In geography , LATITUDE is a geographic coordinate that specifies the
north –south position of a point on the Earth's surface.
CONTENTS * 1 Preliminaries * 2
* 2.1 The graticule on the sphere * 2.2 Named latitudes on the Earth * 2.3 Meridian distance on the sphere * 3
* 3.1 Ellipsoids * 3.2 The geometry of the ellipsoid * 3.3 Geodetic and geocentric latitudes * 3.4 Length of a degree of latitude * 4 Auxiliary latitudes * 4.1 Geocentric latitude * 4.2 Reduced (or parametric) latitude * 4.3 Rectifying latitude * 4.4 Authalic latitude * 4.5 Conformal latitude * 4.6 Isometric latitude * 4.7 Inverse formulae and series * 4.8 Numerical comparison of auxiliary latitudes * 5
* 5.1 Geodetic coordinates
* 5.2
* 6 Astronomical latitude * 7 See also * 8 Notes * 9 References * 10 External links PRELIMINARIES Two levels of abstraction are employed in the definition of latitude
and longitude. In the first step the physical surface is modeled by
the geoid , a surface which approximates the mean sea level over the
oceans and its continuation under the land masses. The second step is
to approximate the geoid by a mathematically simpler reference
surface. The simplest choice for the reference surface is a sphere ,
but the geoid is more accurately modeled by an ellipsoid. The
definitions of latitude and longitude on such reference surfaces are
detailed in the following sections. Lines of constant latitude and
longitude together constitute a graticule on the reference surface.
The latitude of a point on the _actual_ surface is that of the
corresponding point on the reference surface, the correspondence being
along the normal to the reference surface which passes through the
point on the physical surface.
Since there are many different reference ellipsoids , the precise latitude of a feature on the surface is not unique: this is stressed in the ISO standard which states that "without the full specification of the coordinate reference system, coordinates (that is latitude and longitude) are ambiguous at best and meaningless at worst". This is of great importance in accurate applications, such as a Global Positioning System (GPS), but in common usage, where high accuracy is not required, the reference ellipsoid is not usually stated. In English texts the latitude angle, defined below, is usually denoted by the Greek lower-case letter phi (φ or ϕ). It is measured in degrees , minutes and seconds or decimal degrees , north or south of the equator. The precise measurement of latitude requires an understanding of the
gravitational field of the Earth, either to set up theodolites or to
determine
This article relates to coordinate systems for the Earth: it may be extended to cover the Moon, planets and other celestial objects by a simple change of nomenclature. LATITUDE ON THE SPHERE A perspective view of the Earth showing how latitude (φ) and longitude (λ) are defined on a spherical model. The graticule spacing is 10 degrees. THE GRATICULE ON THE SPHERE The graticule is formed by the lines of constant latitude and
constant longitude, which are constructed with reference to the
rotation axis of the Earth. The primary reference points are the poles
where the axis of rotation of the Earth intersects the reference
surface. Planes which contain the rotation axis intersect the surface
at the meridians ; and the angle between any one meridian plane and
that through Greenwich (the
The latitude, as defined in this way for the sphere, is often termed the spherical latitude, to avoid ambiguity with the geodetic latitude and the auxiliary latitudes defined in subsequent sections of this article. NAMED LATITUDES ON THE EARTH The orientation of the Earth at the December solstice. Besides the equator, four other parallels are of significance:
The plane of the Earth's orbit about the Sun is called the ecliptic , and the plane perpendicular to the rotation axis of the Earth is the equatorial plane. The angle between the ecliptic and the equatorial plane is called variously the axial tilt, the obliquity, or the inclination of the ecliptic, and it is conventionally denoted by i. The latitude of the tropical circles is equal to i and the latitude of the polar circles is its complement (90° - _i_). The axis of rotation varies slowly over time and the values given here are those for the current epoch . The time variation is discussed more fully in the article on axial tilt . The figure shows the geometry of a cross-section of the plane
perpendicular to the ecliptic and through the centres of the Earth and
the Sun at the December solstice when the Sun is overhead at some
point of the
On map projections there is no universal rule as to how meridians and
parallels should appear. The examples below show the named parallels
(as red lines) on the commonly used
NORMAL MERCATOR TRANSVERSE MERCATOR MERIDIAN DISTANCE ON THE SPHERE On the sphere the normal passes through the centre and the latitude (φ) is therefore equal to the angle subtended at the centre by the meridian arc from the equator to the point concerned. If the meridian distance is denoted by _m_(_φ_) then m ( ) = 180 R d e g r e e s = R r a d i a n s {displaystyle m(phi )={frac {pi }{180^{circ }}}Rphi _{mathrm {degrees} }=Rphi _{mathrm {radians} }} where R denotes the mean radius of the Earth. R is equal to 6,371 km or 3,959 miles. No higher accuracy is appropriate for R since higher-precision results necessitate an ellipsoid model. With this value for R the meridian length of 1 degree of latitude on the sphere is 111.2 km or 69.1 miles. The length of 1 minute of latitude is 1.853 km or 1.151 miles, and this distance was formerly used as the basis of the nautical mile . LATITUDE ON THE ELLIPSOID ELLIPSOIDS In 1687
Many different reference ellipsoids have been used in the history of
geodesy . In pre-satellite days they were devised to give a good fit
to the geoid over the limited area of a survey but, with the advent of
THE GEOMETRY OF THE ELLIPSOID The shape of an ellipsoid of revolution is determined by the shape of the ellipse which is rotated about its minor (shorter) axis. Two parameters are required. One is invariably the equatorial radius, which is the semi-major axis , a. The other parameter is usually (1) the polar radius or semi-minor axis , b; or (2) the (first) flattening , f; or (3) the eccentricity , e. These parameters are not independent: they are related by f = a b a , e 2 = 2 f f 2 , b = a ( 1 f ) = a 1 e 2 . {displaystyle f={frac {a-b}{a}},qquad e^{2}=2f-f^{2},qquad b=a(1-f)=a{sqrt {1-e^{2}}},.} Many other parameters (see ellipse , ellipsoid ) appear in the study
of geodesy, geophysics and map projections but they can all be
expressed in terms of one or two members of the set a, b, f and e.
Both f and e are small and often appear in series expansions in
calculations; they are of the order 1/300 and 0.08 respectively.
Values for a number of ellipsoids are given in
* a (equatorial radius): 7006637813700000000♠6378137.0 m exactly * 1/_f_ (inverse flattening): 7002298257223563000♠298.257223563 exactly from which are derived * b (polar radius): 7006635675231420000♠6356752.3142 m * _e_2 (eccentricity squared): 6997669437999014000♠0.00669437999014 The difference between the semi-major and semi-minor axes is about 21 km (13 miles) and as fraction of the semi-major axis it equals the flattening; on a computer monitor the ellipsoid could be sized as 300 by 299 pixels. This would barely be distinguishable from a 300-by-300-pixel sphere, so illustrations usually exaggerate the flattening. GEODETIC AND GEOCENTRIC LATITUDES The definition of geodetic latitude (φ) and longitude (λ) on an ellipsoid. The normal to the surface does not pass through the centre, except at the equator and at the poles. The graticule on the ellipsoid is constructed in exactly the same way as on the sphere. The normal at a point on the surface of an ellipsoid does not pass through the centre, except for points on the equator or at the poles, but the definition of latitude remains unchanged as the angle between the normal and the equatorial plane. The terminology for latitude must be made more precise by distinguishing: * GEODETIC LATITUDE: the angle between the normal and the equatorial plane. The standard notation in English publications is φ. This is the definition assumed when the word latitude is used without qualification. The definition must be accompanied with a specification of the ellipsoid. * GEOCENTRIC LATITUDE: the angle between the radius (from centre to the point on the surface) and the equatorial plane. (Figure below ). There is no standard notation: examples from various texts include ψ, q, φ′, _φ_c, _φ_g. This article uses ψ. * SPHERICAL LATITUDE: the angle between the normal to a spherical reference surface and the equatorial plane. * GEOGRAPHIC LATITUDE must be used with care. Some authors use it as a synonym for geodetic latitude whilst others use it as an alternative to the astronomical latitude . * LATITUDE (unqualified) should normally refer to the geodetic latitude. The importance of specifying the reference datum may be illustrated
by a simple example. On the reference ellipsoid for WGS84, the centre
of the
LENGTH OF A DEGREE OF LATITUDE Main article:
In
where _M_(_φ_) is the meridional radius of curvature . The distance from the equator to the pole is m p = m ( 2 ) {displaystyle m_{mathrm {p} }=mleft({frac {pi }{2}}right),} For
The evaluation of the meridian distance integral is central to many
studies in geodesy and map projection. It can be evaluated by
expanding the integral by the binomial series and integrating term by
term: see
{DISPLAYSTYLE PHI } Δ1 lat Δ1 long 0° 110.574 km 111.320 km 15° 110.649 km 107.550 km 30° 110.852 km 96.486 km 45° 111.132 km 78.847 km 60° 111.412 km 55.800 km 75° 111.618 km 28.902 km 90° 111.694 km 0.000 km When the latitude difference is 1 degree, corresponding to π/180 radians, the arc distance is about l a t 1 = a ( 1 e 2 ) 180 ( 1 e 2 sin 2 ) 3 2 {displaystyle Delta _{mathrm {lat} }^{1}={frac {pi aleft(1-e^{2}right)}{180^{circ }left(1-e^{2}sin ^{2}phi right)^{frac {3}{2}}}}} The distance in metres (correct to 0.01 metre) between latitudes
{displaystyle phi } − 0.5 degrees and {displaystyle phi }
+ 0.5 degrees on the
The variation of this distance with latitude (on
A calculator for any latitude is provided by the U.S. Government's
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. AUXILIARY LATITUDES There are six AUXILIARY LATITUDES that have applications to special problems in geodesy, geophysics and the theory of map projections: * Geocentric latitude * Reduced (or parametric) latitude * Rectifying latitude * Authalic latitude * Conformal latitude * Isometric latitude The definitions given in this section all relate to locations on the reference ellipsoid but the first two auxiliary latitudes, like the geodetic latitude, can be extended to define a three-dimensional geographic coordinate system as discussed below . The remaining latitudes are not used in this way; they are used _only_ as intermediate constructs in map projections of the reference ellipsoid to the plane or in calculations of geodesics on the ellipsoid. Their numerical values are not of interest. For example, no one would need to calculate the authalic latitude of the Eiffel Tower. The expressions below give the auxiliary latitudes in terms of the geodetic latitude, the semi-major axis, a, and the eccentricity, e. (For inverses see below .) The forms given are, apart from notational variants, those in the standard reference for map projections, namely "Map projections: a working manual" by J. P. Snyder. Derivations of these expressions may be found in Adams and online publications by Osborne and Rapp. GEOCENTRIC LATITUDE The definition of geodetic (or geographic) and geocentric latitudes. The GEOCENTRIC LATITUDE is the angle between the equatorial plane and the radius from the centre to a point on the surface. The relation between the geocentric latitude (ψ) and the geodetic latitude (φ) is derived in the above references as ( ) = tan 1 ( ( 1 e 2 ) tan ) . {displaystyle psi (phi )=tan ^{-1}left((1-e^{2})tan phi right),.} The geodetic and geocentric latitudes are equal at the equator and at the poles but at other latitudes they differ by a few minutes of arc. Taking the value of the squared eccentricity as 0.0067 (it depends on the choice of ellipsoid) the maximum difference of {displaystyle phi {-}psi } may be shown to be about 11.5 minutes of arc at a geodetic latitude of approximately 45° 6′. REDUCED (OR PARAMETRIC) LATITUDE Definition of the reduced latitude (β) on the ellipsoid. The REDUCED or PARAMETRIC LATITUDE, β, is defined by the radius drawn from the centre of the ellipsoid to that point Q on the surrounding sphere (of radius a) which is the projection parallel to the Earth's axis of a point P on the ellipsoid at latitude φ. It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming them to an equivalent problem for spherical geodesics by using this smaller latitude. Bessel's notation, _u_(_φ_), is also used in the current literature. The reduced latitude is related to the geodetic latitude by: ( ) = tan 1 ( 1 e 2 tan ) {displaystyle beta (phi )=tan ^{-1}left({sqrt {1-e^{2}}}tan phi right)} The alternative name arises from the parameterization of the equation of the ellipse describing a meridian section. In terms of Cartesian coordinates p, the distance from the minor axis, and z, the distance above the equatorial plane, the equation of the ellipse is: p 2 a 2 + z 2 b 2 = 1 . {displaystyle {frac {p^{2}}{a^{2}}}+{frac {z^{2}}{b^{2}}}=1,.} The Cartesian coordinates of the point are parameterized by p = a cos , z = b sin ; {displaystyle p=acos beta ,,qquad z=bsin beta ,;} Cayley suggested the term _parametric latitude_ because of the form of these equations. The reduced latitude is not used in the theory of map projections. Its most important application is in the theory of ellipsoid geodesics. (Vincenty , Karney). RECTIFYING LATITUDE See also:
The RECTIFYING LATITUDE, μ, is the meridian distance scaled so that its value at the poles is equal to 90 degrees or π/2 radians: ( ) = 2 m ( ) m p {displaystyle mu (phi )={frac {pi }{2}}{frac {m(phi )}{m_{mathrm {p} }}}} where the meridian distance from the equator to a latitude φ is (see
and the length of the meridian quadrant from the equator to the pole (the polar distance ) is m p = m ( 2 ) . {displaystyle m_{mathrm {p} }=mleft({frac {pi }{2}}right),.} Using the rectifying latitude to define a latitude on a sphere of radius R = 2 m p {displaystyle R={frac {2m_{mathrm {p} }}{pi }}} defines a projection from the ellipsoid to the sphere such that all
meridians have true length and uniform scale. The sphere may then be
projected to the plane with an equirectangular projection to give a
double projection from the ellipsoid to the plane such that all
meridians have true length and uniform meridian scale. An example of
the use of the rectifying latitude is the Equidistant conic projection
. (Snyder, Section 16). The rectifying latitude is also of great
importance in the construction of the Transverse
AUTHALIC LATITUDE See also:
The AUTHALIC (Greek for same area) latitude, ξ, gives an area-preserving transformation to a sphere. ( ) = sin 1 ( q ( ) q p ) {displaystyle xi (phi )=sin ^{-1}left({frac {q(phi )}{q_{mathrm {p} }}}right)} where q ( ) = ( 1 e 2 ) sin 1 e 2 sin 2 1 e 2 2 e ln ( 1 e sin 1 + e sin ) = ( 1 e 2 ) sin 1 e 2 sin 2 + 1 e 2 e tanh 1 ( e sin ) {displaystyle {begin{aligned}q(phi )&={frac {(1-e^{2})sin phi }{1-e^{2}sin ^{2}phi }}-{frac {1-e^{2}}{2e}}ln left({frac {1-esin phi }{1+esin phi }}right)\ width:48.736ex; height:13.843ex;" alt="{displaystyle {begin{aligned}q(phi )&={frac {(1-e^{2})sin phi }{1-e^{2}sin ^{2}phi }}-{frac {1-e^{2}}{2e}}ln left({frac {1-esin phi }{1+esin phi }}right)\"> q p = q ( 2 ) = 1 1 e 2 2 e ln ( 1 e 1 + e ) = 1 + 1 e 2 e tanh 1 e {displaystyle {begin{aligned}q_{mathrm {p} }=qleft({frac {pi }{2}}right)&=1-{frac {1-e^{2}}{2e}}ln left({frac {1-e}{1+e}}right) width:66.098ex; height:6.176ex;" alt="{displaystyle {begin{aligned}q_{mathrm {p} }=qleft({frac {pi }{2}}right)&=1-{frac {1-e^{2}}{2e}}ln left({frac {1-e}{1+e}}right)"> R q = a q p 2 . {displaystyle R_{q}=a{sqrt {frac {q_{mathrm {p} }}{2}}},.} An example of the use of the authalic latitude is the Albers equal-area conic projection . :§14 CONFORMAL LATITUDE The CONFORMAL LATITUDE, χ, gives an angle-preserving (conformal )
transformation to the sphere. ( ) = 2 tan 1
1 2 2 = 2 tan 1 2 = sin 1
= gd {displaystyle {begin{aligned}chi (phi )&=2tan
^{-1}left^{frac {1}{2}}-{frac {pi }{2}}\&=2tan ^{-1}left-{frac {pi
}{2}}\&=sin ^{-1}left\ margin-bottom: -0.236ex; width:55.488ex;
height:24.509ex;" alt="{displaystyle {begin{aligned}chi (phi )&=2tan
^{-1}left^{frac {1}{2}}-{frac {pi }{2}}\&=2tan ^{-1}left-{frac {pi
}{2}}\&=sin ^{-1}left\">ψ (not to be confused with the geocentric
latitude): it is used in the development of the ellipsoidal versions
of the normal
The isometric latitude ψ is closely related to the conformal latitude χ: ( ) = gd 1 ( ) . {displaystyle psi (phi )=operatorname {gd} ^{-1}chi (phi ),.} INVERSE FORMULAE AND SERIES The formulae in the previous sections give the auxiliary latitude in terms of the geodetic latitude. The expressions for the geocentric and reduced latitudes may be inverted directly but this is impossible in the four remaining cases: the rectifying, authalic, conformal, and isometric latitudes. There are two methods of proceeding. The first is a numerical inversion of the defining equation for each and every particular value of the auxiliary latitude. The methods available are fixed-point iteration and Newton–Raphson root finding. The other, more useful, approach is to express the auxiliary latitude as a series in terms of the geodetic latitude and then invert the series by the method of Lagrange reversion . Such series are presented by Adams who uses Taylor series expansions and gives coefficients in terms of the eccentricity. Osborne derives series to arbitrary order by using the computer algebra package Maxima and expresses the coefficients in terms of both eccentricity and flattening. The series method is not applicable to the isometric latitude and one must use the conformal latitude in an intermediate step. NUMERICAL COMPARISON OF AUXILIARY LATITUDES The following plot shows the magnitude of the difference between the geodetic latitude, (denoted as the "common" latitude on the plot), and the auxiliary latitudes other than the isometric latitude (which diverges to infinity at the poles). In every case the geodetic latitude is the greater. The differences shown on the plot are in arc minutes. The horizontal resolution of the plot fails to make clear that the maxima of the curves are not at 45° but calculation shows that they are within a few arc minutes of 45°. Some representative data points are given in the table following the plot. Note the closeness of the conformal and geocentric latitudes. This was exploited in the days of hand calculators to expedite the construction of map projections. :108 Approximate difference from geodetic latitude (φ) φ Reduced _φ_ − _β_ Authalic _φ_ − _ξ_ Rectifying _φ_ − _μ_ Conformal _φ_ − _χ_ Geocentric _φ_ − _ψ_ 0° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′ 15° 2.91′ 3.89′ 4.37′ 5.82′ 5.82′ 30° 5.05′ 6.73′ 7.57′ 10.09′ 10.09′ 45° 5.84′ 7.78′ 8.76′ 11.67′ 11.67′ 60° 5.06′ 6.75′ 7.59′ 10.12′ 10.13′ 75° 2.92′ 3.90′ 4.39′ 5.85′ 5.85′ 90° 0.00′ 0.00′ 0.00′ 0.00′ 0.00′ LATITUDE AND COORDINATE SYSTEMS The geodetic latitude, or any of the auxiliary latitudes defined on the reference ellipsoid, constitutes with longitude a two-dimensional coordinate system on that ellipsoid. To define the position of an arbitrary point it is necessary to extend such a coordinate system into three dimensions. Three latitudes are used in this way: the geodetic, geocentric and reduced latitudes are used in geodetic coordinates, spherical polar coordinates and ellipsoidal coordinates respectively. GEODETIC COORDINATES _ Geodetic coordinates P(ɸ_,_λ_,_h_) At an arbitrary point P consider the line PN which is normal to the reference ellipsoid. The geodetic coordinates P(_ɸ_,_λ_,_h_) are the latitude and longitude of the point N on the ellipsoid and the distance PN. This height differs from the height above the geoid or a reference height such as that above mean sea level at a specified location. The direction of PN will also differ from the direction of a vertical plumb line. The relation of these different heights requires knowledge of the shape of the geoid and also the gravity field of the Earth. SPHERICAL POLAR COORDINATES _ Geocentric coordinate related to spherical polar coordinates P(r_,_θ_,_λ_) The geocentric latitude ψ is the complement of the polar angle θ in conventional spherical polar coordinates in which the coordinates of a point are P(_r_,_θ_,_λ_) where r is the distance of P from the centre O, θ is the angle between the radius vector and the polar axis and λ is longitude. Since the normal at a general point on the ellipsoid does not pass through the centre it is clear that points on the normal, which all have the same geodetic latitude, will have differing geocentric latitudes. Spherical polar coordinate systems are used in the analysis of the gravity field. ELLIPSOIDAL COORDINATES _ Ellipsoidal coordinates P(u_,_β_,_λ_) The reduced latitude can also be extended to a three-dimensional coordinate system. For a point P not on the reference ellipsoid (semi-axes OA and OB) construct an auxiliary ellipsoid which is confocal (same foci F, F′) with the reference ellipsoid: the necessary condition is that the product ae of semi-major axis and eccentricity is the same for both ellipsoids. Let u be the semi-minor axis (OD) of the auxiliary ellipsoid. Further let β be the reduced latitude of P on the auxiliary ellipsoid. The set (_u_,_β_,_λ_) define the ellipsoid coordinates. :§4.2.2 These coordinates are the natural choice in models of the gravity field for a uniform distribution of mass bounded by the reference ellipsoid. COORDINATE CONVERSIONS The relations between the above coordinate systems, and also
Cartesian coordinates are not presented here. The transformation
between geodetic and Cartesian coordinates may be found in Geographic
coordinate conversion . The relation of Cartesian and spherical polars
is given in
ASTRONOMICAL LATITUDE ASTRONOMICAL LATITUDE (Φ) is the angle between the equatorial plane and the true vertical at a point on the surface. The true vertical, the direction of a plumb line , is also the direction of the gravity acceleration, the resultant of the gravitational acceleration (mass-based) and the centrifugal acceleration at that latitude. Astronomic latitude is calculated from angles measured between the zenith and stars whose declination is accurately known. In general the true vertical at a point on the surface does not exactly coincide with either the normal to the reference ellipsoid or the normal to the geoid. The angle between the astronomic and geodetic normals is usually a few seconds of arc but it is important in geodesy. The reason why it differs from the normal to the geoid is, because the geoid is an idealized, theoretical shape "at mean sea level". Points on the real surface of the earth are usually above or below this idealized geoid surface and here the true vertical can vary slightly. Also, the true vertical at a point at a specific time is influenced by tidal forces, which the theoretical geoid averages out. Astronomical latitude is not to be confused with declination , the coordinate astronomers use in a similar way to specify the angular position of stars north/south of the celestial equator (see equatorial coordinates ), nor with ecliptic latitude , the coordinate that astronomers use to specify the angular position of stars north/south of the ecliptic (see ecliptic coordinates ). SEE ALSO *
NOTES * ^ The current full documentation of ISO 19111 may be purchased
from http://www.iso.org but drafts of the final standard are freely
available at many web sites, one such is available at the following
CSIRO
* ^ The value of this angle today is 23°26′13.2″ (or
23.437°). This figure is provided by Template:
REFERENCES * ^ Newton, Isaac. "Book III Proposition XIX Problem III".
_Philosophiæ Naturalis Principia Mathematica_. Translated by Motte,
Andrew. p. 407.
* ^ "TR8350.2". National Geospatial-Intelligence Agency
publication. p. 3-1.
* ^ _A_ _B_ _C_ _D_ _E_ Torge, W. (2001). _Geodesy_ (3rd ed.). De
Gruyter. ISBN 3-11-017072-8 .
* ^ _A_ _B_ _C_ _D_ _E_ Osborne, Peter (2013). "Chapters 5,6". _The
Mercator Projections_. doi :10.5281/zenodo.35392 . for LaTeX code and
figures.
* ^ _A_ _B_ _C_ _D_ Rapp, Richard H. (1991). "Chapter 3".
_Geometric Geodesy, Part I_. Columbus, OH: Dept. of Geodetic Science
and Surveying, Ohio State Univ.
* ^ "Length of degree calculator". National Geospatial-Intelligence
Agency.
* ^ _A_ _B_ _C_ _D_ _E_ Snyder, John P. (1987). _Map Projections: A
Working Manual_. U.S. Geological Survey Professional Paper 1395.
Washington, DC: United States Government Printing Office.
* ^ _A_ _B_ Adams, Oscar S. (1921). _
* ^ Bessel, F. W. (1825). "Über die Berechnung der geographischen
Langen und Breiten aus geodatischen Vermessungen". _Astron. Nachr_. 4
(86): 241–254. doi :10.1002/asna.201011352 .
TRANSLATION: Karney, C. F. F.; Deakin, R. E. (2010). "The calculation
of longitude and latitude from geodesic measurements". _Astron.
Nachr_. 331 (8): 852–861.
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