How to convert latitude or longitude to meters

Question

If I have a latitude or longitude reading in standard NMEA format is there an easy way   formula to convert that reading to meters  which I can then implement in Java  J9    Edit  Ok seems what I want to do is not possible easily  however what I really want to do is   Say I have a lat and long of a way point and a lat and long of a user is there an easy way to compare them to decide when to tell the user they are within a reasonably close distance of the way point   I realise reasonable is subject but is this easily do-able or still overly maths-y

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You need to convert the coordinates to radians to do the spherical geometry   Once converted  then you can calculate a distance between the two points   The distance then can be converted to any measure you want

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There are many tools that will make this easy   See monjardin s answer for more details about what s involved   However  doing this isn t necessarily difficult   It sounds like you re using Java  so I would recommend looking into something like GDAL   It provides java wrappers for their routines  and they have all the tools required to convert from Lat Lon  geographic coordinates  to UTM  projected coordinate system  or some other reasonable map projection   UTM is nice  because it s meters  so easy to work with   However  you will need to get the appropriate UTM zone for it to do a good job   There are some simple codes available via googling to find an appropriate zone for a lat long pair

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below is from  http   www zipcodeworld com samples distance vbnet html Public Function distance ByVal lat1 As Double  ByVal lon1 As Double                             ByVal lat2 As Double  ByVal lon2 As Double                             Optional ByVal unit As Char    M c  As Double     Dim theta As Double   lon1 - lon2     Dim dist As Double   Math Sin deg2rad lat1     Math Sin deg2rad lat2                                   Math Cos deg2rad lat1     Math Cos deg2rad lat2                                   Math Cos deg2rad theta       dist   Math Acos dist      dist   rad2deg dist      dist   dist   60   1 1515     If unit    K  Then         dist   dist   1 609344     ElseIf unit    N  Then         dist   dist   0 8684     End If     Return dist End Function Public Function Haversine ByVal lat1 As Double  ByVal lon1 As Double                             ByVal lat2 As Double  ByVal lon2 As Double                             Optional ByVal unit As Char    M c  As Double     Dim R As Double   6371  earth radius in km     Dim dLat As Double     Dim dLon As Double     Dim a As Double     Dim c As Double     Dim d As Double     dLat   deg2rad lat2 - lat1      dLon   deg2rad  lon2 - lon1       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      c   2   Math Atan2 Math Sqrt a   Math Sqrt 1 - a       d   R   c     Select Case unit ToString ToUpper         Case  M c             d   d   0 62137119         Case  N c             d   d   0 5399568     End Select     Return d End Function Private Function deg2rad ByVal deg As Double  As Double     Return  deg   Math PI   180 0  End Function Private Function rad2deg ByVal rad As Double  As Double     Return rad   Math PI   180 0 End Function

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Here is a javascript function   function measure lat1  lon1  lat2  lon2       generally used geo measurement function     var R   6378 137     Radius of earth in KM     var dLat   lat2   Math PI   180 - lat1   Math PI   180      var dLon   lon2   Math PI   180 - lon1   Math PI   180      var a   Math sin dLat 2    Math sin dLat 2        Math cos lat1   Math PI   180    Math cos lat2   Math PI   180        Math sin dLon 2    Math sin dLon 2       var c   2   Math atan2 Math sqrt a   Math sqrt 1-a        var d   R   c      return d   1000     meters     Explanation  https   en wikipedia org wiki Haversine formula     The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes

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To convert latitude and longitude in x and y representation you need to decide what type of map projection to use  As for me  Elliptical Mercator seems very well  Here you can find an implementation  in Java too

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Latitudes and longitudes specify points  not distances  so your question is somewhat nonsensical  If you re asking about the shortest distance between two  lat  lon  points  see this Wikipedia article on great-circle distances

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If you want a simple solution then use the Haversine formula as outlined by the other comments  If you have an accuracy sensitive application keep in mind the Haversine formula does not guarantee an accuracy better then 0 5  as it is assuming the earth is a sphere  To consider that Earth is a oblate spheroid consider using Vincenty s formulae  Additionally  I m not sure what radius we should use with the Haversine formula   Equator  6 378 137 km  Polar  6 356 752 km  Volumetric  6 371 0088 km

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Here is the R version of b-h- s function  just in case   measure  lt - function lon1 lat1 lon2 lat2        R  lt - 6378 137                                  radius of earth in Km     dLat  lt -  lat2-lat1  pi 180     dLon  lt -  lon2-lon1  pi 180     a  lt - sin  dLat 2   2   cos lat1 pi 180  cos lat2 pi 180   sin dLon 2   2     c  lt - 2   atan2 sqrt a   sqrt 1-a       d  lt - R   c     return  d   1000                               distance in meters

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There are quite a few ways to calculate this  All of them use aproximations of spherical trigonometry where the radius is the one of the earth   try http   www movable-type co uk scripts latlong html for a bit of methods and code in different languages

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Based on average distance for degress in the Earth   1     111km   Converting this for radians and dividing for meters  take s a magic number for the RAD  in meters  0 000008998719243599958   then   const RAD   0 000008998719243599958  Math sqrt Math pow lat1 - lat2  2    Math pow long1 - long2  2     RAD

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If its sufficiently close you can get away with treating them as coordinates on a flat plane  This works on say  street or city level if perfect accuracy isnt required and all you need is a rough guess on the distance involved to compare with an arbitrary limit

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One nautical mile  1852 meters  is defined as one arcminute of longitude at the equator  However  you need to define a map projection  see also UTM  in which you are working for the conversion to really make sense

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Given you re looking for a simple formula  this is probably the simplest way to do it  assuming that the Earth is a sphere with a circumference of 40075 km   Length in meters of 1   of latitude   always 111 32 km  Length in meters of 1   of longitude   40075 km   cos  latitude     360

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For approximating short distances between two coordinates I used formulas from   http   en wikipedia org wiki Lat-lon   m per deg lat   111132 954 - 559 822   cos  2   latMid     1 175   cos  4   latMid   m per deg lon   111132 954   cos   latMid         In the code below I ve left the raw numbers to show their relation to the formula from wikipedia   double latMid  m per deg lat  m per deg lon  deltaLat  deltaLon dist m   latMid    Lat1 Lat2   2 0      or just use Lat1 for slightly less accurate estimate   m per deg lat   111132 954 - 559 822   cos  2 0   latMid     1 175   cos  4 0   latMid   m per deg lon    3 14159265359 180     6367449   cos   latMid     deltaLat   fabs Lat1 - Lat2   deltaLon   fabs Lon1 - Lon2    dist m   sqrt    pow  deltaLat   m per deg lat 2    pow  deltaLon   m per deg lon   2       The wikipedia entry states that the distance calcs are within 0 6m for 100km longitudinally and 1cm for 100km latitudinally but I have not verified this as anywhere near that accuracy is fine for my use

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The earth is an annoyingly irregular surface  so there is no simple formula to do this exactly  You have to live with an approximate model of the earth  and project your coordinates onto it  The model I typically see used for this is WGS 84  This is what GPS devices usually use to solve the exact same problem   NOAA has some software you can download to help with this on their website

[math] How to convert latitude or longitude to meters?

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