While simpler, the law of cosines is slightly slower than the haversine, in my tests. The choice may be driven by programming language, processor,Ĭoding context, available trig functions (in different languages), etc – and, for very small distances anĮquirectangular approximation may be more suitable. Many geodesy purposes (if not for astronomy). This makes the simpler law of cosines a reasonable 1-line alternative to the haversine formula for ( Note that the geodetic form of the law of cosines is rearranged from theĬanonical one so that the latitude can be used directly, rather than the Gives well-conditioned results down to distances as small as a few metres on the earth’s surface. By my estimate, with this precision,Ĭosines formula (cos c = cos a cos b + sin a sin b cos C) Numbers, which provide 15 significant figures of precision. In fact, JavaScript (and most modern computers & languages) use ‘IEEE 754’ 64-bit floating-point Little to no benefit is obtained by factoring out common terms probably the JIT compiler Using Chrome on an aging Core i5 PC, a distance calculation takes aroundĢ – 5 microseconds (hence around 200,000 – 500,000 per second). (including protection against rounding errors). If atan2 is not available, c could be calculated Magazine in 1984 (“Virtues of the Haversine”): Sinnott explained that the angular separationīetween Mizar and Alcor in Ursa Major – 0☁1′49.69″ – could be accurately calculated in Basic on aįor the curious, c is the angular distance in radians, and a is the square of half Once widely used by navigators, it was described by Roger Sinnott in

‘half-versed-sine’ is (1−cosθ)/2 or sin²(θ/2) as used above. Particularly well-conditioned for numerical computation even at small distances’ – unlike Multiplications) saved navigators from squaring sines, computing square roots,Įtc – arduous and error-prone activities. Haversine/inverse-haversine (and its logarithm, to aid Also, the sin²(θ/2) form of the haversine avoided addition (which entailedĪn anti-log lookup, the addition, and a log lookup). (real) log of a negative number, the ‘versine’ enabled them to keep trig functions in The height of technology for navigator’s calculations used to be log tables. Latitude/longitude in radians – having found that mixing degrees & radians is often the Note in these scripts, I generally use lat/lon for latitude/longitude in degrees, and φ/λ for Note that angles need to be in radians to pass to trig functions!Ĭonst φ1 = lat1 * Math.PI/180 // φ, λ in radiansĬonst a = Math.sin(Δφ/2) * Math.sin(Δφ/2) +Ĭonst c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a)) Φ is latitude, λ is longitude, R is earth’s radius (mean radius = 6,371km) Points – that is, the shortest distance over the earth’s surface – giving an ‘as-the-crow-flies’ĭistance between the points (ignoring any hills they fly over, of course!).

This uses the ‘ haversine’ formula to calculate the great-circle distance between two signed decimal degrees without compass direction, where negative indicates.deg-min-sec suffixed with N/S/E/W (e.g.Great-circle distance between two pointsĮnter the co-ordinates into the text boxes to try out the calculations. Most purposes… [In fact, the earth is very slightly ellipsoidal using a spherical model gives This page presents a variety of calculations for latitude/longitude points, with the formulas andĪll these formulas are for calculations on the basis of a spherical earth (ignoring ellipsoidalĮffects) – which is accurate enough * for Calculate distance and bearing between two Latitude/Longitude points using haversine formula in JavaScript Movable Type ScriptsĬalculate distance, bearing and more between Latitude/Longitude points

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