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| {{about|shortest-distance on a sphere|the shortest distance on an ellipsoid|geodesics on an ellipsoid}}
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| The '''great-circle''' or '''[[Great Circle|orthodromic]] distance''' is the shortest [[distance]] between two [[Point (geometry)|point]]s on the surface of a [[sphere]], measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in [[Euclidean space]] is the length of a straight line between them, but on the sphere there are no straight lines. In [[non-Euclidean geometry]], straight lines are replaced with [[geodesic]]s. Geodesics on the sphere are the ''[[great circle]]s'' (circles on the sphere whose centers coincide with the center of the sphere).
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| Through any two points on a sphere which are not [[antipodal point|directly opposite each other]], there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is the [[Riemannian circle]].
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| Between two points which are directly opposite each other, called ''[[antipodal point]]s'', there are infinitely many great circles, but all great circle arcs between antipodal points have the same length, i.e. half the [[circumference]] of the circle, or <math>\pi r</math>, where ''r'' is the [[radius]] of the [[sphere]].
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| The Earth is nearly spherical (see [[Earth radius]]) so great-circle distance formulas give the distance between points on the surface of the Earth (''[[as the crow flies]]'') correct to within 0.5% or so.<ref>{{citation|title=Admiralty Manual of Navigation, Volume 1|publisher=The Stationery Office|year=1987|isbn=9780117728806|page=10|url=http://books.google.com/books?id=xcy4K5BPyg4C&pg=PA10|quotation=The errors introduced by assuming a spherical Earth based on the international nautical mile are not more than 0.5% for latitude, 0.2% for longitude.}}</ref>
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| ==Formulas==
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| Let <math>\phi_1,\lambda_1</math> and <math>\phi_2,\lambda_2</math> be the geographical [[latitude]] and [[longitude]] of two points 1 and 2, and <math>\Delta\phi,\Delta\lambda</math> their absolute differences; then <math>\Delta\sigma</math>, the [[central angle]] between them, is given by the [[spherical law of cosines]]:
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| :<math>\Delta\sigma=\arccos\bigl(\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\Delta\lambda\bigr).</math>
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| The distance ''d'', i.e. the [[arc length]], for a sphere of radius ''r'' and <math>\Delta\sigma</math> given in
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| :<math>d = r \, \Delta\sigma.</math>
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| ===Computational Formulas===
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| On computer systems with low [[floating point|floating-point precision]], this formula can have large [[rounding error]]s if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle comes out 0.99999999). For modern [[IEEE 754|64-bit floating-point numbers]], the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth.<ref>{{cite web | url=http://www.movable-type.co.uk/scripts/latlong.html | title=Calculate distance, bearing and more between Latitude/Longitude points | accessdate=10 Aug 2013}}</ref> The [[haversine formula]] is [[Condition number|numerically better-conditioned]] for small distances:<ref>R.W. Sinnott, "Virtues of the Haversine", Sky and Telescope, vol. 68, no. 2, 1984, p. 159</ref>
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| :<math>\Delta\sigma
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| =2\arcsin\left(\sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right)+\cos{\phi_1}\cos{\phi_2}\sin^2\left(\frac{\Delta\lambda}{2}\right)}\right).\;\!</math>
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| Historically, the use of this formula was simplified by the availability of tables for the [[haversine]] function: hav(''θ'') = sin<sup>2</sup>(''θ''/2).
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| Although this formula is accurate for most distances on a sphere, it too suffers from rounding errors for the special (and somewhat unusual) case of [[antipodal point]]s (on opposite ends of the sphere). A more complicated formula that is accurate for all distances is the following special case (a sphere, which is an ellipsoid with equal major and minor axes) of the [[Vincenty's formulae|Vincenty formula]] (which more generally is a method to compute distances on ellipsoids):<ref>{{cite journal
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| | last = Vincenty
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| | first = Thaddeus
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| | authorlink = Thaddeus Vincenty
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| | title = Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations
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| | journal = Survey Review
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| | volume = 23
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| | issue = 176
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| | pages = 88–93
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| | publisher = [[Directorate of Overseas Surveys]]
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| | location = Kingston Road, Tolworth, Surrey
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| | date = 1975-04-01
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| | url =http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf
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| | format = [[PDF]]
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| | accessdate = 2008-07-21 }}
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| </ref>
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| :<math>\Delta\sigma=\arctan\left(\frac{\sqrt{\left(\cos\phi_2\sin\Delta\lambda\right)^2+\left(\cos\phi_1\sin\phi_2-\sin\phi_1\cos\phi_2\cos\Delta\lambda\right)^2}}{\sin\phi_1\sin\phi_2+\cos\phi_1\cos\phi_2\cos\Delta\lambda}\right).</math>
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| When programming a computer, one should use the <code>[[atan2]]()</code> function rather than the ordinary arctangent function (<code>atan()</code>), so that <math>\Delta\sigma</math> is placed in the correct quadrant.
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| The determination of the great-circle distance is just part of the more general problem of [[great-circle navigation]] which computes also the azimuths at the end points and intermediate way-points.
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| === Vector version ===
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| Another representation of similar formulas, but using [[n-vector|normal vectors]] instead of latitude/longitude to describe the positions, is found by means of 3D [[vector_calculus#Vector_operations|vector algebra]], i.e. utilizing the [[dot product]], [[cross product]], or a combination:<ref>{{cite journal |last1= Gade |first1= Kenneth |year= 2010 |title= A non-singular horizontal position representation |journal= The Journal of Navigation |publisher= Cambridge University Press |volume= 63 |issue= 3 |pages=395–417 |url=http://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf| format=PDF |doi= 10.1017/S0373463309990415 }}</ref>
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| :<math>\begin{align}
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| \Delta\sigma &=\arccos( \mathbf n_1\cdot \mathbf n_2 ) \\
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| \Delta\sigma &=\arcsin\left( \left| \mathbf n_1\times \mathbf n_2 \right| \right) \\
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| \Delta\sigma &=\arctan\left( \frac{\left| \mathbf n_1\times \mathbf n_2 \right|}{\mathbf n_1\cdot \mathbf n_2} \right) \\
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| \end{align}\,\!</math>
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| where <math>\mathbf n_1</math> and <math>\mathbf n_2</math> are the normals to the ellipsoid at the two positions 1 and 2. Similarly to the equations above based on latitude and longitude, the expression based on arctan is the only one that is well-conditioned [[Inverse trigonometric functions#Practical considerations|for all angles]].
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| === From chord length ===
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| A line through three-dimensional space between points of interest on a spherical Earth is the chord of the great circle between the points. The [[central angle]] between the two points can be determined from the chord length. The great circle distance is proportional to the central angle.
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| The great circle chord length, <math>C_h\,\!</math>, may be calculated as follows for the corresponding unit sphere, by means of [[Cartesian coordinate system|Cartesian subtraction]]:
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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&=\sqrt{(\Delta{X})^2+(\Delta{Y})^2+(\Delta{Z})^2}
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| \end{align}</math>
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| The central angle is:
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| :<math>\Delta\sigma=2\arcsin\left(\frac C2\right).</math>
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| The great circle distance is:
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| :<math>d = r \Delta\sigma.</math> | |
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| ==Radius for spherical Earth==
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| {{main|Earth radius}}
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| The [[geoid|shape of the Earth]] closely resembles a flattened sphere (a [[spheroid]]) with equatorial radius <math>a</math> of 6378.137 km; distance <math>b</math> from the center of the spheroid to each pole is 6356.752 km. When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of <math>b^2/a</math> (which equals the meridian's [[Conic section#Features|semi-latus rectum]]), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius <math>a^2/b</math>, or 6399.594 km, a 1% difference. So as long as we're assuming a spherical Earth, any single formula for distance on the Earth is only guaranteed correct within 0.5% (though we can do better if our formula is only intended to apply to a limited area). A good choice<ref>{{cite journal
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| |last = McCaw
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| |first = G. T.
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| |title = Long lines on the Earth
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| |journal = Empire Survey Review
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| |volume = 1
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| |number = 6
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| |pages = 259–263
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| |year = 1932}}</ref> for the radius is the [[Earth radius#Mean_radius|mean earth radius]], <math>R_1 = \frac13(2a+b) \approx 6371\,\mathrm{km}</math> (for the [[WGS84]] ellipsoid); in the limit of small flattening, this choice minimizes the mean square [[Relative error#Definitions|relative error]] in the estimates for distance.
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| ==See also==
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| *[[Air navigation]]
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| *[[Central angle]]
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| *[[Circumnavigation]]
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| *[[Flight planning]]
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| *[[Geodesy]]
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| *[[Geodetic system]]
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| *[[Geographical distance]]
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| *[[Great-circle navigation]]
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| *[[Haversine formula]]
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| *[[Meridian arc]]
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| *[[Qibla]]
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| *[[Rhumb line]]
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| *[[Spherical Earth]]
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| *[[Spherical geometry]]
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| *[[Spherical trigonometry]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * [http://mathworld.wolfram.com/GreatCircle.html GreatCircle] at [[MathWorld]]
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| {{DEFAULTSORT:Great-Circle Distance}}
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| [[Category:Metric geometry]]
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| [[Category:Spherical trigonometry]]
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Hello, dear friend! My name is Ashly. I smile that I can unify to the entire world. I live in Canada, in the south region. I dream to check out the various nations, to look for familiarized with interesting people.
My blog :: heathrow Massage