View source for Meridian arc

{{short description|Distance along a portion of a meridian, for use in geodesy}}
In [[geodesy]], a '''meridian arc''' measurement is the distance between two points with the same [[longitude]], i.e., a [[arc (geometry)|segment]] of a [[meridian (geography)|meridian]] [[curve (geometry)|curve]] or its length. Two or more such determinations at different locations then specify the shape of the [[reference ellipsoid]] which best approximates the shape of the [[geoid]]. This process is called the determination of the [[figure of the Earth]]. The earliest determinations of the size of a [[spherical Earth]] required a single arc. The latest determinations use [[geodetic astronomy|astro-geodetic]] measurements and the methods of [[satellite geodesy]] to determine the reference ellipsoids.

Those interested in accurate expressions of the meridian arc for the [[WGS84]] ellipsoid should consult the subsection entitled [[#Numerical expressions|numerical expressions]].

==History of measurement{{anchor|History}}==
{{see also|History of geodesy|Earth's circumference#History|Earth's radius#History}}

===Spherical Earth===
{{main|Spherical Earth#History}}
{{further|Earth radius|Earth's circumference}}

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]] in the 9th century. The first realistic value was calculated by [[Alexandria]]n scientist [[Eratosthenes]] about 240 BC. He knew that on the [[summer solstice]] at local noon the [[sun]] goes through the [[zenith]] in the ancient Egyptian city of [[Syene]] (Assuan). He also knew from his own measurements that, at the same moment in his hometown of [[Alexandria]], the [[zenith distance]] was {{sfrac|1|50}} of a full circle (7.2°).

Assuming that Alexandria was due north of Syene, Eratosthenes concluded that the distance between Alexandria and Syene must be {{sfrac|1|50}} of Earth's circumference. Using data from [[caravan (travellers)|caravan]] travels, he estimated the distance to be 5,000 [[stadion (unit of length)|stadia]] (about 500 [[nautical miles]])—which implies a circumference of 252,000 stadia. Assuming the [[Ancient Greek units of measurement#Length|Attic stadion]] (185&nbsp;m) this corresponds to 46,620&nbsp;km, or 16% too great. However, if Eratosthenes used the Egyptian stadion (157.5&nbsp;m) his measurement turns out to be 39,690&nbsp;km, an error of only 1%. Syene is not precisely on the [[Tropic of Cancer]] and not directly south of Alexandria. The sun appears as a disk of 0.5°, and an estimate of the overland distance traveling along the [[Nile]] or through the desert couldn't be more accurate than about 10%.

Eratosthenes' estimation of Earth's size was accepted for nearly two thousand years. A similar method was used by [[Posidonius]] about 150 years later, and slightly better results were calculated in 827 by the [[grade measurement]]{{Citation needed|date=January 2012}} of the Caliph [[Al-Ma'mun]].

===Ellipsoidal Earth===
{{main|Earth ellipsoid}}

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]] 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 Earth|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]], [[French Guiana]] and found that it lost {{frac|2|1|2}} minutes per day compared to its rate at [[Paris]].<ref>{{cite book
  | last = Poynting
  | first = John Henry
  | authorlink =
  |author2=Joseph John Thompson
  | title = A Textbook of Physics, 4th Ed.
  | publisher = Charles Griffin & Co.
  | year = 1907
  | location = London
  | page = 20
  | url = https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA20#
  | doi =
  | id =
  | isbn = }}</ref><ref name="Lenzen1964">{{cite conference
  | first = Lenzen
  | last = Victor F.
  | authorlink =
  |author2=Robert P. Multauf
  | title = Paper 44: Development of gravity pendulums in the 19th century
  | booktitle = United States National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
  | pages = 307
  | publisher = [[Smithsonian Institution Press]]
  | year = 1964
  | location = Washington
  | url = https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA307
  | doi =
  | id =
  | accessdate = 2009-01-28}}</ref>  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]]s than at the [[Equator]].

In 1687, Newton had published in the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' as a proof that the Earth was an oblate [[spheroid]] of [[flattening]] equal to {{sfrac|1|230}}.<ref name=Newton>Isaac Newton: [https://books.google.com/books?id=KaAIAAAAIAAJ&pg=PA405 ''Principia'', Book III, Proposition XIX, Problem III], translated into English by Andrew Motte. A searchable modern translation is available at [http://17centurymaths.com 17centurymaths]. Search the following [http://17centurymaths.com/contents/newton/book3s1.pdf pdf file] for 'spheroid'.</ref> This was disputed by some, but not all, French scientists. A meridian arc of [[Jean Picard]] was extended to a longer arc by [[Giovanni Domenico Cassini]] and his son [[Jacques Cassini]]  over the period 1684–1718.<ref name=clarke>{{cite book
|ref= harv |year=1880
|last=Clarke
|first=Alexander Ross
|authorlink=Alexander Ross Clarke
|title=Geodesy
|url= https://archive.org/details/in.ernet.dli.2015.42772 |publisher=Clarendon Press
|location=Oxford
|oclc=2484948}}. Freely available online at [https://archive.org/details/cu31924004129650 Archive.org] and [https://www.forgottenbooks.com/en/books/Geodesy_10059832 Forgotten Books] ({{ISBN|9781440088650}}). In addition the book has been reprinted by [https://www.bookdepository.com/Geodesy-Alexander-Ross-Clarke/9781293262535 Nabu Press] ({{ISBN|978-1286804131}}), the first chapter covers the history of early surveys.</ref> 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]] (1735) proposed expeditions to Peru ([[Pierre Bouguer|Bouguer]], [[Louis Godin]], [[Charles Marie de La Condamine|de La Condamine]], [[Antonio de Ulloa]], [[Jorge Juan y Santacilia|Jorge Juan]]) and Lapland ([[Pierre Louis Maupertuis|Maupertuis]], [[Alexis Clairaut|Clairaut]], [[Charles Étienne Louis Camus|Camus]], [[Pierre Charles Le Monnier|Le Monnier]], Abbe Outhier, [[Anders Celsius|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.<ref name=clarke/> By 1743, [[Clairaut's theorem]] however had completely supplanted Newton's approach.

By the end of the century, [[Delambre]] had remeasured and extended the French arc from [[Dunkirk]] to the [[Mediterranean]]. 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]] was calculated as {{val|5130762}}&nbsp;[[toise]]s as specified by the standard toise bar in Paris. Defining this distance as exactly {{val|10000000|u=m}} led to the construction of a new standard [[metre]] bar as {{val|0.5130762}}&nbsp;toises.<ref name=clarke/>{{rp|22}}

====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 ellipsoid|Bessel 1830]], Everest 1830, and [[Alexander Ross Clarke|Clarke 1866]].<ref>{{cite book
|ref= harv |year=1866a
|first=Alexander Ross
|last=Clarke
|first2=Henry
|last2=James
|title=Comparisons of the standards of length of England, France, Belgium, Prussia, Russia, India, Australia, made at the Ordnance survey office, Southampton
|publisher=G.E. Eyre and W. Spottiswoode for H.M. Stationery Office
|location=London
|url=http://www.mdz-nbn-resolving.de/urn/resolver.pl?urn=urn:nbn:de:bvb:12-bsb10352632-5
|oclc=906501
|pages=281–87
|postscript=. Appendix on Figure of the Earth.
}}</ref> A comprehensive list of ellipsoids is given under [[Earth ellipsoid#Historical Earth ellipsoids|Earth ellipsoid]].

==Calculation==

The determination of the meridian distance, that is the distance from the equator to a point at a latitude {{mvar|φ}} on the ellipsoid is an important problem in the theory of map projections, particularly the [[transverse Mercator projection]]. Ellipsoids are normally specified in terms of the parameters defined above, {{mvar|a}}, {{mvar|b}}, {{mvar|f}}, but in theoretical work it is useful to define extra parameters, particularly the [[Eccentricity (mathematics)|eccentricity]], {{mvar|e}}, and the third [[flattening]] {{mvar|n}}. Only two of these parameters are independent and there are many relations between them:
:<math>\begin{align}
 f&=\frac{a-b}{a}\,, \qquad  e^2=f(2-f)\,, \qquad n=\frac{a-b}{a+b}=\frac{f}{2-f}\,,\\
b&=a(1-f)=a\sqrt{1-e^2}\,,\qquad  e^2=\frac{4n}{(1+n)^2}\,.
\end{align}</math>

<!--'''Comment''': notation is problematic in this area. One must distinguish between the meridian radius of curvature and the meridian distance. The notation {{math|''M''(''φ'')}} has been used for both. The definitions adopted here are:
*{{math|''M''(''φ'')}} is the meridian radius of curvature.
*{{math|''m''(''φ'')}} is the meridian distance.-->

The [[Earth radius#Meridional|meridian radius of curvature]] can be shown<ref>Rapp, R, (1991): [http://hdl.handle.net/1811/24333 Geometric Geodesy, Part I], &sect;3.5.1, pp. 28–32.</ref><ref name=osborne>{{citation|first=Peter
|last=Osborne
|title=The Mercator Projections
|year=2013
|doi=10.5281/zenodo.35392
|postscript=.  Section 5.6. This reference includes the derivation of curvature formulae from first principles and a proof of Meusnier's theorem. (Supplements: [https://zenodo.org/record/35561 Maxima files] and  [https://zenodo.org/record/35562 Latex code and figures])}}</ref> to be equal to
:<math> M(\varphi) = \frac{a(1 - e^2)}{\left(1 - e^2 \sin^2 \varphi \right)^\frac32},</math>
so that the arc length of an infinitesimal element of the meridian is {{math|''dm'' {{=}} ''M''(''φ'') ''dφ''}} (with {{mvar|φ}} in radians). Therefore, the meridian distance from the equator to latitude {{mvar|φ}} is
:<math>\begin{align}
m(\varphi) &=\int_0^\varphi M(\varphi) \, d\varphi \\
&= a(1 - e^2)\int_0^\varphi \left(1 - e^2 \sin^2 \varphi \right)^{-\frac32} \, d\varphi\,.
\end{align}</math>
The distance formula is simpler when written in terms of the
[[Latitude#Reduced (or parametric) latitude|parametric latitude]],
:<math>m(\varphi) = b\int_0^\beta\sqrt{1 + e'^2\sin^2\beta}\,d\beta\,,</math>
where {{math|tan ''β'' {{=}} (1 − ''f'')tan ''φ''}} and {{math|''e''′<sup>2</sup> {{=}} {{sfrac|''e''<sup>2</sup>|1 − ''e''<sup>2</sup>}}}}.

The distance from the equator to the pole, the quarter meridian, is
:<math>m_\mathrm{p} = m\left(\frac \pi 2\right)\,.</math>

Even though latitude is normally confined to the range {{math|[−{{sfrac|π|2}},{{sfrac|π|2}}]}}, all the formulae given here apply to measuring distance around the complete meridian ellipse (including the anti-meridian). Thus the ranges of {{mvar|φ}}, {{mvar|β}}, and the rectifying latitude {{mvar|μ}}, are unrestricted.

===Relation to elliptic integrals===

The above integral is related to a special case of an [[Elliptic integral#Incomplete elliptic integral of the third kind|incomplete elliptic integral of the third kind]]. In the notation of the online NIST handbook<ref>F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors,
2010, [http://dlmf.nist.gov NIST Handbook of Mathematical Functions] (Cambridge
University Press).</ref> ([http://dlmf.nist.gov/19.2#ii Section 19.2(ii)]),
:<math>m(\varphi)=a\left(1-e^2\right)\,\Pi(\varphi,e^2,e)\,.</math>
It may also be written in terms of [[Elliptic integral#Incomplete elliptic integral of the second kind|incomplete elliptic integrals of the second kind]] (See the NIST handbook [http://dlmf.nist.gov/19.6#iv Section 19.6(iv)]),
:<math>\begin{align}
m(\varphi) &= a\left(E(\varphi,e)-\frac{e^2\sin\varphi\cos\varphi}{\sqrt{1-e^2\sin^2\varphi}}\right) \\
&= a\left(E(\varphi,e)+\frac{d^2}{d\varphi^2}E(\varphi,e)\right) \\
&= b E(\beta, ie')\,.
\end{align}</math>
The quarter meridian can be expressed in terms of the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]],
:<math>m_\mathrm{p}=aE(e)=bE(ie').</math>
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<ref>[http://reference.wolfram.com/mathematica/guide/EllipticIntegrals.html Mathematica guide: Elliptic Integrals]</ref> and Maxima.<ref>[http://maxima.sf.net Maxima], 2009, A computer algebra system, version 5.20.1.</ref>

===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|Euler]]<ref>{{cite journal
|ref = harv |year = 1755
|last = Euler |first = L. |authorlink = Leonhard Euler
|title = Élémens de la trigonométrie sphéroïdique tirés de la méthode des plus grands et plus petits
|trans-title=Elements of spheroidal trigonometry taken from the method of maxima and minima
|language = French
|journal = Mémoires de l'Académie Royale des Sciences de Berlin 1753
|pages = 258&ndash;293
|volume = 9
|url = https://books.google.com/books?id=QIIfAAAAYAAJ&pg=PA258
|postscript = . [https://books.google.com/books?id=QIIfAAAAYAAJ&pg=PA362-IA1 Figures].
}}</ref> derived an expansion in the [[Eccentricity (mathematics)#Ellipses|third eccentricity]] squared.

====Expansions in the eccentricity ({{mvar|e}})====
[[Jean Baptiste Joseph Delambre|Delambre]] in 1799<ref name=delambre>Delambre, J. B. J. (1799): [https://books.google.com/books?id=DBAOAAAAQAAJ&pg=PA72 ''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</ref> derived a widely used expansion on {{math|''e''<sup>2</sup>}},

:<math>m(\varphi)=\frac{b^2}a\left(D_0\varphi+D_2\sin 2\varphi+D_4\sin4\varphi+D_6\sin6\varphi+D_8\sin8\varphi+\cdots\right)\,,</math>
where
:<math>\begin{align}
D_0 &= 1 + \tfrac{3}{4} e^2 + \tfrac{45}{64} e^4 + \tfrac{175}{256} e^6 + \tfrac{11025}{16384} e^8 + \cdots, \\
D_2 &= - \tfrac{3}{8} e^2 - \tfrac{15}{32} e^4 - \tfrac{525}{1024} e^6 - \tfrac{2205}{4096} e^8 - \cdots, \\
D_4 &= \tfrac{15}{256} e^4 + \tfrac{105}{1024} e^6 + \tfrac{2205}{16384} e^8 + \cdots, \\
D_6 &= - \tfrac{35}{3072} e^6 - \tfrac{105}{4096} e^8 - \cdots, \\
D_8 &= \tfrac{315}{131072} e^8 + \cdots.
\end{align}</math>

Rapp<ref>Rapp, R, (1991), §3.6, pp. 36–40.</ref> gives a detailed derivation of this result.  In this article, trigonometric terms of the form {{math|sin 4''φ''}} are interpreted as {{math|sin(4''φ'')}}.

====Expansions in the third flattening ({{mvar|n}})====

Series with considerably faster convergence can be obtained by expanding in terms of the [[Flattening#First, second and third flattening|third flattening]] {{mvar|n}} instead of the eccentricity. They are related by
:<math>e^2 = \frac{4n}{(1+n)^2}\,.</math>

In 1837, [[Friedrich Bessel|Bessel]] obtained one such series,<ref>{{Cite journal | ref = harv| last = Bessel | first = F. W. | authorlink = Friedrich Bessel| doi = 10.1002/asna.18370142301 | title = Bestimmung der Axen des elliptischen Rotationssphäroids, welches den vorhandenen Messungen von Meridianbögen der Erde am meisten entspricht | language = German|trans-title=Estimation of the axes of the ellipsoid through measurements of the meridian arc| journal = Astronomische Nachrichten | volume = 14 | issue = 333 | pages = 333–346| year = 1837 | pmid =  | pmc = | bibcode = 1837AN.....14..333B }}</ref> which was put into a simpler form by  [[Friedrich Robert Helmert|Helmert]],<ref>Helmert, F. R. (1880): [https://books.google.com/books?id=0l0OAAAAYAAJ&pg=PA44 ''Die mathematischen und physikalischen Theorieen der höheren Geodäsie'', Einleitung und 1 Teil], Druck und Verlag von B. G. Teubner, Leipzig, § 1.7, pp. 44–48.  English translation (by the Aeronautical Chart and Information Center, St. Louis) available at  {{DOI|10.5281/zenodo.32050}}</ref><ref>Krüger, L. (1912): ''[https://dx.doi.org/10.2312/GFZ.b103-krueger28 Konforme Abbildung des Erdellipsoids in der Ebene]''. Royal Prussian Geodetic Institute, New Series 52, page 12</ref>

:<math>m(\varphi)=\frac{a+b}2\left(H_0\varphi+H_2\sin 2\varphi+H_4\sin4\varphi+H_6\sin6\varphi+H_8\sin8\varphi+\cdots\right)\,,</math>
with
:<math>\begin{align}
H_0 &= 1 + \tfrac{1}{4} n^2 + \tfrac{1}{64} n^4 + \cdots, \\
H_2 &= - \tfrac{3}{2} n + \tfrac{3}{16} n^3 + \cdots,&
H_6 &= - \tfrac{35}{48} n^3 + \cdots, \\
H_4 &= \tfrac{15}{16} n^2 - \tfrac{15}{64} n^4 - \cdots,\qquad&
H_8 &= \tfrac{315}{512} n^4 - \cdots.
\end{align}</math>

Because {{mvar|n}} changes sign when {{mvar|a}} and {{mvar|b}} are interchanged, and because the initial factor {{math|{{sfrac|1|2}}(''a'' + ''b'')}} is constant under this interchange, half the terms in the expansions of {{math|''H''<sub>2''k''</sub>}} vanish.

The series can be expressed with either {{mvar|a}} or {{mvar|b}} as the initial factor by writing, for example,
:<math>\frac12(a+b) = \frac{a}{1+n} = a(1-n+n^2-n^3+n^4-\cdots)\,,</math>
and expanding the result as a series in {{mvar|n}}. 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]]<ref>J. W. Hager, J.F. Behensky, and B.W. Drew, 1989. Defense Mapping Agency Technical Report TM 8358.2. [http://earth-info.nga.mil/GandG/publications/tm8358.2/TM8358_2.pdf The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS)]</ref> and the [[Ordnance Survey of Great Britain]].<ref name=osgb>[http://www.ordnancesurvey.co.uk/docs/support/guide-coordinate-systems-great-britain.pdf A guide to coordinate systems in Great Britain], Ordnance Survey of Great Britain.</ref>

====Series in terms of the parametric latitude====

In 1825, Bessel <ref name=bessel25>{{cite journal
|ref = {{harvid|Bessel|1825}}
|last = Bessel
|first = F. W.
|title = The calculation of longitude and latitude from geodesic measurements (1825)
|authorlink = Friedrich Bessel
|journal = Astron. Nachr.
|year = 2010
|volume = 331
|number = 8
|pages = 852&ndash;861
|arxiv = 0908.1824
|doi = 10.1002/asna.201011352
|postscript=. English translation of Astron. Nachr. '''4''', 241–254 (1825), §5.
|bibcode = 2010AN....331..852K
}}</ref> derived an expansion of the meridian distance in terms of the parametric latitude {{mvar|β}} in connection with his work on [[geodesics on an ellipsoid|geodesics]],

:<math>m(\varphi)=\frac{a+b}2\left(B_0\beta+B_2\sin 2\beta+B_4\sin4\beta+B_6\sin6\beta+B_8\sin8\beta+\cdots\right)\,,</math>
with
:<math>\begin{align}
B_0 &= 1 + \tfrac{1}{4} n^2 + \tfrac{1}{64} n^4 + \cdots = H_0\,,\\
B_2 &= - \tfrac{1}{2} n + \tfrac{1}{16} n^3 + \cdots, &
B_6 &= - \tfrac{1}{48} n^3 + \cdots, \\
B_4 &= - \tfrac{1}{16} n^2 + \tfrac{1}{64} n^4 + \cdots, \qquad&
B_8 &= - \tfrac{5}{512} n^4 + \cdots.
\end{align}</math>

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

:<math>m(\varphi)=\frac{a+b}2\left(B_0\varphi-B_2\sin2\varphi+B_4\sin4\varphi-B_6\sin6\varphi+B_8\sin8\varphi-\cdots-\frac{2n\sin2\varphi}{\sqrt{1+2n\cos2\varphi+n^2}}\right)\,.</math>

====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<ref name=delambre/> and Bessel<ref name=bessel25/> 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

:<math>B_{2k} =
\begin{cases} c_0\,, & \text{if }k = 0\,, \\[5px]
\dfrac{c_k}{k}\,, & \text{if } k > 0\,,
\end{cases}</math>
where
:<math>c_k = \sum_{j=0}^\infty \frac{(2j-3)!!\, (2j+2k-3)!!}{(2j)!!\, (2j+2k)!!} n^{k+2j}</math>

and {{math|''k''!!}} is the [[double factorial]], extended to negative values via the recursion relation: {{nowrap|(−1)!! {{=}} 1}} and {{nowrap|(−3)!! {{=}} −1}}.

The coefficients in Helmert's series can similarly be expressed generally by
:<math>H_{2k} = (-1)^k (1-2k)(1+2k) B_{2k}\,.</math>

This result was conjected by Helmert<ref>Helmert (1880), §1.11</ref> and proved by Kawase.<ref>Kawase, K. (2011): [http://www.gsi.go.jp/common/000062452.pdf A General Formula for Calculating Meridian Arc Length and its Application to Coordinate Conversion in the Gauss-Krüger Projection], Bulletin of the [[Geospatial Information Authority of Japan]], '''59''', 1–13</ref>

The factor {{math|(1 − 2''k'')(1 + 2''k'')}} results in poorer convergence of the series in terms of {{mvar|φ}} compared to the one in {{mvar|β}}.

The quarter meridian is given by
:<math>m_\mathrm{p} = \frac{\pi(a+b)}4 c_0 = \frac{\pi(a+b)}4 \sum_{j=0}^\infty\left(\frac{(2j-3)!!}{(2j)!!}\right)^2 n^{2j}\,,</math>
a result which was first obtained by Ivory.<ref>{{Cite journal| ref = harv| author1-link = James Ivory (mathematician)| doi = 10.1017/s0080456800030817| title = A new series for the rectification of the ellipsis| journal = [[Transactions of the Royal Society of Edinburgh]] | volume = 4| issue = 2| pages = 177&ndash;190| year = 1798| last1 = Ivory | first1 = J. |url = https://books.google.com/books?id=FaUaqZZYYPAC&pg=PA177}}</ref>

====Numerical expressions====

The trigonometric series given above can be conveniently evaluated using [[Clenshaw algorithm#Geodetic applications|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 {{math|''m''(''φ''<sub>1</sub>) − ''m''(''φ''<sub>2</sub>)}} while maintaining high relative accuracy.

Substituting the values for the semi-major axis and eccentricity of the [[World Geodetic System|WGS84]] ellipsoid gives
:<math>\begin{align}
m(\varphi)&=\left(111\,132.952\,55\,\varphi^{(\circ)}-16\,038.509\,\sin 2\varphi+16.833\,\sin4\varphi-0.022\,\sin6\varphi+0.000\,03\,\sin8\varphi\right)\mbox{ metres} \\
&= \left(111\,132.952\,55\,\beta^{(\circ)}-5\,346.170\,\sin 2\beta-1.122\,\sin4\beta-0.001\,\sin6\beta-0.5\times10^{-6}\,\sin8\beta\right)\mbox{ metres,}
\end{align}</math>
where {{math|''φ''<sup>(</sup>°<sup>)</sup> {{=}} {{sfrac|''φ''|1°}}}} is {{mvar|φ}} expressed in degrees (and similarly for {{math|''β''<sup>(</sup>°<sup>)</sup>}}).

For the WGS84 ellipsoid the quarter meridian is
:<math> m_\mathrm{p}= \frac{\pi(a+b)}4 c_0 =10\,001\,965.729\mbox{ m.}</math>
The perimeter of a meridian ellipse is {{math|4''m''<sub>p</sub> {{=}} 2π(''a'' + ''b'')''c''<sub>0</sub>}}. Therefore, {{math|{{sfrac|1|2}}(''a'' + ''b'')''c''<sub>0</sub>}} is the radius of the circle whose circumference is the same as the perimeter of a meridian ellipse. This defines the [[Earth radius#Rectifying radius|rectifying Earth radius]] as {{val|6367449.146|u=m}}.

On the ellipsoid the exact distance between parallels at {{math|''φ''<sub>1</sub>}} and {{math|''φ''<sub>2</sub>}} is {{math|''m''(''φ''<sub>1</sub>) − ''m''(''φ''<sub>2</sub>)}}. For WGS84 an approximate expression for the distance {{math|Δ''m''}} between the two parallels at &plusmn;0.5° from the circle at latitude {{mvar|φ}} is given by

:<math>\Delta m=(111\,133 - 560\cos 2\varphi)\mbox{ metres.}</math>

==The inverse meridian problem for the ellipsoid==

In some problems, we need to be able to solve the inverse problem: given {{mvar|m}}, determine {{mvar|φ}}. This may be solved by [[Newton's method]], iterating
:<math>\varphi_{i+1} = \varphi_i - \frac{m(\varphi_i) - m}{M(\varphi_i)}\,,</math>
until convergence.  A suitable starting guess is given by {{math|''φ''<sub>0</sub> {{=}} ''μ''}} where
:<math>\mu = \frac{\pi}2 \frac m{m_\mathrm{p}}</math>
is the [[rectifying latitude]].  Note that it there is no need to differentiate the series for {{math|''m''(''φ'')}}, since the formula for the meridian radius of curvature {{math|''M''(''φ'')}} can be used instead.

Alternatively, Helmert's series for the meridian distance can be reverted to give<ref>Helmert (1880), §1.10</ref><ref name=adams1921>Adams, Oscar S (1921). ''Latitude Developments Connected With Geodesy and Cartography, (with tables, including a table for Lambert equal area meridional projection)''. Special Publication No. 67 of the US Coast and Geodetic Survey. A facsimile of this publication is available from the US National Oceanic and Atmospheric Administration ([[NOAA]]) at http://docs.lib.noaa.gov/rescue/cgs_specpubs/QB275U35no671921.pdf, p. 127</ref>

:<math>\varphi = \mu + H'_2\sin2\mu + H'_4\sin4\mu + H'_6\sin6\mu + H'_8\sin8\mu + \cdots</math>
where
:<math>\begin{align}
H'_2 &= \tfrac{3}{2} n - \tfrac{27}{32} n^3 + \cdots,&
H'_6 &= \tfrac{151}{96} n^3 + \cdots, \\
H'_4 &= \tfrac{21}{16} n^2 - \tfrac{55}{32} n^4 + \cdots,\qquad&
H'_8 &= \tfrac{1097}{512} n^4 + \cdots.
\end{align}</math>

Similarly, Bessel's series for {{mvar|m}} in terms of {{mvar|β}} can be reverted to give<ref>Helmert (1880), §5.6</ref>
:<math>\beta = \mu + B'_2\sin2\mu + B'_4\sin4\mu + B'_6\sin6\mu + B'_8\sin8\mu + \cdots,</math>
where
:<math>\begin{align}
B'_2 &= \tfrac{1}{2} n - \tfrac{9}{32} n^3 + \cdots,&
B'_6 &= \tfrac{29}{96} n^3 - \cdots, \\
B'_4 &= \tfrac{5}{16} n^2 - \tfrac{37}{96} n^4 + \cdots,\qquad&
B'_8 &= \tfrac{539}{1536} n^4 - \cdots.
\end{align}</math>

Legendre<ref>{{cite book
|ref = harv |year = 1811
|last = Legendre |first = A. M. |authorlink = Adrien-Marie Legendre
|title = Exercices de Calcul Intégral sur Divers Ordres de Transcendantes et sur les Quadratures
|language = French
|trans-title=Exercises in Integral Calculus
|page=180
|publisher = Courcier
|location = Paris
|url = https://books.google.com/books?id=riIOAAAAQAAJ
|oclc = 312469983
}}</ref> 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 {{mvar|m}} in terms of {{mvar|β}} and its inverse given above play a key role in the solution of the [[geodesics on an ellipsoid|geodesic problem]] with {{mvar|m}} replaced by {{mvar|s}}, the distance along the geodesic, and {{mvar|β}} replaced by {{mvar|σ}}, the arc length on the auxiliary sphere.<ref name=bessel25/><ref>Helmert (1880), Chap. 5</ref> The requisite series extended to sixth order are given by Karney,<ref>{{Cite journal | ref = harv| last1 = Karney | first1 = C. F. F. | doi = 10.1007/s00190-012-0578-z | title = Algorithms for geodesics | journal = Journal of Geodesy | volume = 87 | pages = 43–55| year = 2013| issue = 1| pmid =  | pmc = | postscript = {{open access}} [http://geographiclib.sf.net/geod-addenda.html Addenda].|arxiv = 1109.4448 |bibcode = 2013JGeod..87...43K }}</ref> Eqs. (17) & (21), with {{mvar|ε}} playing the role of {{mvar|n}} and {{mvar|τ}} playing the role of {{mvar|μ}}.

== See also ==
{{columns-list|colwidth=30em|
* [[History of geodesy]]
* [[Geodesy]]
* [[Reference ellipsoid]]
* [[Paris meridian]] (West Europe-Africa Meridian-arc)
* [[French Geodesic Mission]]
* [[Struve Geodetic Arc]]
* [[Torne Valley#French Geodesic Mission]]
* [[Rectifying latitude]]
* [[Geodesics on an ellipsoid]]
}}

== References ==
{{reflist|30em}}

== External links ==
* [http://www.in-dubio-pro-geo.de/?file=ellip/marc0&english=1 Online computation of meridian arcs on different geodetic reference ellipsoids]

{{DEFAULTSORT:Meridian Arc}}
[[Category:Geodesy]]
[[Category:Lines of longitude]]