Calculate the center point of multiple latitude longitude coordinate pairs

Question

Given a set of latitude and longitude points  how can I calculate the latitude and longitude of the center point of that set  aka a point that would center a view on all points    EDIT   Python solution I ve used   Convert lat lon  must be in radians  to Cartesian coordinates for each location  X   cos lat    cos lon  Y   cos lat    sin lon  Z   sin lat   Compute average x  y and z coordinates  x    x1   x2         xn    n y    y1   y2         yn    n z    z1   z2         zn    n  Convert average x  y  z coordinate to latitude and longitude  Lon   atan2 y  x  Hyp   sqrt x   x   y   y  Lat   atan2 z  hyp

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Javascript version of the original function         Get a center latitude longitude from an array of like geopoints        param array data 2 dimensional array of latitudes and longitudes    For Example      data   array           0    gt  array 45 849382  76 322333        1    gt  array 45 843543  75 324143        2    gt  array 45 765744  76 543223        3    gt  array 45 784234  74 542335           function GetCenterFromDegrees data               if    data length  gt  0            return false              var num coords   data length       var X   0 0      var Y   0 0      var Z   0 0       for i   0  i  lt  data length  i             var lat   data i  0    Math PI   180          var lon   data i  1    Math PI   180           var a   Math cos lat    Math cos lon           var b   Math cos lat    Math sin lon           var c   Math sin lat            X    a          Y    b          Z    c             X    num coords      Y    num coords      Z    num coords       var lon   Math atan2 Y  X       var hyp   Math sqrt X   X   Y   Y       var lat   Math atan2 Z  hyp        var newX    lat   180   Math PI       var newY    lon   180   Math PI        return new Array newX  newY

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This is is the same as a weighted average problem where all the weights are the same  and there are two dimensions    Find the average of all latitudes for your center latitude and the average of all longitudes for the center longitude   Caveat Emptor  This is a close distance approximation and the error will become unruly when the deviations from the mean are more than a few miles due to the curvature of the Earth  Remember that latitudes and longitudes are degrees  not really a grid

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I did this task in javascript like below  function GetCenterFromDegrees data          var data     lat 22 281610498720003 lng 70 77577162868579   lat 22 28065743343672 lng 70 77624369747241   lat 22 280860953131217 lng 70 77672113067706   lat 22 281863655593973 lng 70 7762061465462        var num coords   data length      var X   0 0      var Y   0 0      var Z   0 0       for i 0  i lt num coords  i             var lat   data i  lat   Math PI   180          var lon   data i  lng   Math PI   180          var a   Math cos lat    Math cos lon           var b   Math cos lat    Math sin lon           var c   Math sin lat            X    a          Y    b          Z    c             X    num coords      Y    num coords      Z    num coords       lon   Math atan2 Y  X       var hyp   Math sqrt X   X   Y   Y       lat   Math atan2 Z  hyp        var finalLat   lat   180   Math PI      var finalLng    lon   180   Math PI        var finalArray   Array        finalArray push finalLat       finalArray push finalLng       return finalArray

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Java Version if anyone needs it  Constants defined static to not calculate them twice                                                                                                                        Center of geometry defined by coordinates                                                                                                                  private static double pi   Math PI   180  private static double xpi   180   Math PI   public static Coordinate center Coordinate    arr        if  arr length    1            return arr 0             double x   0  y   0  z   0       for  Coordinate c   arr            double latitude   c lat     pi  longitude   c lon     pi          double cl   Math cos latitude    save it as we need it twice         x    cl   Math cos longitude           y    cl   Math sin longitude           z    Math sin latitude              int total   arr length       x   x   total      y   y   total      z   z   total       double centralLongitude   Math atan2 y  x       double centralSquareRoot   Math sqrt x   x   y   y       double centralLatitude   Math atan2 z  centralSquareRoot        return new Coordinate centralLatitude   xpi  centralLongitude   xpi

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Thanks  Here is a C  version of OP s solutions using degrees  It utilises the System Device Location GeoCoordinate class      public static GeoCoordinate GetCentralGeoCoordinate          IList lt GeoCoordinate gt  geoCoordinates                if  geoCoordinates Count    1                        return geoCoordinates Single                       double x   0          double y   0          double z   0           foreach  var geoCoordinate in geoCoordinates                        var latitude   geoCoordinate Latitude   Math PI   180              var longitude   geoCoordinate Longitude   Math PI   180               x    Math Cos latitude    Math Cos longitude               y    Math Cos latitude    Math Sin longitude               z    Math Sin latitude                      var total   geoCoordinates Count           x   x   total          y   y   total          z   z   total           var centralLongitude   Math Atan2 y  x           var centralSquareRoot   Math Sqrt x   x   y   y           var centralLatitude   Math Atan2 z  centralSquareRoot            return new GeoCoordinate centralLatitude   180   Math PI  centralLongitude   180   Math PI

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I used a formula that I got from www geomidpoint com and wrote the following C   implementation  The array and geocoords are my own classes whose functionality should be self-explanatory         midpoints calculated using formula from www geomidpoint com        geocoords geocoords  calcmidpoint  array lt geocoords gt  amp  points              if  points empty     return geocoords           float cart x   0              cart y   0              cart z   0         for  auto amp  point   points                    cart x    cos  point lat rad       cos  point lon rad               cart y    cos  point lat rad       sin  point lon rad               cart z    sin  point lat rad                     cart x    points numelems          cart y    points numelems          cart z    points numelems           geocoords mean         mean lat rad  atan2  cart z  sqrt  pow  cart x  2     pow  cart y  2             mean lon rad  atan2  cart y  cart x            return mean

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Dart Implementation for Flutter to find Center point for Multiple Latitude  Longitude  import math package import  dart math  as math   Latitude and Longitude List List lt LatLng gt  latLongList    LatLng 12 9824  80 0603  LatLng 13 0569 80 2425      LatLng getCenterLatLong List lt LatLng gt  latLongList        double pi   math pi   180      double xpi   180   math pi      double x   0  y   0  z   0       if latLongList length  1                return latLongList 0             for  int i   0  i  lt  latLongList length  i            double latitude   latLongList i  latitude   pi        double longitude   latLongList i  longitude   pi        double c1   math cos latitude         x   x   c1   math cos longitude         y   y   c1   math sin longitude         z   z   math sin latitude              int total   latLongList length      x   x   total      y   y   total      z   z   total       double centralLongitude   math atan2 y  x       double centralSquareRoot   math sqrt x   x   y   y       double centralLatitude   math atan2 z  centralSquareRoot        return LatLng centralLatitude xpi centralLongitude xpi

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The simple approach of just averaging them has weird edge cases with angles when they wrap from 359  back to 0    A much earlier question on SO asked about finding the average of a set of compass angles   An expansion of the approach recommended there for spherical coordinates would be    Convert each lat long pair into a unit-length 3D vector  Sum each of those vectors Normalise the resulting vector Convert back to spherical coordinates

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Dart Flutter Calculate the center point of multiple latitude longitude coordinate pairs  Map lt String  double gt  getLatLngCenter List lt List lt double gt  gt  coords        const LATIDX   0      const LNGIDX   1      double sumX   0      double sumY   0      double sumZ   0       for  var i   0  i  lt  coords length  i            var lat   VectorMath radians coords i  LATIDX          var lng   VectorMath radians coords i  LNGIDX             sum of cartesian coordinates       sumX    Math cos lat    Math cos lng         sumY    Math cos lat    Math sin lng         sumZ    Math sin lat              var avgX   sumX   coords length      var avgY   sumY   coords length      var avgZ   sumZ   coords length          convert average x  y  z coordinate to latitude and longtitude     var lng   Math atan2 avgY  avgX       var hyp   Math sqrt avgX   avgX   avgY   avgY       var lat   Math atan2 avgZ  hyp        return          latitude   VectorMath degrees lat          longitude   VectorMath degrees lng

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Here is the python Version for finding center point  The lat1 and lon1 are latitude and longitude lists  it will retuen the latitude and longitude of center point   def GetCenterFromDegrees lat1 lon1           if  len lat1   lt   0       return false   num coords   len lat1   X   0 0 Y   0 0 Z   0 0  for i in range  len lat1        lat   lat1 i    np pi   180     lon   lon1 i    np pi   180      a   np cos lat    np cos lon      b   np cos lat    np sin lon      c   np sin lat        X    a     Y    b     Z    c   X    num coords Y    num coords Z    num coords  lon   np arctan2 Y  X  hyp   np sqrt X   X   Y   Y  lat   np arctan2 Z  hyp   newX    lat   180   np pi  newY    lon   180   np pi  return newX  newY

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I found this post very useful so here is the solution in PHP  I ve been using this successfully and just wanted to save another dev some time          Get a center latitude longitude from an array of like geopoints        param array data 2 dimensional array of latitudes and longitudes    For Example      data   array           0    gt  array 45 849382  76 322333        1    gt  array 45 843543  75 324143        2    gt  array 45 765744  76 543223        3    gt  array 45 784234  74 542335           function GetCenterFromDegrees  data        if   is array  data   return FALSE        num coords   count  data         X   0 0       Y   0 0       Z   0 0       foreach   data as  coord                 lat    coord 0    pi     180           lon    coord 1    pi     180            a   cos  lat    cos  lon            b   cos  lat    sin  lon            c   sin  lat             X     a           Y     b           Z     c              X     num coords       Y     num coords       Z     num coords        lon   atan2  Y   X        hyp   sqrt  X    X    Y    Y        lat   atan2  Z   hyp        return array  lat   180   pi     lon   180   pi

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Very useful post  I ve implemented this in JavaScript  hereby my code  I ve used this successfully    function rad2degr rad    return rad   180   Math PI    function degr2rad degr    return degr   Math PI   180             param latLngInDeg array of arrays with latitude and longtitude      pairs in degrees  e g    latitude1  longtitude1    latitude2       longtitude2              return array with the center latitude longtitude pairs in       degrees      function getLatLngCenter latLngInDegr        var LATIDX   0      var LNGIDX   1      var sumX   0      var sumY   0      var sumZ   0       for  var i 0  i lt latLngInDegr length  i              var lat   degr2rad latLngInDegr i  LATIDX            var lng   degr2rad latLngInDegr i  LNGIDX               sum of cartesian coordinates         sumX    Math cos lat    Math cos lng           sumY    Math cos lat    Math sin lng           sumZ    Math sin lat              var avgX   sumX   latLngInDegr length      var avgY   sumY   latLngInDegr length      var avgZ   sumZ   latLngInDegr length          convert average x  y  z coordinate to latitude and longtitude     var lng   Math atan2 avgY  avgX       var hyp   Math sqrt avgX   avgX   avgY   avgY       var lat   Math atan2 avgZ  hyp        return   rad2degr lat   rad2degr lng

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In Django this is trivial  and actually works  I had issues with a number of the solutions not correctly returning negatives for latitude    For instance  let s say you are using django-geopostcodes  of which I am the author    from django contrib gis geos import MultiPoint from django contrib gis db models functions import Distance from django geopostcodes models import Locality  qs   Locality objects anything icontains  New York   points    locality point for locality in qs  multipoint   MultiPoint  points  point   multipoint centroid   point is a Django Point instance that can then be used to do things such as retrieve all objects that are within 10km of that centre point   Locality objects filter point  distance lte  point  D km 10          annotate distance Distance  point   point         order by  distance     Changing this to raw Python is trivial   from django contrib gis geos import Point  MultiPoint  points         Point  145 137075  -37 639981        Point  144 137075  -39 639981      multipoint   MultiPoint  points  point   multipoint centroid   Under the hood Django is using GEOS - more details at https   docs djangoproject com en 1 10 ref contrib gis geos

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very nice solutions  just what i needed for my swift project  so here s a swift port  thanks  amp  here s also a playground project   https   github com ppoh71 playgounds tree master centerLocationPoint playground       calculate the center point of multiple latitude longitude coordinate-pairs     import CoreLocation import GLKit  var LocationPoints    CLLocationCoordinate2D       add some points to Location ne  nw  sw  se   it s a rectangle basicaly LocationPoints append CLLocationCoordinate2D latitude  37 627512369999998  longitude  -122 38780611999999   LocationPoints append CLLocationCoordinate2D latitude  37 627512369999998  longitude   -122 43105867   LocationPoints append CLLocationCoordinate2D latitude  37 56502528  longitude  -122 43105867   LocationPoints append CLLocationCoordinate2D latitude  37 56502528  longitude  -122 38780611999999       center func func getCenterCoord LocationPoints   CLLocationCoordinate2D   - gt  CLLocationCoordinate2D       var x Float   0 0      var y Float   0 0      var z Float   0 0       for points in LocationPoints         let lat   GLKMathDegreesToRadians Float points latitude         let long   GLKMathDegreesToRadians Float points longitude             x    cos lat    cos long           y    cos lat    sin long           z    sin lat              x   x   Float LocationPoints count       y   y   Float LocationPoints count       z   z   Float LocationPoints count        let resultLong   atan2 y  x       let resultHyp   sqrt x   x   y   y       let resultLat   atan2 z  resultHyp          let result   CLLocationCoordinate2D latitude  CLLocationDegrees GLKMathRadiansToDegrees Float resultLat     longitude  CLLocationDegrees GLKMathRadiansToDegrees Float resultLong           return result        get the centerpoint var centerPoint   getCenterCoord LocationPoints  print  Latitude    centerPoint latitude    Longitude    centerPoint longitude

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Out of object in PHP  Given array of coordinate pairs  returns center          Calculate center of given coordinates     param  array     coordinates    Each array of coordinate pairs     return array                    Center of coordinates     function getCoordsCenter  coordinates             lats    lons   array        foreach   coordinates as  key   gt   value            array push  lats   value 0            array push  lons   value 1               minlat   min  lats        maxlat   max  lats        minlon   min  lons        maxlon   max  lons        lat    maxlat -    maxlat -  minlat    2        lng    maxlon -    maxlon -  minlon    2       return array  lat    gt   lat   lon    gt   lng       Taken idea from  4

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If you are interested in obtaining a very simplified  center  of the points  for example  to simply center a map to the center of your gmaps polygon   then here s a basic approach that worked for me   public function center          minlat   false       minlng   false       maxlat   false       maxlng   false       data array   json decode  this- gt data  true       foreach   data array as  data element             data coords   explode      data element           if  isset  data coords 1                  if   minlat     false     minlat    data coords 0     else    minlat     data coords 0   lt   minlat     data coords 0     minlat                if   maxlat     false     maxlat    data coords 0     else    maxlat     data coords 0   gt   maxlat     data coords 0     maxlat                if   minlng     false     minlng    data coords 1     else    minlng     data coords 1   lt   minlng     data coords 1     minlng                if   maxlng     false     maxlng    data coords 1     else    maxlng     data coords 1   gt   maxlng     data coords 1     maxlng                         lat    maxlat -    maxlat -  minlat    2        lng    maxlng -    maxlng -  minlng    2       return  lat      lng      This returns the middle lat lng coordinate for the center of a polygon

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In the interest of possibly saving someone a minute or two  here is the solution that was used in Objective-C instead of python  This version takes an NSArray of NSValues that contain MKMapCoordinates  which was called for in my implementation    import  lt MapKit MKGeometry h gt       CLLocationCoordinate2D centerCoordinateForCoordinates  NSArray   coordinateArray       double x   0      double y   0      double z   0       for NSValue  coordinateValue in coordinateArray            CLLocationCoordinate2D coordinate    coordinateValue MKCoordinateValue            double lat   GLKMathDegreesToRadians coordinate latitude           double lon   GLKMathDegreesToRadians coordinate longitude           x    cos lat    cos lon           y    cos lat    sin lon           z    sin lat              x   x    double coordinateArray count      y   y    double coordinateArray count      z   z    double coordinateArray count       double resultLon   atan2 y  x       double resultHyp   sqrt x   x   y   y       double resultLat   atan2 z  resultHyp        CLLocationCoordinate2D result   CLLocationCoordinate2DMake GLKMathRadiansToDegrees resultLat   GLKMathRadiansToDegrees resultLon        return result

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If you wish to take into account the ellipsoid being used you can find the formulae here http   www ordnancesurvey co uk oswebsite gps docs A Guide to Coordinate Systems in Great Britain pdf  see Annexe B  The document contains lots of other useful stuff  B

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Here is the Android version based on  Yodacheese s C  answer using Google Maps api   public static LatLng GetCentralGeoCoordinate List lt LatLng gt  geoCoordinates                if  geoCoordinates size      1                return geoCoordinates get 0              double x   0      double y   0      double z   0       for LatLng geoCoordinate   geoCoordinates                double  latitude   geoCoordinate latitude   Math PI   180          double longitude   geoCoordinate longitude   Math PI   180           x    Math cos latitude    Math cos longitude           y    Math cos latitude    Math sin longitude           z    Math sin latitude              int total   geoCoordinates size         x   x   total      y   y   total      z   z   total       double centralLongitude   Math atan2 y  x       double centralSquareRoot   Math sqrt x   x   y   y       double centralLatitude   Math atan2 z  centralSquareRoot        return new LatLng centralLatitude   180   Math PI  centralLongitude   180   Math PI        in app build gradle add   implementation  com google android gms play-services-maps 17 0 0

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As an appreciation for this thread  here is my little contribution with the implementation in Ruby  hoping that I will save someone a few minutes from their precious time   def self find center locations    number of locations   locations length   return locations first if number of locations    1   x   y   z   0 0  locations each do  station     latitude   station latitude   Math  PI   180    longitude   station longitude   Math  PI   180     x    Math cos latitude    Math cos longitude     y    Math cos latitude    Math sin longitude     z    Math sin latitude   end   x   x number of locations  y   y number of locations  z   z number of locations   central longitude    Math atan2 y  x   central square root   Math sqrt x   x   y   y   central latitude   Math atan2 z  central square root     latitude  central latitude   180   Math  PI    longitude  central longitude   180   Math  PI  end

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If you want all points to be visible in the image  you d want the extrema in latitude and longitude and make sure that your view includes those values with whatever border you want    From Alnitak s answer  how you calculate the extrema may be a little problematic  but if they re a few degrees on either side of the longitude that wraps around  then you ll call the shot and take the right range    If you don t want to distort whatever map that these points are on  then adjust the bounding box s aspect ratio so that it fits whatever pixels you ve allocated to the view but still includes the extrema   To keep the points centered at some arbitrary zooming level  calculate the center of the bounding box that  just fits  the points as above  and keep that point as the center point

[math] Calculate the center point of multiple latitude/longitude coordinate pairs

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