Meridian arc

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.

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, French Guiana and found that it lost 2+1⁄2 minutes per day compared to its rate at Paris.[3][4] 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 poles than at the Equator.

In 1687, Isaac Newton had published in the Principia as a proof that the Earth was an oblate spheroid of flattening equal to 1/230.[5] 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.[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 a prolate spheroid (with an equatorial radius less than the polar radius). To resolve the issue, the French Academy of Sciences (1735) undertook expeditions to Peru (Bouguer, Louis Godin, de La Condamine, Antonio de Ulloa, Jorge Juan) and to 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 from Dunkirk to the Mediterranean Sea (the meridian 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 the Paris Meridian was calculated as 5130762 toises as specified by the standard toise bar in Paris. Defining this distance as exactly 10000000 m led to the construction of a new standard metre bar as 0.5130762 toises.[6]^: 22

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, and Clarke 1866.[7] A comprehensive list of ellipsoids is given under Earth ellipsoid.

Historically a nautical 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 the Royal Yachting Association says in its manual for day 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".[8]

The meridian radius of curvature can be shown to be equal to:[9][10]

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

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

${\displaystyle {\begin{aligned}m(\varphi )&=\int _{0}^{\varphi }M(\varphi )\,d\varphi \\&=a(1-e^{2})\int _{0}^{\varphi }\left(1-e^{2}\sin ^{2}\varphi \right)^{-{\frac {3}{2}}}\,d\varphi \,.\end{aligned}}}$

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

${\displaystyle m(\varphi )=b\int _{0}^{\beta }{\sqrt {1+e'^{2}\sin ^{2}\beta }}\,d\beta \,,}$

where tan β = (1 − f)tan φ and e′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 of φ, β, and the rectifying latitude μ, are unrestricted.

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[11] (Section 19.2(ii)),

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

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

${\displaystyle {\begin{aligned}m(\varphi )&=a\left(E(\varphi ,e)-{\frac {e^{2}\sin \varphi \cos \varphi }{\sqrt {1-e^{2}\sin {}^{\!2}\varphi }}}\right)\\&=a\left(E(\varphi ,e)+{\frac {d^{2}}{d\varphi ^{2}}}E(\varphi ,e)\right)\\&=bE(\beta ,ie')\,.\end{aligned}}}$

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[12] and Maxima.[13]

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.[14]

Delambre in 1799[15] derived a widely used expansion on e²,

${\displaystyle m(\varphi )={\frac {b^{2}}{a}}\left(D_{0}\varphi +D_{2}\sin 2\varphi +D_{4}\sin 4\varphi +D_{6}\sin 6\varphi +D_{8}\sin 8\varphi +\cdots \right)\,,}$

where

${\displaystyle {\begin{aligned}D_{0}&=1+{\tfrac {3}{4}}e^{2}+{\tfrac {45}{64}}e^{4}+{\tfrac {175}{256}}e^{6}+{\tfrac {11025}{16384}}e^{8}+\cdots ,\\[5mu]D_{2}&=-{\tfrac {3}{8}}e^{2}-{\tfrac {15}{32}}e^{4}-{\tfrac {525}{1024}}e^{6}-{\tfrac {2205}{4096}}e^{8}-\cdots ,\\[5mu]D_{4}&={\tfrac {15}{256}}e^{4}+{\tfrac {105}{1024}}e^{6}+{\tfrac {2205}{16384}}e^{8}+\cdots ,\\[5mu]D_{6}&=-{\tfrac {35}{3072}}e^{6}-{\tfrac {105}{4096}}e^{8}-\cdots ,\\[5mu]D_{8}&={\tfrac {315}{131072}}e^{8}+\cdots .\end{aligned}}}$

Richard Rapp gives a detailed derivation of this result.[16]

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

${\displaystyle e^{2}={\frac {4n}{(1+n)^{2}}}\,.}$

In 1837, Friedrich Bessel obtained one such series,[17] which was put into a simpler form by Helmert,[18][19]

${\displaystyle m(\varphi )={\frac {a+b}{2}}\left(H_{0}\varphi +H_{2}\sin 2\varphi +H_{4}\sin 4\varphi +H_{6}\sin 6\varphi +H_{8}\sin 8\varphi +\cdots \right)\,,}$

with

${\displaystyle {\begin{aligned}H_{0}&=1+{\tfrac {1}{4}}n^{2}+{\tfrac {1}{64}}n^{4}+\cdots ,\\H_{2}&=-{\tfrac {3}{2}}n+{\tfrac {3}{16}}n^{3}+\cdots ,&H_{6}&=-{\tfrac {35}{48}}n^{3}+\cdots ,\\H_{4}&={\tfrac {15}{16}}n^{2}-{\tfrac {15}{64}}n^{4}-\cdots ,\qquad &H_{8}&={\tfrac {315}{512}}n^{4}-\cdots .\end{aligned}}}$

Because n changes sign when a and b are interchanged, and because the initial factor 1/2(a + b) is constant under this interchange, half the terms in the expansions of H_2k vanish.

The series can be expressed with either a or b as the initial factor by writing, for example,

${\displaystyle {\tfrac {1}{2}}(a+b)={\frac {a}{1+n}}=a(1-n+n^{2}-n^{3}+n^{4}-\cdots )\,,}$

and expanding the result as a series in 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[20] and the Ordnance Survey of Great Britain.[21]

In 1825, Bessel[22] derived an expansion of the meridian distance in terms of the parametric latitude β in connection with his work on geodesics,

${\displaystyle m(\varphi )={\frac {a+b}{2}}\left(B_{0}\beta +B_{2}\sin 2\beta +B_{4}\sin 4\beta +B_{6}\sin 6\beta +B_{8}\sin 8\beta +\cdots \right)\,,}$

with

${\displaystyle {\begin{aligned}B_{0}&=1+{\tfrac {1}{4}}n^{2}+{\tfrac {1}{64}}n^{4}+\cdots =H_{0}\,,\\B_{2}&=-{\tfrac {1}{2}}n+{\tfrac {1}{16}}n^{3}+\cdots ,&B_{6}&=-{\tfrac {1}{48}}n^{3}+\cdots ,\\B_{4}&=-{\tfrac {1}{16}}n^{2}+{\tfrac {1}{64}}n^{4}+\cdots ,\qquad &B_{8}&=-{\tfrac {5}{512}}n^{4}+\cdots .\end{aligned}}}$

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

${\displaystyle m(\varphi )={\frac {a+b}{2}}\left(B_{0}\varphi -B_{2}\sin 2\varphi +B_{4}\sin 4\varphi -B_{6}\sin 6\varphi +B_{8}\sin 8\varphi -\cdots -{\frac {2n\sin 2\varphi }{\sqrt {1+2n\cos 2\varphi +n^{2}}}}\right)\,.}$

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[15] and Bessel[22] 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

${\displaystyle B_{2k}={\begin{cases}c_{0}\,,&{\text{if }}k=0\,,\\[5px]{\dfrac {c_{k}}{k}}\,,&{\text{if }}k>0\,,\end{cases}}}$

where

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

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

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

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

This result was conjectured by Friedrich Helmert[23] and proved by Kazushige Kawase.[24]

The factor (1 − 2k)(1 + 2k) results in poorer convergence of the series in terms of φ compared to the one in β.

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 m(φ₁) − m(φ₂) while maintaining high relative accuracy.

Substituting the values for the semi-major axis and eccentricity of the WGS84 ellipsoid gives

${\displaystyle {\begin{aligned}m(\varphi )&=\left(111\,132.952\,55\,\varphi ^{(\circ )}-16\,038.509\,\sin 2\varphi +16.833\,\sin 4\varphi -0.022\,\sin 6\varphi +0.000\,03\,\sin 8\varphi \right){\mbox{ metres}}\\&=\left(111\,132.952\,55\,\beta ^{(\circ )}-5\,346.170\,\sin 2\beta -1.122\,\sin 4\beta -0.001\,\sin 6\beta -0.5\times 10^{-6}\,\sin 8\beta \right){\mbox{ metres,}}\end{aligned}}}$

where φ⁽°⁾ = φ/1° is φ expressed in degrees (and similarly for β⁽°⁾).

On the ellipsoid the exact distance between parallels at φ₁ and φ₂ is m(φ₁) − m(φ₂). For WGS84 an approximate expression for the distance Δm between the two parallels at ±0.5° from the circle at latitude φ is given by

${\displaystyle \Delta m=(111\,133-560\cos 2\varphi ){\mbox{ metres.}}}$