Those interested in accurate expressions of the meridian arc for the WGS84 ellipsoid should consult the subsection entitled 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 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). 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 grade measurement of the Caliph Al-Ma'mun.
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.
By the end of the century, Delambre had remeasured and extended the French arc from Dunkirk to the Mediterranean (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 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.: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 1830, Everest 1830, and Clarke 1866. A comprehensive list of ellipsoids is given under Earth ellipsoid.
The determination of the meridian distance, that is the distance from the equator to a point at a latitude ? 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, 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 distance from the equator to the pole, the quarter meridian, is
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.
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.
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, Euler derived an expansion in the third eccentricity squared.
In 1825, Bessel derived an expansion of the meridian distance in terms of the parametric latitude ? in connection with his work on geodesics,
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
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
and k!! is the double factorial, extended to negative values via the recursion relation: and .
The coefficients in Helmert's series can similarly be expressed generally by
This result was conjected by Helmert and proved by Kawase.
The factor (1 - 2k)(1 + 2k) results in poorer convergence of the series in terms of ? compared to the one in ?.
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(?1) - m(?2) while maintaining high relative accuracy.
Substituting the values for the semi-major axis and eccentricity of the WGS84 ellipsoid gives
where ?(°) = is ? expressed in degrees (and similarly for ?(°)).
For the WGS84 ellipsoid the quarter meridian is
The perimeter of a meridian ellipse is 4mp = 2?(a + b)c0. Therefore, (a + b)c0 is the radius of the circle whose circumference is the same as the perimeter of a meridian ellipse. This defines the rectifying Earth radius as .
On the ellipsoid the exact distance between parallels at ?1 and ?2 is m(?1) - m(?2). For WGS84 an approximate expression for the distance ?m between the two parallels at ±0.5° from the circle at latitude ? is given by
The inverse meridian problem for the ellipsoid
In some problems, we need to be able to solve the inverse problem: given m, determine ?. This may be solved by Newton's method, iterating
until convergence. A suitable starting guess is given by ?0 = ? where
is the rectifying latitude. Note that it there is no need to differentiate the series for m(?), since the formula for the meridian radius of curvature M(?) can be used instead.
Alternatively, Helmert's series for the meridian distance can be reverted to give
Similarly, Bessel's series for m in terms of ? can be reverted to give
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 m in terms of ? 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 ? replaced by ?, the arc length on the auxiliary sphere. The requisite series extended to sixth order are given by Karney, Eqs. (17) & (21), with ? playing the role of n and ? playing the role of ?.
^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