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| '''Geographical distance''' is the [[distance]] measured along the surface of the [[earth]]. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of [[latitude]] and [[longitude]]. This distance is an element in solving the [[Geodesy#Geodetic_problems|second (inverse) geodetic problem]].
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| ==Introduction==
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| Calculating the distance between geographical coordinates is based on some level of abstraction; it does not provide an ''exact'' distance, which is unattainable if one attempted to account for every irregularity in the surface of the earth.<ref>http://www.cartography.org.uk/default.asp?contentID=749</ref> Common abstractions for the surface between two geographic points are:
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| *Flat surface;
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| *Spherical surface;
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| *Ellipsoidal surface.
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| All abstractions above ignore changes in elevation. Calculation of distances which account for changes in elevation relative to the idealized surface are not discussed in this article.
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| ===Nomenclature===
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| | |
| Distance, <math>D,\,\!</math> is calculated between two points, <math>P_1\,\!</math> and <math>P_2\,\!</math>. The geographical coordinates of the two points, as (latitude, longitude) pairs, are <math>(\phi_1,\lambda_1)\,\!</math> and <math>(\phi_2,\lambda_2),\,\!</math> respectively. Which of the two points is designated as <math>P_1\,\!</math> is not important for the calculation of distance.
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| | |
| Latitude and longitude coordinates on maps are usually expressed in [[degree (angle)|degree]]s. In the given forms of the formulae below, one or more values ''must'' be expressed in the specified units to obtain the correct result. Where geographic coordinates are used as the argument of a trigonometric function, the values may be expressed in any angular units compatible with the method used to determine the value of the trigonometric function. Many electronic calculators allow calculations of trigonometric functions in either degrees or [[radian]]s. The calculator mode must be compatible with the units used for geometric coordinates.
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| Differences in latitude and longitude are labeled and calculated as follows:
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| :<math>\begin{align}
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| \Delta\phi&=\phi_2-\phi_1;\\
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| \Delta\lambda&=\lambda_2-\lambda_1.
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| \end{align}
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| \,\!</math>
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| It is not important whether the result is positive or negative when used in the formulae below. | |
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| "Mean latitude" is labeled and calculated as follows:
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| :<math>\phi_m=\frac{\phi_1+\phi_2}{2}.\,\!</math>
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| Colatitude is labeled and calculated as follows:
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| :For latitudes expressed in radians:
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| ::<math>\theta=\frac{\pi}{2}-\phi;\,\!</math>
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| :For latitudes expressed in degrees:
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| ::<math>\theta=90^\circ-\phi.\,\!</math>
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| | |
| Unless specified otherwise, the [[Earth radius|radius]] of the earth for the calculations below is:
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| :<math>R\,\!</math> = 6,371.009 kilometers = 3,958.761 statute miles = 3,440.069 [[nautical mile]]s.
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| <math>D_\,\!</math> = Distance between the two points, as measured along the surface of the earth and in the same units as the value used for radius unless specified otherwise.
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| ===Singularities and discontinuity of latitude/longitude===
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| Longitude has [[Mathematical singularity|singularities]] at the [[Geographical pole|Poles]] (longitude is undefined) and a [[Discontinuity (mathematics)|discontinuity]] at the ±[[180th meridian|180° meridian]]. Also, planar projections of the [[Circle of latitude|circles of constant latitude]] are highly curved near the Poles. Hence, the above equations for [[Delta (letter)|delta]] latitude/longitude (<math>\Delta\phi\!</math>, <math>\Delta\lambda\!</math>) and mean latitude (<math>\phi_m\!</math>) may not give the expected answer for positions near the Poles or the ±180° meridian. Consider e.g. the value of <math>\Delta\lambda\!</math> (“east displacement”) when <math>\lambda_1\!</math> and <math>\lambda_2\!</math> are on either side of the ±180° meridian, or the value of <math>\phi_m\!</math> (“mean latitude”) for the two positions (<math>\phi_1\!</math>=89°, <math>\lambda_1\!</math>=45°) and (<math>\phi_2\!</math>=89°, <math>\lambda_2\!</math>=−135°).
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| If a calculation based on latitude/longitude should be valid for all Earth positions, it should be verified that the discontinuity and the Poles are handled correctly. Another solution is to use [[n-vector|''n''-vector]] instead of latitude/longitude, since this [[horizontal position representation|representation]] does not have discontinuities or singularities.
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| ==Flat-surface formulae==
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| A planar approximation for the surface of the earth may be useful over small distances. The accuracy of distance calculations using this approximation become increasingly inaccurate as:
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| * The separation between the points becomes greater;
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| * A point becomes closer to a geographic pole.
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| The shortest distance between two points in plane is a straight line. The [[Pythagorean theorem]] is used to calculate the distance between points in a plane.
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| Even over short distances, the accuracy of geographic distance calculations which assume a flat Earth depend on the method by which the latitude and longitude coordinates have been projected onto the plane. The projection of global latitude and longitude coordinates onto a plane is the realm of [[cartography]].
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| The formulae presented in this section provide varying degrees of accuracy.
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| ===Spherical Earth projected to a plane===
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| This formula takes into account the variation in distance between meridians with latitude:
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| :<math>D=R\sqrt{(\Delta\phi)^2+(\cos(\phi_m)\Delta\lambda)^2}{\color{white}\frac{\big|}{.}}\,\!</math>
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| :where:
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| ::<math>\Delta\phi\,\!</math> and <math>\Delta\lambda\,\!</math> are in radians;
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| ::<math>\phi_m\,\!</math> must be in units compatible with the method used for determining <math>\cos(\phi_m).\,\!</math>
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| :To convert latitude or longitude to radians use
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| ::<math> 1^\circ = (\pi/180)\,\mathrm{radians}.</math>
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| :Note: This approximation is very fast and produces fairly accurate result for small distances {{Citation needed|date=October 2010}}. Also, when ordering locations by distance, such as in a database query, it is much faster to order by squared distance, eliminating the need for computing the square root. | |
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| ===Ellipsoidal Earth projected to a plane===
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| The [[Federal Communications Commission|FCC]] prescribes essentially the following formulae in 47 CFR 73.208 for distances not exceeding 475 km /295 miles:<ref>http://edocket.access.gpo.gov/cfr_2005/octqtr/pdf/47cfr73.208.pdf</ref>
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| :<math>
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| D=(R/100)\sqrt{(K_1\Delta\phi)^2+(K_2\Delta\lambda)^2};{\color{white}\frac{\big|}{.}}\,\!</math>
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| :where
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| ::<math>D\,\!</math> = Distance in kilometers;
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| ::<math>\Delta\phi\,\!</math> and <math>\Delta\lambda\,\!</math> are in degrees;
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| ::<math>\phi_m\,\!</math> must be in units compatible with the method used for determining <math>\cos(\phi_m);\,\!</math>
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| ::<math>\begin{align}
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| K_1&=111.13209-0.56605\cos(2\phi_m)+0.00120\cos(4\phi_m);\\
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| K_2&=111.41513\cos(\phi_m)-0.09455\cos(3\phi_m)+0.00012\cos(5\phi_m).\end{align}\,\!</math>
| |
| | |
| :It may be interesting to note that:
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| ::<math>K_1=M\frac{\pi}{180}\,\!</math> = kilometers per degree of latitude difference;
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| ::<math>K_2=\cos(\phi_m)N\frac{\pi}{180}\,\!</math> = kilometers per degree of longitude difference;
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| ::where <math>M\,\!</math> and <math>N\,\!</math> are the '''''m'''eridional'' and its perpendicular, or "'''''n'''ormal''", [[Radius_of_curvature_(applications)#Principal_radii_of_curvature|radii of curvature]] (the expressions in the FCC formula are derived from the [[binomial series]] expansion form of <math>M\,\!</math> and <math>N\,\!</math>, set to the ''Clarke 1866'' [[reference ellipsoid]]).
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| | |
| ===Polar coordinate flat-Earth formula===
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| :<math>D=R\sqrt{\theta^2_1\;\boldsymbol{+}\;\theta^2_2\;\mathbf{-}\;2\theta_1\theta_2\cos(\Delta\lambda)};{\color{white}\frac{\big|}{.}}\,\!</math>
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| :where the colatitude values are in radians. For a latitude measured in degrees, the colatitude in radians may be calculated as follows: <math>\theta=\frac{\pi}{180}(90^\circ-\phi).\,\!</math>
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| ==Spherical-surface formulae==
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| {{main|Great-circle distance}}
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| If we are willing to accept a possible error of 0.5%, we can use formulas of [[spherical trigonometry]] on the sphere that best approximates the surface of the earth.
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| The shortest distance along the surface of a sphere between two points on the surface is along the great-circle which contains the two points.
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| The [[great-circle distance]] article gives the formula for calculating the distance along a great-circle on a sphere about the size of the Earth. That article includes an example of the calculation.
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| ===Tunnel distance===
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| A tunnel between points on Earth is defined by a line through three-dimensional space between the points of interest.
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| The great circle chord length may be calculated as follows for the corresponding unit sphere:
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| :<math>\begin{align}
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| &\Delta{X}=\cos(\phi_2)\cos(\lambda_2) - \cos(\phi_1)\cos(\lambda_1);\\
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| &\Delta{Y}=\cos(\phi_2)\sin(\lambda_2) - \cos(\phi_1)\sin(\lambda_1);\\
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| &\Delta{Z}=\sin(\phi_2) - \sin(\phi_1);\\
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| &C_h=\sqrt{(\Delta{X})^2 + (\Delta{Y})^2 + (\Delta{Z})^2}.\end{align}
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| </math>
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| | |
| The tunnel distance between points on the surface of a spherical Earth is
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| <math>D = R C_h</math>. For short distances (<math>D\ll R</math>), this underestimates the great circle distance by <math>D(D/R)^2/24</math>.
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| | |
| ==Ellipsoidal-surface formulae==
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| {{Main|Geodesics on an ellipsoid}}
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| | |
| [[File:Long geodesic on an oblate ellipsoid.svg|thumb|
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| Geodesic on an oblate ellipsoid]]
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| An ellipsoid approximates the surface of the earth much better than a
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| sphere or a flat surface does. The shortest distance along the surface
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| of an ellipsoid between two points on the surface is along the
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| [[geodesic]]. Geodesics follow more complicated paths than great
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| circles and in particular, they usually don't return to their starting
| |
| positions after one circuit of the earth. This is illustrated in the
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| figure on the right where ''f'' is taken to be 1/50 to accentuate the
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| effect. Finding the geodesic between two points on the earth, the
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| so-called [[inverse geodetic problem]], was the focus of many
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| mathematicians and geodesists over the course of the 18th and 19th
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| centuries with major contributions by
| |
| Clairaut,<ref>
| |
| {{cite journal
| |
| |ref = harv
| |
| |last = Clairaut
| |
| |first = A. C.
| |
| |year = 1735
| |
| |title = Détermination géometrique de la perpendiculaire à la méridienne tracée par M. Cassini
| |
| |trans_title = Geometrical determination of the perpendicular to the meridian drawn by Jacques Cassini
| |
| |language = French
| |
| |journal = Mémoires de l'Académie Royale des Sciences de Paris 1733
| |
| |pages = 406–416
| |
| |url = http://books.google.com/books?id=GOAEAAAAQAAJ&pg=PA406
| |
| |authorlink = Alexis Claude Clairaut
| |
| }}
| |
| </ref> | |
| Legendre,<ref>
| |
| {{cite journal
| |
| |ref = harv
| |
| |last = Legendre
| |
| |first = A. M.
| |
| |year = 1806
| |
| |title = Analyse des triangles tracées sur la surface d'un sphéroïde
| |
| |trans_title = Analysis of spheroidal triangles
| |
| |language = French
| |
| |journal = Mémoires de l'Institut National de France
| |
| |number = 1st semester
| |
| |pages = 130–161
| |
| |url = http://books.google.com/books?id=EnVFAAAAcAAJ&pg=PA130
| |
| |authorlink = Adrien-Marie Legendre
| |
| }}</ref>
| |
| Bessel,<ref>
| |
| {{cite journal
| |
| | ref = {{harvid|Bessel|1825}}
| |
| | last1 = Bessel | first1 = F. W.
| |
| | author1-link = Friedrich Bessel
| |
| | year = 2010
| |
| | doi = 10.1002/asna.201011352
| |
| | title = The calculation of longitude and latitude from geodesic measurements
| |
| | journal = Astronomische Nachrichten
| |
| | volume = 331 | issue = 8 | pages = 852–861
| |
| | arxiv = 0908.1824
| |
| | origyear = 1825
| |
| | others = . Translated by C. F. F. Karney & R. E. Deakin
| |
| | postscript = . English translation of [http://adsabs.harvard.edu/full/1825AN......4..241B ''Astron. Nachr.'' '''4''', 241–254 (1825)]. [http://geographiclib.sourceforge.net/bessel-errata.html Errata].
| |
| }}</ref>
| |
| and Helmert.<ref> | |
| {{cite book
| |
| |ref = {{harvid|Helmert|1880}}
| |
| |last = Helmert
| |
| |first = F. R.
| |
| |year = 1964
| |
| |origyear = 1880
| |
| |title = Mathematical and Physical Theories of Higher Geodesy
| |
| |volume = 1
| |
| |publisher = Aeronautical Chart and Information Center
| |
| |location = St. Louis
| |
| |url = http://geographiclib.sf.net/geodesic-papers/helmert80-en.html
| |
| |authorlink = Friedrich Robert Helmert
| |
| |postscript = . English translation of [http://books.google.com/books?id=qt2CAAAAIAAJ ''Die Mathematischen und Physikalischen Theorieen der Höheren Geodäsie''], Vol. 1 (Teubner, Leipzig, 1880).
| |
| }}</ref>
| |
| Rapp<ref>
| |
| {{cite techreport
| |
| |first=R. H. |last=Rapp
| |
| |title=Geometric Geodesy, Part II
| |
| |institution=Ohio State University
| |
| |date=March 1993
| |
| |url=http://hdl.handle.net/1811/24409
| |
| |accessdate=2011-08-01
| |
| }}
| |
| </ref>
| |
| provides a good summary of this work.
| |
| | |
| Methods for computing the geodesic distance are widely available in
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| [[geographical information systems]], software libraries, standalone
| |
| utilities, and online tools. The most widely used algorithm is by
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| Vincenty,<ref>
| |
| {{cite journal
| |
| |first=T. |last=Vincenty |authorlink=Thaddeus Vincenty
| |
| |title=Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of nested equations
| |
| |journal=Survey Review
| |
| |volume=23 |issue=176 |date=April 1975 |pages=88–93
| |
| |url=http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf |accessdate=2009-07-11
| |
| |postscript = . Addendum: Survey Review '''23''' (180): 294 (1976).
| |
| }}</ref>
| |
| who uses a series which is accurate to third order in the flattening of
| |
| the ellipsoid, i.e., about 0.5 mm; however, the algorithm fails to
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| converge for points that are nearly [[Antipodes|anti-podal]]. (For
| |
| details, see [[Vincenty's formulae]].) This defect is cured in the
| |
| algorithm given by
| |
| Karney,<ref>
| |
| {{cite doi
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| |10.1007/s00190-012-0578-z
| |
| |noedit
| |
| }}
| |
| </ref>
| |
| who employs series which are accurate to sixth order in the flattening.
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| This results in an algorithm which is accurate to full double precision
| |
| and which converges for arbitrary pairs of points on the earth. This | |
| algorithm is implemented in GeographicLib.<ref>
| |
| {{cite web
| |
| |url = http://geographiclib.sourceforge.net
| |
| |last = Karney
| |
| |first = C. F. F.
| |
| |title = GeographicLib
| |
| |version = 1.32
| |
| |year = 2013
| |
| }}
| |
| </ref>
| |
| | |
| The exact methods above are feasible when carrying out calculations on a
| |
| computer. They are intended to give millimeter accuracy on lines of any
| |
| length; we can use simpler formulas if we don't need millimeter
| |
| accuracy, or if we do need millimeter accuracy but the line is short.
| |
| Rapp,<ref name=rapp91>
| |
| {{cite report
| |
| |last = Rapp
| |
| |first = R, H,
| |
| |title = Geometric Geodesy, Part I
| |
| |year = 1991
| |
| |publisher = Ohio Start Univ.
| |
| |url = http://hdl.handle.net/1811/24333
| |
| }}</ref> Chap. 6, describes the [[Louis Puissant|Puissant]] method,
| |
| the Gauss mid-latitude method, and the Bowring method.<ref>
| |
| {{cite journal
| |
| |last = Bowring
| |
| |first = B. R.
| |
| |title = The direct and inverse problems for short geodesics lines on the ellipsoid
| |
| |journal = Surveying and Mapping
| |
| |volume = 41
| |
| |number = 2
| |
| |year = 1981
| |
| |pages = 135–141
| |
| }}
| |
| </ref>
| |
| | |
| ===Lambert's formula for long lines===
| |
| | |
| Lambert's formulae<ref>
| |
| {{cite journal
| |
| |last = Lambert
| |
| |first = W. D
| |
| |title = The distance between two widely separated points on the surface of the earth
| |
| |journal = J. Washington Academy of Sciences
| |
| |year = 1942
| |
| |volume = 32
| |
| |number = 5
| |
| |pages = 125–130
| |
| }}
| |
| </ref>
| |
| give accuracy on the order of 10 meters over thousands of kilometers. First convert the latitudes <math> \scriptstyle \phi_1</math>, <math> \scriptstyle \phi_2</math> of the two points to [[Latitude#Reduced_(or_parametric)_latitude|reduced latitudes]] <math> \scriptstyle \beta_1</math>, <math> \scriptstyle \beta_2</math>
| |
| | |
| <math> \tan \beta = (1 - f) \tan \phi,</math>
| |
| | |
| where <math>f</math> is the flattening.
| |
| Then calculate the [[central angle]] <math> \sigma</math> in radians between two points <math> (\beta_1 , \; \lambda_1)</math> and <math> (\beta_2 , \; \lambda_2)</math> on a sphere in [[Great-circle distance|the usual way]] ([[spherical law of cosines|law of cosines]] or [[haversine formula]]), with longitudes <math> \lambda_1 \; </math> and <math> \lambda_2 \; </math> being the same on the sphere as on the spheroid.<br><br>
| |
| | |
| <math>P = \frac { \beta_1 + \beta_2 }{2} \qquad Q = \frac {\beta_2 - \beta_1}{2}</math><br><br>
| |
| | |
| <math>X = ( \sigma - \sin \sigma) \frac {\sin^2 P \cos^2 Q}{ \cos^2 \frac { \sigma}{2}} \qquad \qquad Y = ( \sigma + \sin \sigma) \frac {\cos^2 P \sin^2 Q}{ \sin^2 \frac { \sigma}{2}}</math><br><br>
| |
| | |
| <math>\mathrm{distance} = a \bigl( \sigma - \tfrac f2 (X + Y) \bigr) </math>
| |
| | |
| where <math>a</math> is the equatorial radius of the chosen spheroid.
| |
| | |
| On the [[GRS 80]] spheroid Lambert's formula is off by
| |
| | |
| :0 North 0 West to 40 North 120 West, 12.6 meters
| |
| :0N 0W to 40N 60W, 6.6 meters
| |
| :40N 0W to 40N 60W, 0.85 meter
| |
| | |
| ===Bowring's method for short lines===
| |
| | |
| Bowring maps the points to a sphere of radius ''R′'', with latitude and longitude represented as φ′ and λ′. Define
| |
| :<math>A = \sqrt{1 + e'^2\cos^4 \phi_1}, \quad B = \sqrt{1 + e'^2\cos^2 \phi_1},</math>
| |
| where the second eccentricity squared is
| |
| :<math> e'^2 = \frac{a^2 - b^2}{b^2} = \frac{f(2-f)}{(1-f)^2}.</math>
| |
| The spherical radius is
| |
| :<math>R' = \frac{\sqrt{1 + e'^2 }}{B^2} a.</math>
| |
| (The [[Gaussian curvature]] of the ellipsoid at φ<sub>1</sub> is 1/''R′''<sup>2</sup>.)
| |
| The spherical coordinates are given by
| |
| :<math>\begin{align}
| |
| \phi_1' &= \tan^{-1}(\tan\phi/B),\\
| |
| \Delta\phi' &= \frac{\Delta \phi}{B}\biggl[1 + \frac{3 e'^2 }{4 B^2}(\Delta \phi) \sin (2 \phi_1 + \tfrac23 \Delta \phi )\biggr],\\
| |
| \Delta\lambda' &= A\Delta\lambda,
| |
| \end{align}
| |
| </math>
| |
| where <math>\Delta\phi=\phi_2-\phi_1</math>, <math>\Delta\phi'=\phi_2'-\phi_1'</math>,
| |
| <math>\Delta\lambda=\lambda_2-\lambda_1</math>, <math>\Delta\lambda'=\lambda_2'-\lambda_1'</math>. The resulting problem on the sphere may be solved using the techniques for [[great-circle navigation]] to give approximations for the spheroidal distance and bearing. Detailed formulas are given by Rapp,<ref name=rapp91/> §6.5.
| |
| | |
| ==See also==
| |
| *[[Earth radius]]
| |
| *[[Spherical Earth]]
| |
| *[[Great-circle distance]]
| |
| *[[Great-circle navigation]]
| |
| *[[Vincenty's formulae]]
| |
| *[[Meridian arc]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| * {{cite news
| |
| |first1=Frank
| |
| |last1=Ivis
| |
| |url=http://nesug.org/Proceedings/nesug06/dm/da15.pdf
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| |title= Calculating Geographic Distance: Concepts and Methods
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| |year=2006
| |
| }}
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| *An [http://geographiclib.sourceforge.net/cgi-bin/GeodSolve online geodesic calculator] (based on GeographicLib).
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| *An [http://geographiclib.sourceforge.net/geodesic-papers/biblio.html online geodesic bibliography].
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| *Tripstance tool - [http://www.tripstance.com Online distance calculator].
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| [[Category:Cartography]]
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| [[Category:Earth]]
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| [[Category:Geodesy]]
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