Meridian arc

In geodesy, a meridian arc measurement is a highly accurate determination of the distance between two points with the same longitude. 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 astro-geodetic measurements and the methods of satellite geodesy to determine the reference ellipsoids.

The Earth as a sphere

Early estimations of Earth's radius are recorded from Egypt in 240 BC, and from Baghdad caliphs in the 9th century, but it was the Alexandrian scientist Eratosthenes who first calculated the circumference a reasonably good approximate value for the radius. 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 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 1/50 of Earth's circumference. Using data from caravan travels, he estimated the distance to be 5000 stadia (about 500 nautical miles)—which implies a circumference of 252,000 stadia. Assuming the Attic stadion (185 m) this corresponds to 46,620 km, or 16% too great. However, if Eratosthenes used the Egyptian stadion (157.5 m) his measurement turns out to be 39,690 km, an error of only 1%. Considering geometry and ancient conditions, a 16% error is more probable:^{[citation needed]} 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 AD 827 by the Gradmessung of the Caliph al-Ma'mun.

The Earth as an ellipsoid

Comment: early literature uses the term oblate spheroid to describe a sphere "squashed at the poles". Modern literature uses the term "ellipsoid of revolution" although the qualifying words "of revolution" are usually dropped. An ellipsoid which is not an ellipsoid of revolution is called a tri-axial ellipsoid. Spheroid and ellipsoid are used interchangeably in this article.

The method of determining the ellipsoid

High precision land surveys can be used determine the distance between two places at "almost" the same longitude by measuring a base line and a chain of triangles. (Suitable stations for the end points are rarely at the same longitude). The distance Δ along the meridian from one end point to a point at the same latitude as the second end point is then calculated by trigonometry. The surface distance Δ is reduced to Δ', the corresponding distance at mean sea level. The intermediate distances to points on the meridian at the same latitudes as other stations of the survey may also be calculated.

     The geographic latitudes of both end points, φ_s (standpoint) and φ_f (forepoint) and possibly at other points are determined by astrogeodesy, observing the zenith distances of sufficient numbers of stars. If latitudes are measured at end points only, the radius of curvature at the mid-point of the meridian arc can be calculated from R = Δ'/(|φ_s-φ_f|). A second meridian arc will allow the derivation of two parameters required to specify a reference ellipsoid. Longer arcs with intermediate latitude determinations can completely determine the ellipsoid. In practice multiple arc measurements are used to determine the ellipsoid parameters by the method of least squares. The parameters determined are usually the semi-major axis, \(a\), and either the semi-minor axis, \(b\), or the inverse flattening \(1/f\), (where the flattening is  \(f=(a-b)/a\)).

The eighteenth century

In 1687 Newton had published in the Principia a proof that the earth was an oblate spheroid (of inverse flattening equal to 230). This was contended by some, but not all, French scientists. A meridian arc of Picard was extended to a longer arc by Cassini (J.D.) over the period 1684–1718. 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). (The history of the meridian arc from 1600 to 1880 is fully covered in the first chapter of Geodesy by Alexander Ross Clarke.).

     To resolve the issue, the French Academy of Sciences (1735) proposed expeditions to Peru (Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan ) and Lappland (Maupertuis, Clairaut, Camus, Le Monnier, Abbe Outhier, Celsius). (The expedition to Peru is described on the page French Geodesic Mission and that to Lappland is described on the page Torne Valley.) The resulting measurements at equatorial and polar latitudes confirmed that the earth was best modelled by an oblate spheroid, supporting Newton.

     By the end of the century the French arc had been remeasured and extended from Dunkirk to the Mediterranean. (By Delambre). 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 5130762 toise (as specified by the standard toise bar in Paris). Defining this distance as exactly 10,000,000 m led to the construction of a new standard metre bar as 0.5130762 toise. (See Clarke, pp18–22).

The nineteenth and twentieth centuries

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 Airy 1830, Bessel 1830, Everest 1830, Clarke 1866 and Hayford 1910 (also adopted as International 1924. A comprehensive list of ellipsoids is given under Figure of the Earth.

     Geodesy no longer uses simple meridian arcs, but complex networks with hundreds of fixed points linked by the methods of satellite geodesy.

Meridian distance on the ellipsoid

The determination of the meridian distance, that is the distance from the equator to a point at a latitude \(\varphi\) 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, \(a\), \(b\), \(1/f\),  but in theoretical work it is useful to define extra parameters, particularly the eccentricity, \(e\), and the third flattening \(n\). Only two of these parameters are independent and there are many relations between them:

\[ \begin{align} f&=\frac{a-b}{a}, \qquad e^2=f(2-f), \qquad n=\frac{a-b}{a+b}=\frac{f}{2-f}\\ b&=a(1-f)=a(1-e^2)^{1/2},\qquad e^2=\frac{4n}{(1+n)^2}. \end{align} \]

The radius of curvature is defined as

\[ M(\varphi) = \frac{a(1 - e^2)}{\left (1 - e^2 \sin^2 \varphi \right )^{3/2}},\]

so that the arc length of an infinitesimal element of the meridian is \(M(\varphi) \, d\varphi\) (with \(\varphi\) in radians). Therefore the meridian distance from the equator to latitude \(\varphi\) is

\[ \begin{align} m(\varphi) &=\int_0^\varphi M(\varphi) \, d\varphi = a(1 - e^2)\int_0^\varphi \left (1 - e^2 \sin^2 \varphi \right )^{-3/2} \, d\varphi. \end{align} \]

The distance from the equator to the pole, the polar distance, is

\[ m_p = m(\pi/2).\, \]

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 handbook[9] (Chapter 19.2),

\[ m(\varphi)=a\big(1-e^2\big)\,\Pi(\varphi,e,e). \]

It may also be written in terms of incomplete elliptic integrals of the second kind by a change of integration variable from latitude \(\varphi\) to \(\theta\), the complement of the reduced latitude:

\[ \theta(\varphi)=\cot{}^{-1}\left(\sqrt{1-e^2}\tan\varphi\right). \]

The result is

\[ m(\varphi) = m_p-aE\big(\theta(\varphi),e\big) \]

where,

\[ E\big(\theta,e\big) =\int_0^{\theta} \sqrt{1 - e^2 \sin^2 \theta} \,d\theta , \]

and the polar distance is given by a complete elliptic integral of the second kind.

\[ m_p =aE(e)=a\int_0^{\pi/2} \sqrt{1 - e^2 \sin^2 \theta}\,d\theta . \]

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 in eccentricity

The above integral may be approximated by a truncated series in the square of the eccentricity (approximately 1/150) by expanding the integrand in a binomial series. Setting \(s=\sin\varphi\,\!\),

\[ \left (1 - e^2 \sin^2 \varphi \right )^{-3/2} =1+b_2 e^2s^2+b_4 e^4s^4+b_6 e^6s^6+b_8e^8s^8+\cdots, \]

where

\[ b_2=\frac{3}{2},\qquad b_4=\frac{15}{8},\qquad b_6=\frac{35}{16},\qquad b_8=\frac{315}{128}. \]

Using simple trigonometric identities the powers of \(\sin\varphi\,\!\) may be reduced to combinations of factors of \(\cos 2p\varphi\,\!\). Collecting terms with the same cosine factors and integrating gives the following series, first given by Delambre in 1799.

\[ m(\varphi)=A_0\varphi+A_2\sin 2\varphi+A_4\sin4\varphi +A_6\sin6\varphi+A_8\sin8\varphi+\cdots, \]

where

\[ \begin{align} A_0 &= \quad a(1-e^2) \left(1+\frac{3}{4}e^2+\frac{45}{64}e^4+\frac{175}{256}e^6+\frac{11025}{16384}e^8 \right) \\ A_2 &= -\frac{a(1-e^2)}{2}\left(\frac{3}{4}e^2+\frac{15}{16}e^4+\frac{525}{512}e^6+\frac{2205}{2048}e^8 \right)\\ A_4 &= \quad\frac{a(1-e^2)}{4}\left(\frac{15}{64}e^4+\frac{105}{256}e^6+\frac{2205}{4096}e^8\right)\\ A_6 &= -\frac{a(1-e^2)}{6}\left(\frac{35}{512}e^6+\frac{315}{2048}e^8\right)\\ A_8 &= \quad\frac{a(1-e^2)}{8}\left(\frac{315}{16384}e^8\right) \end{align} \]

The numerical values for the semi-major axis and eccentricity of the WGS84 ellipsoid give, in metres,

\[ m(\varphi)=6367449.146\varphi -16038.509\sin 2\varphi +16.833\sin4\varphi -0.022\sin6\varphi +0.00003\sin8\varphi \]

The first four terms have been rounded to the nearest millimetre whilst the eighth order term gives rise to sub-millimetre corrections. Tenth order series are employed in modern "wide zone" implementations of the transverse Mercator projection. (See Stuifbergen).

For the WGS84 ellipsoid the distance from equator to pole is given (in metres) by

\[ m_p= \frac{1}{2}\pi A_0=10,001,965.729. \]

The perimeter of a meridian ellipse is \(4m_p=2\pi A_0\). Therefore \(A_0\) is the radius of the circle whose circumference is the same as the perimeter of a meridian ellipse. This defines the mean Earth radius as 6,367,449.146 m. Dividing \(m_p\) by 90 gives the mean value of the meridian distance per degree of latitude as 111,132 m.

On the ellipsoid the exact distance between parallels at \(\varphi_1\) and \(\varphi_2\) is \(m(\varphi_1)-m(\varphi_2)\). For WGS84 an approximate expression for the distance \(\Delta m\) between the two parallels at one half of a degree from the circle at latitude \(\varphi\) is given (in metres) by

\[ \Delta m=111,132 -559\cos 2\varphi. \]

Series in third flattening (n)

The third flattening is related to the eccentricity by

\[ e^2 = \frac{4n}{(1+n)^2}= 4n(1-2n+3n^2-4n^3+\cdots). \]

With this substition the integral for the meridian distance becomes

\[ m(\varphi) = \int_0^\varphi\frac{a(1-n)^2(1+n)}{\left (1 + 2n \cos 2\varphi + n^2 \right )^{3/2}} \, d\varphi.\]

This integral has been expanded in several ways, all of which can be related to the Delambre series.

Bessel

In 1837 Bessel expanded this integral in a series of the form:

\[ m(\varphi)=a(1-n)^2(1+n)\left[D_0\varphi-D_2\sin 2\varphi+D_4\sin4\varphi-D_6\sin6\varphi+\cdots\right], \, \]

where

\[ \begin{align} D_0 &= 1+\frac{9}{4}n^2+\frac{225}{64}n^4+\cdots, \qquad\qquad& D_4 &= \frac{15}{16}n^2+\frac{105}{64}n^4+\cdots,\\ D_2 &= \frac{3}{2}n+\frac{45}{16}n^3+\frac{525}{128}n^5+\cdots, & D_6 &= \frac{35}{48}n^3+\frac{315}{256}n^5+\cdots. \end{align} \]

Since n is approximately one quarter of the value of the squared eccentricity, the above series for the coefficients converge 16 times as fast as the Delambre series.

Helmert

In 1880 Helmert extended and simplified the above series by rewriting

\[ (1-n)^2(1+n)=\frac{1}{1+n}(1-n^2)^2 \]

and expanding the numerator terms.

\[ m(\varphi)=\frac{a}{1+n}\left[H_0\varphi-H_2\sin 2\varphi+H_4\sin4\varphi-H_6\sin6\varphi+H_8\sin8\varphi+\cdots\right] \]

with

\[ \begin{align} H_0&=1+\frac{n^2}{4}+\frac{n^4}{64}+\cdots\qquad\qquad\qquad & H_6&=\frac{35}{48}n^3+\cdots\\[8pt] H_2&=\frac{3}{2}\left(n-\frac{n^3}{8}+\cdots\right) & H_8&=\frac{315}{512}n^4+\cdots\\[8pt] H_4&=\frac{15}{16}\left(n^2-\frac{n^4}{4}+\cdots\right) \end{align} \]

UTM

Despite the simplicity and fast convergence of Helmert's expansion the U.S. Defense Mapping Agency adopted the fully expanded form of the Bessel series reported by Hinks in 1927. This expansion is important, despite the poorer convergence of series in \(n\), because it is used in the definition of UTM.

\[ m(\varphi) = B_0\varphi + B_2\sin 2\varphi + B_4\sin4\varphi + B_6\sin6\varphi + B_8\sin8\varphi + \cdots, \]

where the coefficients are given to order n⁵ by

\[ \begin{align} B_0 &= \quad a\left(1-n+\frac{5}{4}n^2-\frac{5}{4}n^3+\frac{81}{64}n^4-\frac{81}{64}n^5+\cdots \right),\\[8pt] B_2 &= - \frac{3}{2}a\left(n-n^2+\frac{7}{8}n^3-\frac{7}{8}n^4+\frac{55}{64}n^5-\cdots \right),\\[8pt] B_4 &= \quad \frac{15}{16} a\left(n^2-n^3+\frac{3}{4}n^4-\frac{3}{4}n^5+\cdots \right),\\[8pt] B_6 &= - \frac{35}{48} a\left(n^3-n^4+\frac{11}{16}n^5-\cdots \right),\\[8pt] B_8 &= \quad \frac{315}{512} a\left(n^4-n^5+\cdots \right). \end{align} \]

OSGB

A similar fully expanded series of slow convergence was adopted by the Ordnance Survey of Great Britain.

Generalized series

The above series, to eighth order in eccentricity or fourth order in third flattening, are adequate for most practical applications. Each can be written quite generally. For example, Kazushige Kawase (2009) derived following general formula.:

\[m(\varphi)=\frac{a}{1+n}\sum_{j=0}^\infty\left(\prod_{k=1}^j\varepsilon_k\right)^2\left\{\varphi+\sum_{\ell=1}^{2j}\left(\frac{1}{\ell}-4\ell\right)\sin 2\ell\varphi\prod_{m=1}^\ell\varepsilon_{j+(-1)^m\lfloor m/2\rfloor}^{(-1)^m}\right\}\]

where

\[\varepsilon_i=\frac{3n}{2i}-n. \]

Truncating the summation at j = 2 gives Helmert's approximation.

Approximations

The polar distance may be approximated by the Thomas Muir's formula: \[m_p=\int_0^{\pi/2}\!M(\varphi)\,d\varphi\;\approx\frac\pi2\left[\frac{a^{3/2}+b^{3/2}}{2}\right]^{2/3}\,\!.\]

See also

• History of geodesy
• Erdmessung, Reference ellipsoid
• French Geodesic Mission
• Struve Geodetic Arc
• Torne Valleyde:Meridianbogen