the meridian arc length is simply the

circular arc length Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circula ...

. On an ellipsoid of revolution, for short meridian arcs, their length can be approximated using the

Earth's meridional radius of curvature Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...

and the circular arc formulation. For longer arcs, the length follows from the subtraction of two ''meridian distances'', the distance from the equator to a point at a latitude . This is an important problem in the theory of map projections, particularly the transverse Mercator projection. The main ellipsoidal parameters are, , , , but in theoretical work it is useful to define extra parameters, particularly the eccentricity, , and the third flattening . Only two of these parameters are independent and there are many relations between them: :

\begin
 f&=\frac\,, \qquad  e^2=f(2-f)\,, \qquad n=\frac=\frac\,,\\
b&=a(1-f)=a\sqrt\,,\qquad  e^2=\frac\,.
\end

Definition

The meridian radius of curvature can be shown to be equal to: Section 5.6. This reference includes the derivation of curvature formulae from first principles and a proof of Meusnier's theorem. (Supplements
Maxima files
and
Latex code and figures
:

M(\varphi) = \frac,

The arc length of an infinitesimal element of the meridian is (with in radians). Therefore, the meridian distance from the equator to latitude is :

\begin
m(\varphi) &=\int_0^\varphi M(\varphi) \, d\varphi \\
&= a(1 - e^2)\int_0^\varphi \left(1 - e^2 \sin^2 \varphi \right)^ \, d\varphi\,.
\end

The distance formula is simpler when written in terms of the parametric latitude, :

m(\varphi) = b\int_0^\beta\sqrt\,d\beta\,,

where and . Even though latitude is normally confined to the range , all the formulae given here apply to measuring distance around the complete meridian ellipse (including the anti-meridian). Thus the ranges of , , and the rectifying latitude , are unrestricted.

Relation to elliptic integrals

The above integral is related to a special case of an incomplete elliptic integral of the third kind. In the notation of the online

NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...

handbook
Section 19.2(ii)
, :

m(\varphi)=a\left(1-e^2\right)\,\Pi(\varphi,e^2,e)\,.

It may also be written in terms of incomplete elliptic integrals of the second kind (See the NIST handboo
Section 19.6(iv)
, :

\begin
m(\varphi) &= a\left(E(\varphi,e)-\frac\right) \\
&= a\left(E(\varphi,e)+\fracE(\varphi,e)\right) \\
&= b E(\beta, ie')\,.
\end

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 and Maxima.

Series expansions

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 the third eccentricity squared.

Expansions in the eccentricity ()

Delambre Jean Baptiste Joseph, chevalier Delambre (19 September 1749 – 19 August 1822) was a French mathematician, astronomer, historian of astronomy, and geodesist. He was also director of the Paris Observatory, and author of well-known books on th ...

in 1799Delambre, J. B. J. (1799)
''Méthodes Analytiques pour la Détermination d'un Arc du Méridien''; précédées d'un mémoire sur le même sujet par A. M. Legendre
De L'Imprimerie de Crapelet, Paris, 72–73 derived a widely used expansion on , :

m(\varphi)=\fraca\left(D_0\varphi+D_2\sin 2\varphi+D_4\sin4\varphi+D_6\sin6\varphi+D_8\sin8\varphi+\cdots\right)\,,

where :

D_8 &= \tfrac e^8 + \cdots. \end

Richard Rapp gives a detailed derivation of this result.

Expansions in the third flattening ()

Series with considerably faster convergence can be obtained by expanding in terms of the third flattening instead of the eccentricity. They are related by :

e^2 = \frac\,.

In 1837, Friedrich Bessel obtained one such series, which was put into a simpler form by Helmert, :

m(\varphi)=\frac2\left(H_0\varphi+H_2\sin 2\varphi+H_4\sin4\varphi+H_6\sin6\varphi+H_8\sin8\varphi+\cdots\right)\,,

with :

\begin
H_0 &= 1 + \tfrac n^2 + \tfrac n^4 + \cdots, \\
H_2 &= - \tfrac n + \tfrac n^3 + \cdots,&
H_6 &= - \tfrac n^3 + \cdots, \\
H_4 &= \tfrac n^2 - \tfrac n^4 - \cdots,\qquad&
H_8 &= \tfrac n^4 - \cdots.
\end

Because changes sign when and are interchanged, and because the initial factor is constant under this interchange, half the terms in the expansions of vanish. The series can be expressed with either or as the initial factor by writing, for example, :

\tfrac12(a+b) = \frac = a(1-n+n^2-n^3+n^4-\cdots)\,,

and expanding the result as a series in . Even though this results in more slowly converging series, such series are used in the specification for the transverse Mercator projection by the National Geospatial Intelligence Agency and the Ordnance Survey of Great Britain.A guide to coordinate systems in Great Britain
Ordnance Survey of Great Britain.

Series in terms of the parametric latitude

In 1825, Bessel English translation of Astron. Nachr. 4, 241–254 (1825), §5. derived an expansion of the meridian distance in terms of the parametric latitude in connection with his work on geodesics, :

m(\varphi)=\frac2\left(B_0\beta+B_2\sin 2\beta+B_4\sin4\beta+B_6\sin6\beta+B_8\sin8\beta+\cdots\right)\,,

with :

\begin
B_0 &= 1 + \tfrac n^2 + \tfrac n^4 + \cdots = H_0\,,\\
B_2 &= - \tfrac n + \tfrac n^3 + \cdots, &
B_6 &= - \tfrac n^3 + \cdots, \\
B_4 &= - \tfrac n^2 + \tfrac n^4 + \cdots, \qquad&
B_8 &= - \tfrac n^4 + \cdots.
\end

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 the geographic latitude as :

m(\varphi)=\frac2\left(B_0\varphi-B_2\sin2\varphi+B_4\sin4\varphi-B_6\sin6\varphi+B_8\sin8\varphi-\cdots-\frac\right)\,.

Generalized series

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. Delambre and Bessel both wrote their series in a form that allows them to be generalized to arbitrary order. The coefficients in Bessel's series can expressed particularly simply :

\dfrac\,, & \text k > 0\,, \end

where :

c_k = \sum_^\infty \frac n^

and is the

double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...

, extended to negative values via the recursion relation: and . The coefficients in Helmert's series can similarly be expressed generally by :

H_ = (-1)^k (1-2k)(1+2k) B_\,.

This result was conjectured by Friedrich Helmert and proved by Kazushige Kawase. The factor results in poorer convergence of the series in terms of compared to the one in .

Numerical expressions

The trigonometric series given above can be conveniently evaluated using Clenshaw 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 difference while maintaining high relative accuracy. Substituting the values for the semi-major axis and eccentricity of the WGS84 ellipsoid gives :

\begin
m(\varphi)&=\left(111\,132.952\,55\,\varphi^-16\,038.509\,\sin 2\varphi+16.833\,\sin4\varphi-0.022\,\sin6\varphi+0.000\,03\,\sin8\varphi\right)\mbox \\
&= \left(111\,132.952\,55\,\beta^-5\,346.170\,\sin 2\beta-1.122\,\sin4\beta-0.001\,\sin6\beta-0.5\times10^\,\sin8\beta\right)\mbox
\end

where is expressed in degrees (and similarly for ). On the ellipsoid the exact distance between parallels at and is . For WGS84 an approximate expression for the distance between the two parallels at ±0.5° from the circle at latitude is given by :

\Delta m=(111\,133 - 560\cos 2\varphi)\mbox

Quarter meridian

The distance from the equator to the pole, the quarter meridian (analogous to the

quarter-circle A circular sector, also known as circle sector or disk sector (symbol: ⌔), is the portion of a disk (a closed region bounded by a circle) enclosed by two radii and an arc, where the smaller area is known as the ''minor sector'' and the larger ...

), also known as the Earth quadrant, is :

m_\mathrm = m\left(\frac \pi 2\right)\,.

It was part of the historical

definition of the metre The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...

and of the

. The quarter meridian can be expressed in terms of the complete elliptic integral of the second kind, :

m_\mathrm=aE(e)=bE(ie').

where

e, e'

are the first and second eccentricities. The quarter meridian is also given by the following generalized series: :

m_\mathrm = \frac4 c_0 = \frac4 \sum_^\infty\left(\frac\right)^2 n^\,,

(For the formula of ''c''₀, see section #Generalized series above.) This result was first obtained by James Ivory. The numerical expression for the quarter meridian on the WGS84 ellipsoid is :

m_\mathrm=10\,001\,965.729\mbox

The polar Earth's circumference is simply four times quarter meridian: :

C_p=4m_p

The perimeter of a meridian ellipse can also be rewritten in the form of a rectifying circle perimeter, . Therefore, the

rectifying Earth radius Earth radius (denoted as ''R''🜨 or R_E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid, the radius ranges from a maximum of nearly (equatorial radius, deno ...

is: :

M_r=0.5(a+b)/c_0

It can be evaluated as .

The inverse meridian problem for the ellipsoid

In some problems, we need to be able to solve the inverse problem: given , determine . This may be solved by

Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...

, iterating :

\varphi_ = \varphi_i - \frac\,,

until convergence. A suitable starting guess is given by where :

\mu = \frac2 \frac m

is the

rectifying latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

. Note that it there is no need to differentiate the series for , since the formula for the meridian radius of curvature can be used instead. Alternatively, Helmert's series for the meridian distance can be reverted to giveAdams, Oscar S (1921)
''Latitude Developments Connected With Geodesy and Cartography''
US Coast and Geodetic Survey Special Publication No. 67. p. 127. :

\varphi = \mu + H'_2\sin2\mu + H'_4\sin4\mu + H'_6\sin6\mu + H'_8\sin8\mu + \cdots

where :

\begin
H'_2 &= \tfrac n - \tfrac n^3 + \cdots,&
H'_6 &= \tfrac n^3 + \cdots, \\
H'_4 &= \tfrac n^2 - \tfrac n^4 + \cdots,\qquad&
H'_8 &= \tfrac n^4 + \cdots.
\end

Similarly, Bessel's series for in terms of can be reverted to give :

\beta = \mu + B'_2\sin2\mu + B'_4\sin4\mu + B'_6\sin6\mu + B'_8\sin8\mu + \cdots,

where :

\begin
B'_2 &= \tfrac n - \tfrac n^3 + \cdots,&
B'_6 &= \tfrac n^3 - \cdots, \\
B'_4 &= \tfrac n^2 - \tfrac n^4 + \cdots,\qquad&
B'_8 &= \tfrac n^4 - \cdots.
\end

Adrien-Marie Legendre showed that the distance along a geodesic on an spheroid is the same as the distance along the perimeter of an ellipse. For this reason, the expression for in terms of and its inverse given above play a key role in the solution of the geodesic problem with replaced by , the distance along the geodesic, and replaced by , the arc length on the auxiliary sphere. The requisite series extended to sixth order are given by Charles Karney,
Addenda
Eqs. (17) & (21), with playing the role of and playing the role of .

References

External links

Online computation of meridian arcs on different geodetic reference ellipsoids
{{DEFAULTSORT:Meridian Arc Geodesy Meridians (geography) History of measurement

History of measurement

Ellipsoidal Earth

17th and 18th centuries

19th century

The nautical mile

Calculation

Definition

Relation to elliptic integrals

Series expansions

Expansions in the eccentricity ()

Expansions in the third flattening ()

Series in terms of the parametric latitude

Generalized series

Numerical expressions

Quarter meridian

The inverse meridian problem for the ellipsoid

See also

References

External links