Great - circle distance - wikipedia Great - circle distance This article is about shortest - distance on a sphere. For the shortest distance on an ellipsoid, see geodesics on an ellipsoid. A diagram illustrating great - circle distance ( drawn in red ) between two points on a sphere, P and Q. Two antipodal points, u and v, are also shown. The great - circle distance or orthodromic distance is the shortest distance between two points 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 spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called great circles. Through any two points on a sphere that are not 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 called a Riemannian circle in Riemannian geometry. Between two points that are directly opposite each other, called antipodal points, there are infinitely many great circles, and all great circle arcs between antipodal points have a length of half the circumference of the circle, or π r ( displaystyle pi r ), where r is the radius of the sphere. The Earth is nearly spherical ( see Earth radius ), so great - circle distance formulas give the distance between points on the surface of the Earth correct to within about 0.5 %. ( See Arc length § Arcs of great circles on the Earth. ) Contents 1 Formulas 1.1 Computational formulas 1.2 Vector version 1.3 From chord length 2 Radius for spherical Earth 3 See also 4 References and notes 5 External links Formulas An illustration of the central angle, Δσ, between two points, P and Q. λ and φ are the longitudinal and latitudinal angles of P respectively Let φ 1, λ 1 ( displaystyle phi _ ( 1 ), lambda _ ( 1 ) ) and φ 2, λ 2 ( displaystyle phi _ ( 2 ), lambda _ ( 2 ) ) be the geographical latitude and longitude in radians of two points 1 and 2, and Δ φ, Δ λ ( displaystyle Delta phi, Delta lambda ) be their absolute differences ; then Δ σ ( displaystyle Delta sigma ), the central angle between them, is given by the spherical law of cosines if one of the poles is used as an auxiliary third point on the sphere : Δ σ = arccos ⁡ ( sin ⁡ φ 1 ⋅ sin ⁡ φ 2 + cos ⁡ φ 1 ⋅ cos ⁡ φ 2 ⋅ cos ⁡ ( Δ λ ) ). ( displaystyle Delta sigma = arccos ( \bigl ( ) sin phi _ ( 1 ) cdot sin phi _ ( 2 ) + cos phi _ ( 1 ) cdot cos phi _ ( 2 ) cdot cos ( Delta lambda ) ( \bigr ) ). ) The distance d, i.e. the arc length, for a sphere of radius r and Δ σ ( displaystyle Delta sigma ) given in radians d = r Δ σ. ( displaystyle d = r , Delta sigma. ) Computational formulas On computer systems with low floating - point precision, the spherical law of cosines formula can have large rounding errors 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 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. The haversine formula is numerically better - conditioned for small distances : Δ σ = 2 arcsin ⁡ sin 2 ⁡ ( Δ φ 2 ) + cos ⁡ φ 1 ⋅ cos ⁡ φ 2 ⋅ sin 2 ⁡ ( Δ λ 2 ). ( displaystyle Delta sigma = 2 arcsin ( sqrt ( sin ^ ( 2 ) left ( ( \frac ( Delta phi ) ( 2 ) ) \right ) + cos ( phi _ ( 1 ) ) cdot cos ( phi _ ( 2 ) ) cdot sin ^ ( 2 ) left ( ( \frac ( Delta lambda ) ( 2 ) ) \right ) ) ). ; ! ) Historically, the use of this formula was simplified by the availability of tables for the haversine function : hav ( θ ) = sin ( θ / 2 ). 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 points ( on opposite ends of the sphere ). A more complicated formula that is accurate for all distances is the following special case of the Vincenty formula for an ellipsoid with equal major and minor axes : Δ σ = arctan ⁡ ( cos ⁡ φ 2 ⋅ sin ⁡ ( Δ λ ) ) 2 + ( cos ⁡ φ 1 ⋅ sin ⁡ φ 2 − sin ⁡ φ 1 ⋅ cos ⁡ φ 2 ⋅ cos ⁡ ( Δ λ ) ) 2 sin ⁡ φ 1 ⋅ sin ⁡ φ 2 + cos ⁡ φ 1 ⋅ cos ⁡ φ 2 ⋅ cos ⁡ ( Δ λ ). ( displaystyle Delta sigma = arctan ( \frac ( sqrt ( left ( cos phi _ ( 2 ) cdot sin ( Delta lambda ) \right ) ^ ( 2 ) + left ( cos phi _ ( 1 ) cdot sin phi _ ( 2 ) - sin phi _ ( 1 ) cdot cos phi _ ( 2 ) cdot cos ( Delta lambda ) \right ) ^ ( 2 ) ) ) ( sin phi _ ( 1 ) cdot sin phi _ ( 2 ) + cos phi _ ( 1 ) cdot cos phi _ ( 2 ) cdot cos ( Delta lambda ) ) ). ) When programming a computer, one should use the atan2 ( ) function rather than the ordinary arctangent function ( atan ( ) ), so that Δ σ ( displaystyle Delta sigma ) is placed in the correct quadrant. The determination of the great - circle distance is just part of the more general problem of great - circle navigation, which also computes the azimuths at the end points and intermediate way - points. Vector version Another representation of similar formulas, but using normal vectors instead of latitude / longitude to describe the positions, is found by means of 3D vector algebra, i.e. utilizing the dot product, cross product, or a combination : Δ σ = arccos ⁡ ( n 1 ⋅ n 2 ) Δ σ = arcsin ⁡ n 1 × n 2 Δ σ = arctan ⁡ n 1 × n 2 n 1 ⋅ n 2 ( displaystyle ( \begin ( aligned ) Delta sigma & = arccos ( mathbf ( n ) _ ( 1 ) cdot mathbf ( n ) _ ( 2 ) ) Delta sigma & = arcsin left mathbf ( n ) _ ( 1 ) \times mathbf ( n ) _ ( 2 ) \right Delta sigma & = arctan ( \frac ( left mathbf ( n ) _ ( 1 ) \times mathbf ( n ) _ ( 2 ) \right ) ( mathbf ( n ) _ ( 1 ) cdot mathbf ( n ) _ ( 2 ) ) ) end ( aligned ) ) , ! ) where n 1 ( displaystyle mathbf ( n ) _ ( 1 ) ) and n 2 ( displaystyle mathbf ( n ) _ ( 2 ) ) 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 for all angles. The expression based on arctan requires the magnitude of the cross product over the dot product. From chord length 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. The great circle chord length, C h ( displaystyle C_ ( h ) , ! ), may be calculated as follows for the corresponding unit sphere, by means of Cartesian subtraction : Δ X = cos ⁡ φ 2 ⋅ cos ⁡ λ 2 − cos ⁡ φ 1 ⋅ cos ⁡ λ 1 ; Δ Y = cos ⁡ φ 2 ⋅ sin ⁡ λ 2 − cos ⁡ φ 1 ⋅ sin ⁡ λ 1 ; Δ Z = sin ⁡ φ 2 − sin ⁡ φ 1 ; C = ( Δ X ) 2 + ( Δ Y ) 2 + ( Δ Z ) 2 ( displaystyle ( \begin ( aligned ) Delta ( X ) & = cos phi _ ( 2 ) cdot cos lambda _ ( 2 ) - cos phi _ ( 1 ) cdot cos lambda _ ( 1 ) ; Delta ( Y ) & = cos phi _ ( 2 ) cdot sin lambda _ ( 2 ) - cos phi _ ( 1 ) cdot sin lambda _ ( 1 ) ; Delta ( Z ) & = sin phi _ ( 2 ) - sin phi _ ( 1 ) ; C& = ( sqrt ( ( Delta ( X ) ) ^ ( 2 ) + ( Delta ( Y ) ) ^ ( 2 ) + ( Delta ( Z ) ) ^ ( 2 ) ) ) end ( aligned ) ) ) The central angle is : Δ σ = 2 arcsin ⁡ C 2. ( displaystyle Delta sigma = 2 arcsin ( \frac ( C ) ( 2 ) ). ) The great circle distance is : d = r Δ σ. ( displaystyle d = r Delta sigma. ) Radius for spherical Earth Equatorial ( a ), polar ( b ) and mean Earth radii as defined in the 1984 World Geodetic System revision ( not to scale ) Main article : Earth radius The shape of the Earth closely resembles a flattened sphere ( a spheroid ) with equatorial radius a ( displaystyle a ) of 6378.137 km ; distance b ( displaystyle b ) 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 b 2 / a ( displaystyle b ^ ( 2 ) / a ) ( which equals the meridian 's semi-latus rectum ), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius a 2 / b ( displaystyle a ^ ( 2 ) / b ), or 6399.594 km, a 1 % difference. So long as a spherical Earth is assumed, any single formula for distance on the Earth is only guaranteed correct within 0.5 % ( though better accuracy is possible if the formula is only intended to apply to a limited area ). A good choice for the radius is the mean earth radius, R 1 = 1 3 ( 2 a + b ) ≈ 6371 k m ( displaystyle R_ ( 1 ) = ( \frac ( 1 ) ( 3 ) ) ( 2a + b ) approx 6371 , mathrm ( km ) ) ( for the WGS84 ellipsoid ) ; in the limit of small flattening, this choice minimizes the mean square relative error in the estimates for distance. See also Air navigation Isoazimuthal Central angle Circumnavigation Flight planning Geodesy Geodesics on an ellipsoid Geodetic system Geographical distance Great - circle navigation Haversine formula Meridian arc Rhumb line Spherical Earth Spherical geometry Spherical trigonometry References and notes Jump up ^ Admiralty Manual of Navigation, Volume 1, The Stationery Office, 1987, p. 10, ISBN 9780117728806, 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. Jump up ^ Kells, Lyman M. ; Kern, Willis F. ; Bland, James R. ( 1940 ). Plane And Spherical Trigonometry. McGraw Hill Book Company, Inc. pp. 323 -- 326. Retrieved July 13, 2018. Jump up ^ `` Calculate distance, bearing and more between Latitude / Longitude points ''. Retrieved 10 Aug 2013. Jump up ^ Sinnott, Roger W. ( August 1984 ). `` Virtues of the Haversine ''. Sky and Telescope. 68 ( 2 ) : 159. Jump up ^ Vincenty, Thaddeus ( 1975 - 04 - 01 ). `` Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations '' ( PDF ). Survey Review. Kingston Road, Tolworth, Surrey : Directorate of Overseas Surveys. 23 ( 176 ) : 88 -- 93. doi : 10.1179 / sre. 1975.23. 176.88. Retrieved 2008 - 07 - 21. Jump up ^ Gade, Kenneth ( 2010 ). `` A non-singular horizontal position representation '' ( PDF ). The Journal of Navigation. Cambridge University Press. 63 ( 3 ) : 395 -- 417. doi : 10.1017 / S0373463309990415. Jump up ^ McCaw, G.T. ( 1932 ). `` Long lines on the Earth ''. Empire Survey Review. 1 ( 6 ) : 259 -- 263. doi : 10.1179 / sre. 1932.1. 6.259. External links GreatCircle at MathWorld Retrieved from `` https://en.wikipedia.org/w/index.php?title=Great-circle_distance&oldid=850122938 '' Categories : Metric geometry Spherical trigonometry Talk Contents About Wikipedia Català Čeština Deutsch Eesti Ελληνικά Español Euskara Français Íslenska Italiano Magyar 日本 語 Polski Português Русский Slovenčina Slovenščina Українська 中文 10 more Edit links This page was last edited on 13 July 2018, at 19 : 13 ( UTC ). Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: the shortest distance between two points on the earth's surface is
Answer: great-circle distance