Calculate distance between two latitude-longitude points Haversine formula

Question

How do I calculate the distance between two points specified by latitude and longitude   For clarification  I d like the distance in kilometers  the points use the WGS84 system and I d like to understand the relative accuracies of the approaches available

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This link might be helpful to you  as it details the use of the Haversine formula to calculate the distance   Excerpt        This script  in Javascript  calculates great-circle distances between the two points       that is  the shortest distance over the earth   s surface     using the      Haversine    formula    function getDistanceFromLatLonInKm lat1 lon1 lat2 lon2      var R   6371     Radius of the earth in km   var dLat   deg2rad lat2-lat1       deg2rad below   var dLon   deg2rad lon2-lon1      var a        Math sin dLat 2    Math sin dLat 2        Math cos deg2rad lat1     Math cos deg2rad lat2          Math sin dLon 2    Math sin dLon 2           var c   2   Math atan2 Math sqrt a   Math sqrt 1-a       var d   R   c     Distance in km   return d     function deg2rad deg      return deg    Math PI 180

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If you want the driving distance route  posting it here because this is the first result for the distance between two points on google but for most people the driving distance is more useful   you can use Google Maps Distance Matrix Service  getDrivingDistanceBetweenTwoLatLong origin  destination      return new Observable subscriber   gt      let service   new google maps DistanceMatrixService      service getDistanceMatrix              origins   new google maps LatLng origin lat  origin long          destinations   new google maps LatLng destination lat  destination long          travelMode   DRIVING          response  status    gt          if  status     google maps DistanceMatrixStatus OK            console log  Error    status           subscriber error  error  status  status  status            else           console log response           try             let valueInMeters   response rows 0  elements 0  distance value            let valueInKms   valueInMeters   1000            subscriber next valueInKms             subscriber complete                     catch error            subscriber error  error  error  status  status

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I ve created this small Javascript LatLng object  might be useful for somebody   var latLng1   new LatLng 5  3   var latLng2   new LatLng 6  7   var distance   latLng1 distanceTo latLng2      Code          latLng point     param  Number  lat     param  Number  lng     returns  LatLng      constructor     function LatLng lat lng        this lat   parseFloat lat       this lng   parseFloat lng        this   cache          LatLng prototype         toString  function             return  this lat  this lng  join                              calculate distance in km to another latLng  with caching         param  LatLng  latLng         returns  Number  distance in km             distanceTo  function latLng            var cacheKey   latLng toString            if cacheKey in this   cache                return this   cache cacheKey                         the fastest way to calculate the distance  according to this jsperf test             http   jsperf com haversine-salvador 8            http   stackoverflow com questions 27928         var deg2rad   0 017453292519943295         Math PI   180         var lat1   this lat   deg2rad          var lng1   this lng   deg2rad          var lat2   latLng lat   deg2rad          var lng2   latLng lng   deg2rad          var a                  1 - Math cos lat2 - lat1                  1 - Math cos lng2 - lng1     Math cos lat1    Math cos lat2                  2          var distance   12742   Math asin Math sqrt a       Diameter of the earth in km  2   6371              cache the distance         this   cache cacheKey    distance           return distance

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function getDistanceFromLatLonInKm lat1 lon1 lat2 lon2 units      var R   6371     Radius of the earth in km   var dLat   deg2rad lat2-lat1       deg2rad below   var dLon   deg2rad lon2-lon1      var a        Math sin dLat 2    Math sin dLat 2        Math cos deg2rad lat1     Math cos deg2rad lat2          Math sin dLon 2    Math sin dLon 2           var c   2   Math atan2 Math sqrt a   Math sqrt 1-a       var d   R   c     var miles   d   1 609344    if   units     km        return d      else   return miles       Chuck s solution  valid for miles also

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There could be a simpler solution  and more correct  The perimeter of earth is 40 000Km at the equator  about 37 000 on Greenwich  or any longitude  cycle  Thus   pythagoras   function  lat1  lon1  lat2  lon2       function sqr x   return x   x      function cosDeg x   return Math cos x   Math PI   180 0        var earthCyclePerimeter   40000000 0   cosDeg  lat1   lat2    2 0      var dx    lon1 - lon2    earthCyclePerimeter   360 0     var dy   37000000 0    lat1 - lat2    360 0      return Math sqrt sqr dx    sqr dy         I agree that it should be fine-tuned as  I myself said that it s an ellipsoid  so the radius to be multiplied by the cosine varies  But it s a bit more accurate  Compared with Google Maps and it did reduce the error significantly

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Python implimentation Origin is the center of the contiguous United States   from haversine import haversine origin    39 50  98 35  paris    48 8567  2 3508  haversine origin  paris  miles True    To get the answer in kilometers simply set miles false

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Here s another converted to Ruby code   include Math  Note  from to    lat  long   def get distance in km from  to    radians   lambda    deg  deg   Math PI   180     radius   6371   Radius of the earth in kilometer   dLat   radians to 0 -from 0     dLon   radians to 1 -from 1      cosines product   Math sin dLat 2    Math sin dLat 2    Math cos radians from 0      Math cos radians to 1      Math sin dLon 2    Math sin dLon 2     c   2   Math atan2 Math sqrt cosines product   Math sqrt 1-cosines product      return radius   c   Distance in kilometer end

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I made a custom function in R to calculate haversine distance km  between two spatial points using functions available in R base package  custom hav dist  lt - function lat1  lon1  lat2  lon2    R  lt - 6371 Radian factor  lt - 0 0174533 lat 1  lt -  90-lat1  Radian factor lat 2  lt -  90-lat2  Radian factor diff long  lt - lon1-lon2  Radian factor  distance in km  lt - 6371 acos  cos lat 1  cos lat 2                       sin lat 1  sin lat 2  cos diff long    rm lat1  lon1  lat2  lon2  return distance in km     Sample output custom hav dist 50 31 19 08 54 14 19 39   1  426 3987  PS  To calculate distances in miles  substitute R in function  6371  with 3958 756  and for nautical miles  use 3440 065

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If you are using python  pip install geopy from geopy distance import geodesic   origin    30 172705  31 526725      latitude  longitude  don t confuse destination    30 288281  31 732326   print geodesic origin  destination  meters     23576 805481751613 print geodesic origin  destination  kilometers     23 576805481751613 print geodesic origin  destination  miles     14 64994773134371

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One of the main challenges to calculating distances - especially large ones - is accounting for the curvature of the Earth  If only the Earth were flat  calculating the distance between two points would be as simple as for that of a straight line  The Haversine formula includes a constant  it s the R variable below  that represents the radius of the Earth  Depending on whether you are measuring in miles or kilometers  it would equal 3956 mi or 6367 km respectively  The basic formula is   dlon   lon2 - lon1 dlat   lat2 - lat1 a    sin dlat 2   2   cos lat1    cos lat2     sin dlon 2   2 c   2   atan2  sqrt a   sqrt 1-a    distance   R   c  where R is the radius of the Earth    R   6367 km OR 3956 mi        lat1  lon1  The Latitude and Longitude of point 1  in decimal degrees       lat2  lon2  The Latitude and Longitude of point 2  in decimal degrees       unit  The unit of measurement in which to calculate the results where        M  is statute miles  default        K  is kilometers       N  is nautical miles  Sample function distance lat1  lon1  lat2  lon2  unit        try           var radlat1   Math PI   lat1   180         var radlat2   Math PI   lat2   180         var theta   lon1 - lon2         var radtheta   Math PI   theta   180         var dist   Math sin radlat1    Math sin radlat2    Math cos radlat1    Math cos radlat2    Math cos radtheta           dist   Math acos dist          dist   dist   180   Math PI         dist   dist   60   1 1515         if  unit     quot K quot                 dist   dist   1 609344                   if  unit     quot N quot                 dist   dist   0 8684                   return dist       catch  err            console log err

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The functions needed for an accurate calculation of distance between lat-long points are complex  and the pitfalls are many  I would not recomend haversine or other spherical solutions due to the big inaccuracies  the earth is not a perfect sphere   The vincenty formula is better  but will in some cases throw errors  even when coded correctly   Instead of coding the functions yourself I suggest using geopy which have implemented the very accurate geographiclib for distance calculations  paper from author      pip install geopy from geopy distance import geodesic NY    40 71278 -74 00594  Beijing    39 90421 116 40739  print  WGS84    geodesic NY  Beijing  km   WGS84 is Standard print  Intl24    geodesic NY  Beijing  ellipsoid  Intl 1924   km   geopy includes different ellipsoids print  Custom ellipsoid    geodesic NY  Beijing  ellipsoid  6377   6356   1   297    km   custom ellipsoid   supported ellipsoids   model             major  km    minor  km      flattening   WGS-84           6378 137     6356 7523142   1   298 257223563    GRS-80           6378 137     6356 7523141   1   298 257222101    Airy  1830       6377 563396  6356 256909    1   299 3249646    Intl 1924        6378 388     6356 911946    1   297 0    Clarke  1880     6378 249145  6356 51486955  1   293 465    GRS-67           6378 1600    6356 774719    1   298 25    The only drawback with this library is that it doesn t support vectorized calculations  For vectorized calculations you can use the new geovectorslib    pip install geovectorslib from geovectorslib import inverse print inverse lats1 lons1 lats2 lons2   s12      lats and lons are numpy arrays  Geovectorslib is very accurate and extremly fast  I haven t found a solution for changing ellipsoids though  The WGS84 ellipsoid is used as standard  which is the best choice for most uses

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This script  in PHP  calculates distances between the two points   public static function getDistanceOfTwoPoints  source   dest   unit  K              lat1    source 0            lon1    source 1            lat2    dest 0            lon2    dest 1             theta    lon1 -  lon2           dist   sin deg2rad  lat1     sin deg2rad  lat2      cos deg2rad  lat1     cos deg2rad  lat2     cos deg2rad  theta             dist   acos  dist            dist   rad2deg  dist            miles    dist   60   1 1515           unit   strtoupper  unit            if   unit     K                 return   miles   1 609344                     else if   unit     M                         return   miles   1 609344   1000                     else if   unit     N                 return   miles   0 8684                      else               return  miles

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Here s a simple javascript function that may be useful from this link   somehow related but we re using google earth javascript plugin instead of maps  function getApproximateDistanceUnits point1  point2         var xs   0      var ys   0       xs   point2 getX   - point1 getX        xs   xs   xs       ys   point2 getY   - point1 getY        ys   ys   ys       return Math sqrt xs   ys       The units tho are not in distance but in terms of a ratio relative to your coordinates   There are other computations related you can substitute for the getApproximateDistanceUnits function link here  Then I use this function  to see if a latitude longitude is within the radius  function isMapPlacemarkInRadius point1  point2  radi        if  point1  amp  amp  point2            return getApproximateDistanceUnits point1  point2   lt   radi        else           return 0            point may be defined as      getPoint   function lati  longi            var location                 x  0              y  0              getX  function     return location x                 getY  function     return location y                       location x   lati          location y   longi           return location           then you can do your thing to see if a point is within a region with a radius say      put it on the map if within the range of a specified radi assuming 100 000 000 units         var iconpoint   Map getPoint pp latitude  pp longitude           var centerpoint   Map getPoint Settings CenterLatitude  Settings CenterLongitude              approx  200 units to show only half of the globe from the default center radius         if  isMapPlacemarkInRadius centerpoint  iconpoint  120                 addPlacemark pp latitude  pp longitude  pp name                     else               otherSidePlacemarks push                   latitude  pp latitude                  longitude  pp longitude                  name  pp name

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I post here my working example   List all points in table having distance between a designated point  we use a random point - lat 45 20327  long 23 7806  less than 50 KM  with latitude  amp  longitude  in MySQL  the table fields are coord lat and coord long    List all having DISTANCE lt 50  in Kilometres  considered Earth radius 6371 KM    SELECT denumire   6371   acos  cos  radians 45 20327      cos  radians  coord lat       cos  radians  23 7806   - radians coord long      sin  radians 45 20327      sin  radians coord lat       AS distanta  FROM obiective  WHERE coord lat lt  gt         AND coord long lt  gt     HAVING distanta lt 50  ORDER BY distanta desc   The above example was tested in MySQL 5 0 95 and 5 5 16  Linux

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Java implementation in according Haversine formula   double calculateDistance double latPoint1  double lngPoint1                            double latPoint2  double lngPoint2        if latPoint1    latPoint2  amp  amp  lngPoint1    lngPoint2            return 0d             final double EARTH RADIUS   6371 0    km value         converting to radians     latPoint1   Math toRadians latPoint1       lngPoint1   Math toRadians lngPoint1       latPoint2   Math toRadians latPoint2       lngPoint2   Math toRadians lngPoint2        double distance   Math pow Math sin  latPoint2 - latPoint1    2 0   2                 Math cos latPoint1    Math cos latPoint2                Math pow Math sin  lngPoint2 - lngPoint1    2 0   2       distance   2 0   EARTH RADIUS   Math asin Math sqrt distance         return distance    km value

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Had an issue with math deg in LUA     if anyone knows a fix please clean up this code     In the meantime here s an implementation of the Haversine in LUA  use this with Redis    function calcDist lat1  lon1  lat2  lon2      lat1  lat1 0 0174532925     lat2  lat2 0 0174532925     lon1  lon1 0 0174532925     lon2  lon2 0 0174532925      dlon   lon2-lon1     dlat   lat2-lat1      a   math pow math sin dlat 2  2    math cos lat1    math cos lat2    math pow math sin dlon 2  2      c   2   math asin math sqrt a       dist   6371   c      -- multiply by 0 621371 to convert to miles     return dist end   cheers

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To calculate the distance between two points on a sphere you need to do the Great Circle calculation    There are a number of C C   libraries to help with map projection at MapTools if you need to reproject your distances to a flat surface   To do this you will need the projection string of the various coordinate systems   You may also find MapWindow a useful tool to visualise the points   Also as its open source its a useful guide to how to use the proj dll library  which appears to be the core open source projection library

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Here is a java implementation of the Haversine formula   public final static double AVERAGE RADIUS OF EARTH KM   6371  public int calculateDistanceInKilometer double userLat  double userLng    double venueLat  double venueLng         double latDistance   Math toRadians userLat - venueLat       double lngDistance   Math toRadians userLng - venueLng        double a   Math sin latDistance   2    Math sin latDistance   2          Math cos Math toRadians userLat     Math cos Math toRadians venueLat           Math sin lngDistance   2    Math sin lngDistance   2        double c   2   Math atan2 Math sqrt a   Math sqrt 1 - a         return  int   Math round AVERAGE RADIUS OF EARTH KM   c        Note that here we are rounding the answer to the nearest km

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In Mysql use the following function pass the parameters as using  POINT LONG LAT   CREATE FUNCTION  distance  a POINT  b POINT   RETURNS double     DETERMINISTIC BEGIN  RETURN  GLength  LineString   PointFromWKB a     PointFromWKB b       100000  -- To Make the distance in meters  END

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JAVA     public Double getDistanceBetweenTwoPoints Double latitude1  Double longitude1  Double latitude2  Double longitude2        final int RADIUS EARTH   6371       double dLat   getRad latitude2 - latitude1       double dLong   getRad longitude2 - longitude1        double a   Math sin dLat   2    Math sin dLat   2    Math cos getRad latitude1     Math cos getRad latitude2     Math sin dLong   2    Math sin dLong   2       double c   2   Math atan2 Math sqrt a   Math sqrt 1 - a        return  RADIUS EARTH   c    1000             private Double getRad Double x        return x   Math PI   180

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function getDistanceFromLatLonInKm position1  position2         use strict       var deg2rad   function  deg    return deg    Math PI   180              R   6371          dLat   deg2rad position2 lat - position1 lat           dLng   deg2rad position2 lng - position1 lng           a   Math sin dLat   2    Math sin dLat   2                Math cos deg2rad position1 lat                 Math cos deg2rad position1 lat                 Math sin dLng   2    Math sin dLng   2           c   2   Math atan2 Math sqrt a   Math sqrt 1 - a        return R   c     console log getDistanceFromLatLonInKm       lat  48 7931459  lng  1 9483572        lat  48 827167  lng  2 2459745

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Here is the SQL Implementation to calculate the distance in km   SELECT UserId    3959   acos  cos  radians  your latitude here       cos  radians latitude       cos  radians longitude  - radians  your longitude here       sin  radians  your latitude here        sin  radians latitude        AS distance FROM user HAVING distance  lt  5  ORDER BY distance LIMIT 0   5    For further details in the implementation by programming langugage  you can just go through the php script given here

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For those looking for an Excel formula based on WGS-84  amp  GRS-80 standards    ACOS COS RADIANS 90-Lat1   COS RADIANS 90-Lat2   SIN RADIANS 90-Lat1   SIN RADIANS 90-Lat2   COS RADIANS Long1-Long2    6371   Source

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Here s the accepted answer implementation ported to Java in case anyone needs it   package com project529 garage util           Mean radius      private static double EARTH RADIUS   6371          Returns the distance between two sets of latitudes and longitudes in meters      lt p  gt     Based from the following JavaScript SO answer     http   stackoverflow com questions 27928 calculate-distance-between-two-latitude-longitude-points-haversine-formula     which is based on https   en wikipedia org wiki Haversine formula  error rate   0 55        public double getDistanceBetween double lat1  double lon1  double lat2  double lon2        double dLat   toRadians lat2 - lat1       double dLon   toRadians lon2 - lon1        double a   Math sin dLat   2    Math sin dLat   2                Math cos toRadians lat1     Math cos toRadians lat2                         Math sin dLon   2    Math sin dLon   2       double c   2   Math atan2 Math sqrt a   Math sqrt 1 - a        double d   EARTH RADIUS   c       return d     public double toRadians double degrees        return degrees    Math PI   180

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I don t like adding yet another answer  but the Google maps API v 3 has spherical geometry  and more   After converting your WGS84 to decimal degrees you can do this    lt script src  http   maps google com maps api js sensor false amp libraries geometry  type  text javascript  gt  lt  script gt     distance   google maps geometry spherical computeDistanceBetween      new google maps LatLng fromLat  fromLng        new google maps LatLng toLat  toLng      No word about how accurate Google s calculations are or even what model is used  though it does say  spherical  rather than  geoid    By the way  the  straight line  distance will obviously be different from the distance if one travels on the surface of the earth which is what everyone seems to be presuming

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All the above answers assumes the earth is a sphere  However  a more accurate approximation would be that of an oblate spheroid   a  6378 137 equitorial radius in km b  6356 752 polar radius in km  def Distance lat1  lons1  lat2  lons2       lat1 math radians lat1      lons1 math radians lons1      R1      a  2  math cos lat1    2     b  2  math sin lat1    2     a math cos lat1    2  b math sin lat1    2    0 5  radius of earth at lat1     x1 R math cos lat1  math cos lons1      y1 R math cos lat1  math sin lons1      z1 R math sin lat1       lat2 math radians lat2      lons2 math radians lons2      R1      a  2  math cos lat2    2     b  2  math sin lat2    2     a math cos lat2    2  b math sin lat2    2    0 5  radius of earth at lat2     x2 R math cos lat2  math cos lons2      y2 R math cos lat2  math sin lons2      z2 R math sin lat2       return   x1-x2   2  y1-y2   2  z1-z2   2   0 5

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The haversine is definitely a good formula for probably most cases  other answers already include it so I am not going to take the space  But it is important to note that no matter what formula is used  yes not just one   Because of the huge range of accuracy possible as well as the computation time required  The choice of formula requires a bit more thought than a simple no brainer answer    This posting from a person at nasa  is the best one I found at discussing the options  http   www cs nyu edu visual home proj tiger gisfaq html  For example  if you are just sorting rows by distance in a 100 miles radius  The flat earth formula will be much faster than the haversine   HalfPi   1 5707963  R   3956     the radius gives you the measurement unit    a   HalfPi - latoriginrad  b   HalfPi - latdestrad  u   a   a   b   b  v   - 2   a   b   cos longdestrad - longoriginrad   c   sqrt abs u   v    return R   c    Notice there is just one cosine and one square root  Vs 9 of them on the Haversine formula

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Here is my java implementation for calculation distance via decimal degrees after some search  I used mean radius of world  from wikipedia  in km  If you want result miles then use world radius in miles   public static double distanceLatLong2 double lat1  double lng1  double lat2  double lng2       double earthRadius   6371 0d     KM  use mile here if you want mile result    double dLat   toRadian lat2 - lat1     double dLng   toRadian lng2 - lng1      double a   Math pow Math sin dLat 2   2                Math cos toRadian lat1     Math cos toRadian lat2                Math pow Math sin dLng 2   2      double c   2   Math atan2 Math sqrt a   Math sqrt 1-a       return earthRadius   c     returns result kilometers    public static double toRadian double degrees       return  degrees   Math PI    180 0d

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You can calculate it by using Haversine formula which is   a   sin    f 2    cos f1    cos f2    sin      2  c   2    atan2  va  v 1-a    d   R    c   An example to calculate distance between two points is given below  Suppose i have to calculate distance between New Delhi to London  so how can i use this formula    New delhi co-ordinates  28 7041   N  77 1025   E London co-ordinates  51 5074   N  0 1278   W  var R   6371e3     metres var f1   28 7041 toRadians    var f2   51 5074 toRadians    var  f    51 5074-28 7041  toRadians    var       0 1278-77 1025  toRadians     var a   Math sin  f 2    Math sin  f 2            Math cos f1    Math cos f2            Math sin    2    Math sin    2   var c   2   Math atan2 Math sqrt a   Math sqrt 1-a     var d   R   c     metres d   d 1000     km

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You can use the build in CLLocationDistance to calculate this   CLLocation  location1     CLLocation alloc  initWithLatitude latitude1 longitude longitude1   CLLocation  location2     CLLocation alloc  initWithLatitude latitude2 longitude longitude2    self distanceInMetersFromLocation location1 toLocation location2   -  int distanceInMetersFromLocation  CLLocation  location1 toLocation  CLLocation  location2       CLLocationDistance distanceInMeters    location1 distanceFromLocation location2       return distanceInMeters      In your case if you want kilometers just divide by 1000

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Here is the implementation VB NET  this implementation will give you the result in KM or Miles based on an Enum value you pass   Public Enum DistanceType     Miles     KiloMeters End Enum  Public Structure Position     Public Latitude As Double     Public Longitude As Double End Structure  Public Class Haversine      Public Function Distance Pos1 As Position                               Pos2 As Position                               DistType As DistanceType  As Double          Dim R As Double   If  DistType   DistanceType Miles   3960  6371           Dim dLat As Double   Me toRadian Pos2 Latitude - Pos1 Latitude           Dim dLon As Double   Me toRadian Pos2 Longitude - Pos1 Longitude           Dim a As Double   Math Sin dLat   2    Math Sin dLat   2    Math Cos Me toRadian Pos1 Latitude     Math Cos Me toRadian Pos2 Latitude     Math Sin dLon   2    Math Sin dLon   2           Dim c As Double   2   Math Asin Math Min 1  Math Sqrt a             Dim result As Double   R   c          Return result      End Function      Private Function toRadian val As Double  As Double          Return  Math PI   180    val      End Function  End Class

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there is a good example in here to calculate distance with PHP http   www geodatasource com developers php     function distance  lat1   lon1   lat2   lon2   unit           theta    lon1 -  lon2        dist   sin deg2rad  lat1     sin deg2rad  lat2      cos deg2rad  lat1     cos deg2rad  lat2     cos deg2rad  theta          dist   acos  dist         dist   rad2deg  dist         miles    dist   60   1 1515        unit   strtoupper  unit         if   unit     K              return   miles   1 609344          else if   unit     N               return   miles   0 8684          else             return  miles

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Dart lang   import  dart math  show cos  sqrt  asin   double calculateDistance LatLng l1  LatLng l2      const p   0 017453292519943295    final a   0 5 -       cos  l2 latitude - l1 latitude    p    2         cos l1 latitude   p              cos l2 latitude   p               1 - cos  l2 longitude - l1 longitude    p               2    return 12742   asin sqrt a

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Here is a C  Implementation   static class DistanceAlgorithm       const double PIx   3 141592653589793      const double RADIUS   6378 16            lt summary gt          Convert degrees to Radians          lt  summary gt           lt param name  x  gt Degrees lt  param gt           lt returns gt The equivalent in radians lt  returns gt      public static double Radians double x                return x   PIx   180                  lt summary gt          Calculate the distance between two places           lt  summary gt           lt param name  lon1  gt  lt  param gt           lt param name  lat1  gt  lt  param gt           lt param name  lon2  gt  lt  param gt           lt param name  lat2  gt  lt  param gt           lt returns gt  lt  returns gt      public static double DistanceBetweenPlaces          double lon1          double lat1          double lon2          double lat2                double dlon   Radians lon2 - lon1           double dlat   Radians lat2 - lat1            double a    Math Sin dlat   2    Math Sin dlat   2     Math Cos Radians lat1     Math Cos Radians lat2      Math Sin dlon   2    Math Sin dlon   2            double angle   2   Math Atan2 Math Sqrt a   Math Sqrt 1 - a            return angle   RADIUS

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FSharp version  using miles    let radialDistanceHaversine location1 location2   float                    let degreeToRadian degrees   degrees   System Math PI   180 0                 let earthRadius   3959 0                 let deltaLat   location2 Latitude - location1 Latitude   gt  degreeToRadian                 let deltaLong   location2 Longitude - location1 Longitude   gt  degreeToRadian                 let a                        deltaLat   2 0   gt  sin     2 0                        location1 Latitude   gt  degreeToRadian   gt  cos                         location2 Latitude   gt  degreeToRadian   gt  cos                         deltaLong   2 0   gt  sin     2 0                 atan2  a   gt  sqrt   1 0 - a   gt  sqrt                    2 0                   earthRadius

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This is a simple PHP function that will give a very reasonable approximation  under   -1  error margin       lt  php function distance  lat1   lon1   lat2   lon2          pi80   M PI   180       lat1     pi80       lon1     pi80       lat2     pi80       lon2     pi80        r   6372 797     mean radius of Earth in km      dlat    lat2 -  lat1       dlon    lon2 -  lon1       a   sin  dlat   2    sin  dlat   2    cos  lat1    cos  lat2    sin  dlon   2    sin  dlon   2        c   2   atan2 sqrt  a   sqrt 1 -  a         km    r    c         echo   lt br  gt    km      return  km      gt    As said before  the earth is NOT a sphere   It is like an old  old baseball that Mark McGwire decided to practice with - it is full of dents and bumps   The simpler calculations  like this  treat it like a sphere   Different methods may be more or less precise according to where you are on this irregular ovoid AND how far apart your points are  the closer they are the smaller the absolute error margin    The more precise your expectation  the more complex the math   For more info  wikipedia geographic distance

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I condensed the computation down by simplifying the formula   Here it is in Ruby   include Math earth radius mi   3959 radians   lambda    deg  deg   PI   180   coord radians   lambda    c     lat   gt  radians c  lat     lng   gt  radians c  lng          from to      lat   gt   latitude in degrees    lng   gt   longitude in degrees    def haversine distance from  to    from  to   coord radians from   coord radians to    cosines product   cos to  lat     cos from  lat     cos from  lng  - to  lng     sines product   sin to  lat     sin from  lat     return earth radius mi   acos cosines product   sines product  end

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In the other answers an implementation in r is missing   Calculating the distance between two point is quite straightforward with the distm function from the geosphere package   distm p1  p2  fun   distHaversine    where   p1   longitude latitude for point s  p2   longitude latitude for point s    type of distance calculation fun   distCosine   distHaversine   distVincentySphere   distVincentyEllipsoid    As the earth is not perfectly spherical  the Vincenty formula for ellipsoids is probably the best way to calculate distances  Thus in the geosphere package you use then   distm p1  p2  fun   distVincentyEllipsoid      Off course you don t necessarily have to use geosphere package  you can also calculate the distance in base R with a function   hav dist  lt - function long1  lat1  long2  lat2      R  lt - 6371   diff long  lt -  long2 - long1    diff lat  lt -  lat2 - lat1    a  lt - sin diff lat 2  2   cos lat1    cos lat2    sin diff long 2  2   b  lt - 2   asin pmin 1  sqrt a       d   R   b   return d

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As pointed out  an accurate calculation should take into account that the earth is not a perfect sphere  Here are some comparisons of the various algorithms offered here   geoDistance 50 5 58 3  Haversine  899 km Maymenn  833 km Keerthana  897 km google maps geometry spherical computeDistanceBetween    900 km  geoDistance 50 5 -58 -3  Haversine  12030 km Maymenn  11135 km Keerthana  10310 km google maps geometry spherical computeDistanceBetween    12044 km  geoDistance  05  005  058  003  Haversine  0 9169 km Maymenn  0 851723 km Keerthana  0 917964 km google maps geometry spherical computeDistanceBetween    0 917964 km  geoDistance  05 80  058 80 3  Haversine  33 37 km Maymenn  33 34 km Keerthana  33 40767 km google maps geometry spherical computeDistanceBetween    33 40770 km   Over small distances  Keerthana s algorithm does seem to coincide with that of Google Maps  Google Maps does not seem to follow any simple algorithm  suggesting that it may be the most accurate method here   Anyway  here is a Javascript implementation of Keerthana s algorithm   function geoDistance lat1  lng1  lat2  lng2       const a   6378 137     equitorial radius in km     const b   6356 752     polar radius in km      var sq   x   gt   x x       var sqr   x   gt  Math sqrt x       var cos   x   gt  Math cos x       var sin   x   gt  Math sin x       var radius   lat   gt  sqr  sq a a cos lat   sq b b sin lat     sq a cos lat   sq b sin lat           lat1   lat1   Math PI   180      lng1   lng1   Math PI   180      lat2   lat2   Math PI   180      lng2   lng2   Math PI   180       var R1   radius lat1       var x1   R1 cos lat1  cos lng1       var y1   R1 cos lat1  sin lng1       var z1   R1 sin lat1        var R2   radius lat2       var x2   R2 cos lat2  cos lng2       var y2   R2 cos lat2  sin lng2       var z2   R2 sin lat2        return sqr sq x1-x2  sq y1-y2  sq z1-z2

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I needed to calculate a lot of distances between the points for my project  so I went ahead and tried to optimize the code  I have found here  On average in different browsers my new implementation runs 2 times faster than the most upvoted answer   function distance lat1  lon1  lat2  lon2      var p   0 017453292519943295        Math PI   180   var c   Math cos    var a   0 5 - c  lat2 - lat1    p  2              c lat1   p    c lat2   p                1 - c  lon2 - lon1    p   2     return 12742   Math asin Math sqrt a       2   R  R   6371 km     You can play with my jsPerf and see the results here   Recently I needed to do the same in python  so here is a python implementation   from math import cos  asin  sqrt  pi  def distance lat1  lon1  lat2  lon2       p   pi 180     a   0 5 - cos  lat2-lat1  p  2   cos lat1 p    cos lat2 p     1-cos  lon2-lon1  p   2     return 12742   asin sqrt a    2 R asin      And for the sake of completeness  Haversine on wiki

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Here is a typescript implementation of the Haversine formula  static getDistanceFromLatLonInKm lat1  number  lon1  number  lat2  number  lon2  number   number       var deg2Rad   deg   gt            return deg   Math PI   180             var r   6371     Radius of the earth in km     var dLat   deg2Rad lat2 - lat1          var dLon   deg2Rad lon2 - lon1       var a           Math sin dLat   2    Math sin dLat   2            Math cos deg2Rad lat1     Math cos deg2Rad lat2             Math sin dLon   2    Math sin dLon   2       var c   2   Math atan2 Math sqrt a   Math sqrt 1 - a        var d   r   c     Distance in km     return d

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here is an example in postgres sql  in km  for miles version  replace 1 609344 by 0 8684 version   CREATE OR REPLACE FUNCTION public geodistance alat float  alng float  blat    float  blng  float    RETURNS float AS  BODY  DECLARE     v distance float  BEGIN      v distance   asin  sqrt              sin radians blat-alat  2  2                                           sin radians blng-alng  2  2                        cos radians alat                         cos radians blat                                             cast  7926 3352  as float    cast  1 609344  as float          RETURN v distance  END   BODY  language plpgsql VOLATILE SECURITY DEFINER  alter function geodistance alat float  alng float  blat float  blng float  owner to postgres

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As this is the most popular discussion of the topic I ll add my experience from late 2019-early 2020 here  To add to the existing answers - my focus was to find an accurate AND fast  i e  vectorized  solution  Let s start with what is mostly used by answers here - the Haversine approach  It is trivial to vectorize  see example in python below  def haversine lat1  lon1  lat2  lon2        quot  quot  quot      Calculate the great circle distance between two points     on the earth  specified in decimal degrees       All args must be of equal length      Distances are in meters           Ref      https   stackoverflow com questions 29545704 fast-haversine-approximation-python-pandas     https   ipython readthedocs io en stable interactive magics html      quot  quot  quot      Radius   6 371e6     lon1  lat1  lon2  lat2   map np radians   lon1  lat1  lon2  lat2        dlon   lon2 - lon1     dlat   lat2 - lat1      a   np sin dlat 2 0   2   np cos lat1    np cos lat2    np sin dlon 2 0   2      c   2   np arcsin np sqrt a       s12   Radius   c            initial azimuth in degrees     y   np sin lon2-lon1    np cos lat2      x   np cos lat1  np sin lat2  - np sin lat1  np cos lat2  np cos dlon      azi1   np arctan2 y  x  180  math pi      return   s12  s12   azi1   azi1   Accuracy-wise  it is least accurate  Wikipedia states 0 5  of relative deviation on average without any sources  My experiments show less of a deviation  Below is the comparison ran on 100 000 random points vs my library  which should be accurate to millimeter levels  np random seed 42  lats1   np random uniform -90 90 100000  lons1   np random uniform -180 180 100000  lats2   np random uniform -90 90 100000  lons2   np random uniform -180 180 100000  r1   inverse lats1  lons1  lats2  lons2  r2   haversine lats1  lons1  lats2  lons2  print  quot Max absolute error    4 2f m quot  format np max r1  s12  -r2  s12      print  quot Mean absolute error    4 2f m quot  format np mean r1  s12  -r2  s12      print  quot Max relative error    4 2f   quot  format np max  r2  s12   r1  s12  -1  100    print  quot Mean relative error    4 2f   quot  format np mean  r2  s12   r1  s12  -1  100     Output  Max absolute error  26671 47m Mean absolute error  -2499 84m Max relative error  0 55  Mean relative error  -0 02   So on average 2 5km deviation on 100 000 random pairs of coordinates  which may be good for majority of cases  Next option is Vincenty s formulae which is accurate up to millimeters  depending on convergence criteria and can be vectorized as well  It does have the issue with convergence near antipodal points  You can make it converge at those points by relaxing convergence criteria  but accuracy drops to 0 25  and more  Outside of antipodal points Vincenty will provide results close to Geographiclib within relative error of less than 1 e-6 on average  Geographiclib  mentioned here  is really the current golden standard  It has several implementations and fairly fast  especially if you are using C   version  Now  if you are planning to use Python for anything above 10k points I d suggest to consider my vectorized implementation  I created a geovectorslib library with vectorized Vincenty routine for my own needs  which uses Geographiclib as fallback for near antipodal points  Below is the comparison vs Geographiclib for 100k points  As you can see it provides up to 20x improvement for inverse and 100x for direct methods for 100k points and the gap will grow with number of points  Accuracy-wise it will be within 1 e-5 rtol of Georgraphiclib  Direct method for 100 000 points 94 9 ms    25 ms per loop  mean    std  dev  of 7 runs  1 loop each  9 79 s    1 4 s per loop  mean    std  dev  of 7 runs  1 loop each   Inverse method for 100 000 points 1 5 s    504 ms per loop  mean    std  dev  of 7 runs  1 loop each  24 2 s    3 91 s per loop  mean    std  dev  of 7 runs  1 loop each

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Thanks very much for all this  I used the following code in my Objective-C iPhone app     const double PIx   3 141592653589793  const double RADIO   6371     Mean radius of Earth in Km  double convertToRadians double val        return val   PIx   180     - double kilometresBetweenPlace1  CLLocationCoordinate2D  place1 andPlace2  CLLocationCoordinate2D  place2            double dlon   convertToRadians place2 longitude - place1 longitude           double dlat   convertToRadians place2 latitude - place1 latitude            double a     pow sin dlat   2   2    cos convertToRadians place1 latitude      cos convertToRadians place2 latitude     pow sin dlon   2   2           double angle   2   asin sqrt a             return angle   RADIO      Latitude and Longitude are in decimal  I didn t use min   for the asin   call as the distances that I m using are so small that they don t require it   It gave incorrect answers until I passed in the values in Radians - now it s pretty much the same as the values obtained from Apple s Map app  -   Extra update   If you are using iOS4 or later then Apple provide some methods to do this so the same functionality would be achieved with   - double kilometresBetweenPlace1  CLLocationCoordinate2D  place1 andPlace2  CLLocationCoordinate2D  place2        MKMapPoint  start  finish        start   MKMapPointForCoordinate place1       finish   MKMapPointForCoordinate place2        return MKMetersBetweenMapPoints start  finish    1000

[algorithm] Calculate distance between two latitude-longitude points? (Haversine formula)

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