The purpose of measuring meridian arcs is to determine a
figure of the Earth.
One or more measurements of meridian arcs can be used to infer the shape of the
reference ellipsoid 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 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 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).
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
Baghdad in the 9th century. The first realistic value was calculated by
Alexandrian scientist
Eratosthenes 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).[1] 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,[2] attributed to the Caliph
Al-Ma'mun.[citation needed]
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, 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.
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 5130762toises 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
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 1841, Everest 1830, and
Clarke 1866.[7] A comprehensive list of ellipsoids is given under
Earth ellipsoid.
The nautical mile
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]
On a sphere, the meridian arc length is simply the
circular arc length.
On an ellipsoid of revolution, for short meridian arcs, their length can be approximated using the
Earth's meridional radius of curvature 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 φ.
This is an important problem in the theory of map projections, particularly the
transverse Mercator projection.
The main ellipsoidal parameters are, a, b, f, but in theoretical work it is useful to define extra parameters, particularly the
eccentricity, e, and the third
flatteningn. Only two of these parameters are independent and there are many relations between them:
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
The distance formula is simpler when written in terms of the
parametric latitude,
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 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]
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.[14]
Expansions in the eccentricity (e)
Delambre in 1799[15] derived a widely used expansion on e2,
where
Richard Rapp gives a detailed derivation of this result.[16]
Expansions in the third flattening (n)
Series with considerably faster convergence can be obtained by expanding in terms of the
third flatteningn instead of the eccentricity. They are related by
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 H2k vanish.
The series can be expressed with either a or b as the initial factor by writing, for example,
In 1825, Bessel[22] derived an expansion of the meridian distance in terms of the
parametric latitudeβ in connection with his work on
geodesics,
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 the
geodetic latitude as
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[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
where
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
The extra factor (1 − 2k)(1 + 2k) originates from the additional expansion of appearing in the above formula and results in poorer convergence of the series in terms of φ compared to the one in β.
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 m(φ1) − m(φ2) while maintaining high relative accuracy.
Substituting the values for the semi-major axis and eccentricity of the
WGS84 ellipsoid gives
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
Quarter meridian
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The
perimeter of a meridian ellipse can also be rewritten in the form of a rectifying circle perimeter, Cp = 2πMr. Therefore, the
rectifying Earth radius is:
It can be evaluated as 6367449.146 m.
The inverse meridian problem for the ellipsoid
In some problems, we need to be able to solve the inverse problem: given m, determine φ. This may be solved by
Newton's method, iterating
until convergence. A suitable starting guess is given by φ0 = μ where
is the
rectifying latitude. Note that it there is no need to differentiate the series for m(φ), since the formula for the meridian radius of curvature M(φ) can be used instead.
Alternatively, Helmert's series for the meridian distance can be reverted to give[26][27]
where
Similarly, Bessel's series for m in terms of β can be reverted to give[28]
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.[29] For this reason, the expression for m in terms of β and its inverse given above play a key role in the solution of the
geodesic problem with m replaced by s, the distance along the geodesic, and β replaced by σ, the arc length on the auxiliary sphere.[22][30] The requisite series extended to sixth order are given by Charles Karney,[31] Eqs. (17) & (21), with ε playing the role of n and τ playing the role of μ.
^Osborne, Peter (2013), The Mercator Projections,
doi:
10.5281/zenodo.35392 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)