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ayisyen </a></li> <li><a href="/w/index.php?title=Gcf:Mercator&action=edit&redlink=1" class="new" title="Gcf:Mercator (page does not exist)"> kréyòl gwadloupéyen </a></li> <li><a href="/w/index.php?title=Ku:Mercator&action=edit&redlink=1" class="new" title="Ku:Mercator (page does not exist)"> kurdî </a></li> <li><a href="/w/index.php?title=Lv:Mercator&action=edit&redlink=1" class="new" title="Lv:Mercator (page does not exist)"> latviešu </a></li> <li><a href="/w/index.php?title=Lb:Mercator&action=edit&redlink=1" class="new" title="Lb:Mercator (page does not exist)"> Lëtzebuergesch </a></li> <li><a href="/w/index.php?title=Lt:Mercator&action=edit&redlink=1" class="new" title="Lt:Mercator (page does not exist)"> lietuvių </a></li> <li><a href="/w/index.php?title=Hu:Mercator&action=edit&redlink=1" class="new" title="Hu:Mercator (page does not exist)"> magyar </a></li> <li><a href="/w/index.php?title=NL:Mercator&action=edit&redlink=1" class="new" title="NL:Mercator (page does not exist)"> Nederlands </a></li> <li><a href="/w/index.php?title=No:Mercator&action=edit&redlink=1" class="new" title="No:Mercator (page does not exist)"> norsk </a></li> <li><a href="/w/index.php?title=Nn:Mercator&action=edit&redlink=1" class="new" title="Nn:Mercator (page does not exist)"> norsk nynorsk </a></li> <li><a href="/w/index.php?title=Oc:Mercator&action=edit&redlink=1" class="new" title="Oc:Mercator (page does not exist)"> occitan </a></li> <li><a href="/w/index.php?title=Pl:Mercator&action=edit&redlink=1" class="new" title="Pl:Mercator (page does not exist)"> polski </a></li> <li><a href="/w/index.php?title=Pt:Mercator&action=edit&redlink=1" class="new" title="Pt:Mercator (page does not exist)"> português </a></li> <li><a href="/w/index.php?title=Ro:Mercator&action=edit&redlink=1" class="new" title="Ro:Mercator (page does not exist)"> română </a></li> <li><a href="/w/index.php?title=Sc:Mercator&action=edit&redlink=1" class="new" title="Sc:Mercator (page does not exist)"> sardu </a></li> <li><a href="/w/index.php?title=Sq:Mercator&action=edit&redlink=1" class="new" title="Sq:Mercator (page does not exist)"> shqip </a></li> <li><a href="/w/index.php?title=Sk:Mercator&action=edit&redlink=1" class="new" title="Sk:Mercator (page does not exist)"> slovenčina </a></li> <li><a href="/w/index.php?title=Sl:Mercator&action=edit&redlink=1" class="new" title="Sl:Mercator (page does not exist)"> slovenščina </a></li> <li><a href="/w/index.php?title=Sr-latn:Mercator&action=edit&redlink=1" class="new" title="Sr-latn:Mercator (page does not exist)"> srpski (latinica) </a></li> <li><a href="/w/index.php?title=Fi:Mercator&action=edit&redlink=1" class="new" title="Fi:Mercator (page does not exist)"> suomi </a></li> <li><a href="/w/index.php?title=Sv:Mercator&action=edit&redlink=1" class="new" title="Sv:Mercator (page does not exist)"> svenska </a></li> <li><a href="/w/index.php?title=Tl:Mercator&action=edit&redlink=1" class="new" title="Tl:Mercator (page does not exist)"> Tagalog </a></li> <li><a href="/w/index.php?title=Vi:Mercator&action=edit&redlink=1" class="new" title="Vi:Mercator (page does not exist)"> Tiếng Việt </a></li> <li><a href="/w/index.php?title=Tr:Mercator&action=edit&redlink=1" class="new" title="Tr:Mercator (page does not exist)"> Türkçe </a></li> <li><a href="/w/index.php?title=Diq:Mercator&action=edit&redlink=1" class="new" title="Diq:Mercator (page does not exist)"> Zazaki </a></li> <li><a href="/w/index.php?title=El:Mercator&action=edit&redlink=1" class="new" title="El:Mercator (page does not exist)"> Ελληνικά </a></li> <li><a href="/w/index.php?title=Be:Mercator&action=edit&redlink=1" class="new" title="Be:Mercator (page does not exist)"> беларуская </a></li> <li><a href="/w/index.php?title=Bg:Mercator&action=edit&redlink=1" class="new" title="Bg:Mercator (page does not exist)"> български </a></li> <li><a href="/w/index.php?title=Mk:Mercator&action=edit&redlink=1" class="new" title="Mk:Mercator (page does not exist)"> македонски </a></li> <li><a href="/w/index.php?title=Mn:Mercator&action=edit&redlink=1" class="new" title="Mn:Mercator (page does not exist)"> монгол </a></li> <li><a href="/w/index.php?title=RU:Mercator&action=edit&redlink=1" class="new" title="RU:Mercator (page does not exist)"> русский </a></li> <li><a href="/w/index.php?title=Sr:Mercator&action=edit&redlink=1" class="new" title="Sr:Mercator (page does not exist)"> српски / srpski </a></li> <li><a href="/w/index.php?title=Uk:Mercator&action=edit&redlink=1" class="new" title="Uk:Mercator (page does not exist)"> українська </a></li> <li><a href="/w/index.php?title=Hy:Mercator&action=edit&redlink=1" class="new" title="Hy:Mercator (page does not exist)"> հայերեն </a></li> <li><a href="/w/index.php?title=He:Mercator&action=edit&redlink=1" class="new" title="He:Mercator (page does not exist)"> עברית </a></li> <li><a href="/w/index.php?title=Ar:Mercator&action=edit&redlink=1" class="new" title="Ar:Mercator (page does not exist)"> العربية </a></li> <li><a href="/w/index.php?title=Skr:Mercator&action=edit&redlink=1" class="new" title="Skr:Mercator (page does not exist)"> سرائیکی </a></li> <li><a href="/w/index.php?title=Fa:Mercator&action=edit&redlink=1" class="new" title="Fa:Mercator (page does not exist)"> فارسی </a></li> <li><a href="/w/index.php?title=Pnb:Mercator&action=edit&redlink=1" class="new" title="Pnb:Mercator (page does not exist)"> پنجابی </a></li> <li><a href="/w/index.php?title=Ps:Mercator&action=edit&redlink=1" class="new" title="Ps:Mercator (page does not exist)"> پښتو </a></li> <li><a href="/w/index.php?title=Ne:Mercator&action=edit&redlink=1" class="new" title="Ne:Mercator (page does not exist)"> नेपाली </a></li> <li><a href="/w/index.php?title=Mr:Mercator&action=edit&redlink=1" class="new" title="Mr:Mercator (page does not exist)"> मराठी </a></li> <li><a href="/w/index.php?title=Hi:Mercator&action=edit&redlink=1" class="new" title="Hi:Mercator (page does not exist)"> हिन्दी </a></li> <li><a href="/w/index.php?title=Bn:Mercator&action=edit&redlink=1" class="new" title="Bn:Mercator (page does not exist)"> বাংলা </a></li> <li><a href="/w/index.php?title=Pa:Mercator&action=edit&redlink=1" class="new" title="Pa:Mercator (page does not exist)"> ਪੰਜਾਬੀ </a></li> <li><a href="/w/index.php?title=Ta:Mercator&action=edit&redlink=1" class="new" title="Ta:Mercator (page does not exist)"> தமிழ் </a></li> <li><a href="/w/index.php?title=Ml:Mercator&action=edit&redlink=1" class="new" title="Ml:Mercator (page does not exist)"> മലയാളം </a></li> <li><a href="/w/index.php?title=Si:Mercator&action=edit&redlink=1" class="new" title="Si:Mercator (page does not exist)"> සිංහල </a></li> <li><a href="/w/index.php?title=Th:Mercator&action=edit&redlink=1" class="new" title="Th:Mercator (page does not exist)"> ไทย </a></li> <li><a href="/w/index.php?title=My:Mercator&action=edit&redlink=1" class="new" title="My:Mercator (page does not exist)"> မြန်မာဘာသာ </a></li> <li><a href="/w/index.php?title=Ko:Mercator&action=edit&redlink=1" class="new" title="Ko:Mercator (page does not exist)"> 한국어 </a></li> <li><a href="/w/index.php?title=Ka:Mercator&action=edit&redlink=1" class="new" title="Ka:Mercator (page does not exist)"> ქართული </a></li> <li><a href="/w/index.php?title=Tzm:Mercator&action=edit&redlink=1" class="new" title="Tzm:Mercator (page does not exist)"> ⵜⴰⵎⴰⵣⵉⵖⵜ </a></li> <li><a href="/w/index.php?title=Zh-hans:Mercator&action=edit&redlink=1" class="new" title="Zh-hans:Mercator (page does not exist)"> 中文（简体） </a></li> <li><a href="/w/index.php?title=Zh-hant:Mercator&action=edit&redlink=1" class="new" title="Zh-hant:Mercator (page does not exist)"> 中文（繁體） </a></li> <li><a href="/w/index.php?title=JA:Mercator&action=edit&redlink=1" class="new" title="JA:Mercator (page does not exist)"> 日本語 </a></li> <li><a href="/w/index.php?title=Yue:Mercator&action=edit&redlink=1" class="new" title="Yue:Mercator (page does not exist)"> 粵語 </a></li></div> </div> <div class="nomobile" style="background:#EEE;padding:1px"> <div id="ExpandLanguages" class="hlist" style="font-weight:bold;max-height:10000000px;text-align:center"> <ul><li><a href="#top">Other languages...</a></li></ul> </div> </div> </div> <dl><dd>For the OSM editor program, see <a href="/wiki/Merkaartor" title="Merkaartor">Merkaartor</a>.</dd></dl> This article describes algorithms for <a href="https://en.wikipedia.org/wiki/Mercator_projection" class="extiw" title="wikipedia:Mercator projection">Mercator projection</a>. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Spherical_Pseudo-Mercator_projection">1 Spherical Pseudo-Mercator projection</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#JavaScript_(or_ActionScript)">1.1 JavaScript (or ActionScript)</a></li> <li class="toclevel-2 tocsection-3"><a href="#C">1.2 C</a></li> <li class="toclevel-2 tocsection-4"><a href="#C#">1.3 C#</a></li> <li class="toclevel-2 tocsection-5"><a href="#D">1.4 D</a></li> <li class="toclevel-2 tocsection-6"><a href="#Excel">1.5 Excel</a></li> <li class="toclevel-2 tocsection-7"><a href="#Java">1.6 Java</a></li> <li class="toclevel-2 tocsection-8"><a href="#LibreOffice_Calc_(3.5)">1.7 LibreOffice Calc (3.5)</a></li> <li class="toclevel-2 tocsection-9"><a href="#Perl">1.8 Perl</a></li> <li class="toclevel-2 tocsection-10"><a href="#PHP">1.9 PHP</a></li> <li class="toclevel-2 tocsection-11"><a href="#PostGIS_/_SQL">1.10 PostGIS / SQL</a></li> <li class="toclevel-2 tocsection-12"><a href="#Python">1.11 Python</a></li> <li class="toclevel-2 tocsection-13"><a href="#Rust">1.12 Rust</a></li> <li class="toclevel-2 tocsection-14"><a href="#Swift">1.13 Swift</a></li> <li class="toclevel-2 tocsection-15"><a href="#Go_(Golang)">1.14 Go (Golang)</a></li> </ul> </li> <li class="toclevel-1 tocsection-16"><a href="#Elliptical_(true)_Mercator_Projection">2 Elliptical (true) Mercator Projection</a> <ul> <li class="toclevel-2 tocsection-17"><a href="#JavaScript_(or_ActionScript)_implementation">2.1 JavaScript (or ActionScript) implementation</a></li> <li class="toclevel-2 tocsection-18"><a href="#C_implementation">2.2 C implementation</a></li> <li class="toclevel-2 tocsection-19"><a href="#C#_implementation">2.3 C# implementation</a></li> <li class="toclevel-2 tocsection-20"><a href="#Java_implementation">2.4 Java implementation</a></li> <li class="toclevel-2 tocsection-21"><a href="#PHP_implementation">2.5 PHP implementation</a></li> <li class="toclevel-2 tocsection-22"><a href="#Python_implementation">2.6 Python implementation</a></li> <li class="toclevel-2 tocsection-23"><a href="#sdlBasic_implementation">2.7 sdlBasic implementation</a></li> <li class="toclevel-2 tocsection-24"><a href="#bash-script_(bc)_implementation">2.8 bash-script (bc) implementation</a></li> <li class="toclevel-2 tocsection-25"><a href="#VBA_for_Office_implementation">2.9 VBA for Office implementation</a></li> </ul> </li> <li class="toclevel-1 tocsection-26"><a href="#Links">3 Links</a></li> </ul> </div> <h2>Spherical Pseudo-Mercator projection</h2> Most of OSM, including the main tiling system, uses a Pseudo-Mercator projection where the Earth is modelized as if it was a perfect a sphere. Combined with the zoom level, the system is known as a <a rel="nofollow" class="external text" href="https://en.wikipedia.org/wiki/Web_Mercator"><bdi>Web Mercator</bdi></a> on Wikipedia. This produces a fast approximation to the truer, but heavier elliptical projection, where the Earth would be projected on a more accurate ellipsoid (flattened on poles). As a consequence, direct mesurements of distances in this projection will be approximative, except on the Equator, and the aspect ratios on the rendered map for true squares measured on the surface on Earth will slightly change with latitude and angles not so precisely preserved by this spherical projection. <h3>JavaScript (or ActionScript)</h3> The following functions take or return their results in degrees:<div class="mw-highlight mw-highlight-lang-javascript mw-content-ltr" dir="ltr"><pre>RAD2DEG = 180 / Math.PI; PI_4 = Math.PI / 4; /* The following functions take or return their results in degrees */ function y2lat(y) { return (Math.atan(Math.exp(y / RAD2DEG)) / PI_4 - 1) * 90; } function x2lon(x) { return x; } function lat2y(lat) { return Math.log(Math.tan((lat / 90 + 1) * PI_4 )) * RAD2DEG; } function lon2x(lon) { return lon; } </pre></div>The following take or return their results in Meters and Lat/Lon (similar to Python implementation below):<div class="mw-highlight mw-highlight-lang-javascript mw-content-ltr" dir="ltr"><pre>PI = Math.PI; RAD2DEG = 180/PI; DEG2RAD = PI/180; R = 6378137.0 function y2lat(y) { return (2 * Math.atan( Math.exp(y / R)) - PI/2 ) * RAD2DEG } function x2lon(x) { return RAD2DEG * (x/R); } function lat2y(lat) { return Math.log(Math.tan(PI/4 + lat * DEG2RAD / 2)) * R} function lon2x(lon) { return lon * DEG2RAD * R; } </pre></div> <h3>C</h3> <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>#include <math.h> #define DEG2RAD(a) ((a) / (180 / M_PI)) #define RAD2DEG(a) ((a) * (180 / M_PI)) #define EARTH_RADIUS 6378137 /* The following functions take their parameter and return their result in degrees */ double y2lat_d(double y) { return RAD2DEG( atan(exp( DEG2RAD(y) )) * 2 - M_PI/2 ); } double x2lon_d(double x) { return x; } double lat2y_d(double lat) { return RAD2DEG( log(tan( DEG2RAD(lat) / 2 + M_PI/4 )) ); } double lon2x_d(double lon) { return lon; } /* The following functions take their parameter in something close to meters, along the equator, and return their result in degrees */ double y2lat_m(double y) { return RAD2DEG(2 * atan(exp( y/EARTH_RADIUS)) - M_PI/2); } double x2lon_m(double x) { return RAD2DEG( x/EARTH_RADIUS ); } /* The following functions take their parameter in degrees, and return their result in something close to meters, along the equator */ double lat2y_m(double lat) { return log(tan( DEG2RAD(lat) / 2 + M_PI/4 )) * EARTH_RADIUS; } double lon2x_m(double lon) { return DEG2RAD(lon) * EARTH_RADIUS; } </pre></div> <h3>C#</h3> <div class="mw-highlight mw-highlight-lang-csharp mw-content-ltr" dir="ltr"><pre> public double YToLatitude(double y) { return System.Math.Atan(System.Math.Exp( y / 180 * System.Math.PI )) / System.Math.PI * 360 - 90; } public double LatitudeToY (double latitude) { return System.Math.Log(System.Math.Tan( (latitude + 90) / 360 * System.Math.PI )) / System.Math.PI * 180; } </pre></div> <h3>D</h3> <div class="mw-highlight mw-highlight-lang-d mw-content-ltr" dir="ltr"><pre>double lon2x(double lon) { import std.math; return PI*(lon/180.0) * 6378137.0; } double lat2y(double lat) { import std.math; return log(tan(PI_4+PI*(lat/360.0)))*6378137.0; } double x2lon(double x) { import std.math; return 180*(x/6378137)/PI; } double y2lat(double y) { import std.math; return 180*(2*atan(exp(y/6378137))-PI_2)/PI; } </pre></div> <h3>Excel</h3> I have simply converted the above script formulas into Excel formulas (values in A1 or B2): --<a href="/wiki/User:Krza" title="User:Krza">Krza</a> 23:36, 19 July 2008 (UTC) <table class="wikitable" style="text-align:center"> <tbody><tr> <td>  </td> <th scope="col">1</th> <th scope="col">2 </th></tr> <tr> <th scope="row">A </th> <td>y</td> <td>lat: <tt>= ARCTAN(EXP(A1/180*PI())/PI()*360-90</tt> </td></tr> <tr> <th scope="row">B </th> <td>y: <tt>= LOG(TAN((B2+90)/360*PI()))/PI()*180</tt></td> <td>lat </td></tr></tbody></table> Note: this may work for older versions of Excel, but in Excel 2010, ARCTAN() is called ATAN(), and you need to use LN() instead of LOG() to get the natural logarithm Excel formulas for converting Lat/Lon to Meters Lat to Y (expecting Lat in A1):<div class="mw-highlight mw-content-ltr" dir="ltr"><pre>=LN(TAN((PI()/4)+(RADIANS(A1)/2)))*6378137</pre></div>Lon to X (expecting Lon in B1):<div class="mw-highlight mw-content-ltr" dir="ltr"><pre>=RADIANS(B1)*6378137</pre></div> <h3>Java</h3> <div class="mw-highlight mw-highlight-lang-java mw-content-ltr" dir="ltr"><pre>import java.lang.Math; public class SphericalMercator { public static final double RADIUS = 6378137.0; /* in meters on the equator */ /* These functions take their length parameter in meters and return an angle in degrees */ public static double y2lat(double aY) { return Math.toDegrees(Math.atan(Math.exp(aY / RADIUS)) * 2 - Math.PI/2); } public static double x2lon(double aX) { return Math.toDegrees(aX / RADIUS); } /* These functions take their angle parameter in degrees and return a length in meters */ public static double lat2y(double aLat) { return Math.log(Math.tan(Math.PI / 4 + Math.toRadians(aLat) / 2)) * RADIUS; } public static double lon2x(double aLong) { return Math.toRadians(aLong) * RADIUS; } } </pre></div> <h3>LibreOffice Calc (3.5)</h3> lon value in A1, lat value in A2 <div class="mw-highlight mw-highlight-lang-text mw-content-ltr" dir="ltr"><pre> lon2x: = A1 * (PI()/180) * 6378137 x2lon: = A1 / (PI()/180) / 6378137 lat2y: = LN(TAN(A2 * (PI()/180)/2 + PI()/4)) * 6378137 y2lat: = (2*ATAN(EXP(A2/6378137)) - PI()/2) / (PI()/180 </pre></div> <h3>Perl</h3> <div class="mw-highlight mw-highlight-lang-perl mw-content-ltr" dir="ltr"><pre>use Geo::Mercator; my ($x, $y) = mercate($lat, $long) </pre></div> <h3>PHP</h3> <div class="mw-highlight mw-highlight-lang-php mw-content-ltr" dir="ltr"><pre>function lon2x($lon) { return deg2rad($lon) * 6378137.0; } function lat2y($lat) { return log(tan(M_PI_4 + deg2rad($lat) / 2.0)) * 6378137.0; } function x2lon($x) { return rad2deg($x / 6378137.0); } function y2lat($y) { return rad2deg(2.0 * atan(exp($y / 6378137.0)) - M_PI_2); } </pre></div> <h3>PostGIS / SQL</h3> The SRID 900913 is commonly used for our projection but is not formally assigned. It is not included in the default projections supplied with PostGIS. It can be added into your database via: [<a rel="nofollow" class="external text" href="http://trac.openstreetmap.org/browser/applications/utils/export/osm2pgsql/900913.sql">900913.sql</a>] <div class="mw-highlight mw-highlight-lang-sql mw-content-ltr" dir="ltr"><pre> INSERT INTO spatial_ref_sys (srid, auth_name, auth_srid, srtext, proj4text) VALUES (900913,'EPSG',900913,'PROJCS["WGS84 / Simple Mercator",GEOGCS["WGS 84", DATUM["WGS_1984",SPHEROID["WGS_1984", 6378137.0, 298.257223563]],PRIMEM["Greenwich", 0.0], UNIT["degree", 0.017453292519943295],AXIS["Longitude", EAST],AXIS["Latitude", NORTH]], PROJECTION["Mercator_1SP_Google"],PARAMETER["latitude_of_origin", 0.0], PARAMETER["central_meridian", 0.0],PARAMETER["scale_factor", 1.0],PARAMETER["false_easting", 0.0], PARAMETER["false_northing", 0.0],UNIT["m", 1.0],AXIS["x", EAST], AXIS["y", NORTH],AUTHORITY["EPSG","900913"]]', '+proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +no_defs'); </pre></div> <h3>Python</h3> <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>import math R = 6378137.0 def y2lat(y): return math.degrees(2 * math.atan(math.exp (y / R)) - math.pi / 2.0) def lat2y(lat): return math.log(math.tan(math.pi / 4 + math.radians(lat) / 2)) * R def x2lng(x): return math.degrees(x / R) def lon2x(lon): return math.radians(lon) * R </pre></div> <h3>Rust</h3> <div class="mw-highlight mw-highlight-lang-rust mw-content-ltr" dir="ltr"><pre>// length of semi-major axis of the WGS84 ellipsoid, i.e. radius at equator const EARTH_RADIUS_KM: f64 = 6_378.137; pub fn lon2x(lon: f64) -> f64 { EARTH_RADIUS_KM * 1000. * lon.to_radians() } pub fn x2lon(x: f64) -> f64 { (x / (EARTH_RADIUS_KM * 1000.)).to_degrees() } pub fn lat2y(lat: f64) -> f64 { ((lat.to_radians() / 2. + std::f64::consts::PI / 4.).tan()).log(std::f64::consts::E) * EARTH_RADIUS_KM * 1000. } pub fn y2lat(y: f64) -> f64 { (2. * ((y / (EARTH_RADIUS_KM * 1000.)).exp()).atan() - std::f64::consts::PI / 2.) .to_degrees() } </pre></div> <h3>Swift</h3> <div class="mw-highlight mw-highlight-lang-swift mw-content-ltr" dir="ltr"><pre>class SphericalMercator { private let radius: Double = 6378137.0; /* in meters on the equator */ /* These functions take their length parameter in meters and return an angle in degrees */ func y2lat(aY: Double) -> Double { (atan(exp(aY / radius)) * 2 - Double.pi/2).radiansToDegrees } func x2lon(aX: Double) -> Double { (aX / radius).radiansToDegrees; } /* These functions take their angle parameter in degrees and return a length in meters */ func lat2y(aLat: Double) -> Double { log(tan(Double.pi / 4 + (aLat.degreesToRadians / 2))) * radius } func lon2x(aLong: Double) -> Double { (aLong).degreesToRadians * radius; } } extension FloatingPoint { var degreesToRadians: Self { self * .pi / 180 } var radiansToDegrees: Self { self * 180 / .pi } } </pre></div> <h3>Go (Golang)</h3> <div class="mw-highlight mw-highlight-lang-go mw-content-ltr" dir="ltr"><pre>package main import "math" const R = 6378137.0 func Radians(deg float64) float64 { return deg * math.Pi / 180 } func Degrees(rad float64) float64 { return rad * 180 / math.Pi } func Y2lat(y float64) float64 { return Degrees(2*math.Atan(math.Exp(y/R)) - math.Pi/2) } func Lat2y(lat float64) float64 { return R * math.Log(math.Tan(math.Pi/4+Radians(lat)/2)) } func X2lat(x float64) float64 { return Degrees(x / R) } func Lon2x(lon float64) float64 { return R * Radians(lon) } </pre></div> FIXME: add spherical code in other languages here <h2>Elliptical (true) Mercator Projection</h2> This projection gives more accurate aspect ratios for objects anywhere on Earth, and respects their angles with higher precision. However, this projection is not commonly used on most maps used on the OSM websites and in editors. <h3>JavaScript (or ActionScript) implementation</h3> From a posting by <a href="/wiki/User:Crschmidt" title="User:Crschmidt">Christopher Schmidt</a> to the dev list on 2nd December 2006: So, for everyone's elucidation, here is the way to convert from latitude and longitude to a simple (elliptical) Mercator projection, in Javascript: <div class="mw-highlight mw-highlight-lang-javascript mw-content-ltr" dir="ltr"><pre>function deg_rad(ang) { return ang * (Math.PI/180.0) } function merc_x(lon) { var r_major = 6378137.000; return r_major * deg_rad(lon); } function merc_y(lat) { if (lat > 89.5) lat = 89.5; if (lat < -89.5) lat = -89.5; var r_major = 6378137.000; var r_minor = 6356752.3142; var temp = r_minor / r_major; var es = 1.0 - (temp * temp); var eccent = Math.sqrt(es); var phi = deg_rad(lat); var sinphi = Math.sin(phi); var con = eccent * sinphi; var com = .5 * eccent; con = Math.pow((1.0-con)/(1.0+con), com); var ts = Math.tan(.5 * (Math.PI*0.5 - phi))/con; var y = 0 - r_major * Math.log(ts); return y; } function merc(x,y) { return [merc_x(x),merc_y(y)]; } </pre></div> LatLon to/from Mercator converting class based on script above, and proj4 implementation: <div class="mw-highlight mw-highlight-lang-javascript mw-content-ltr" dir="ltr"><pre>var Conv=({ r_major:6378137.0,//Equatorial Radius, WGS84 r_minor:6356752.314245179,//defined as constant f:298.257223563,//1/f=(a-b)/a , a=r_major, b=r_minor deg2rad:function(d) { var r=d*(Math.PI/180.0); return r; }, rad2deg:function(r) { var d=r/(Math.PI/180.0); return d; }, ll2m:function(lon,lat) //lat lon to mercator { //lat, lon in rad var x=this.r_major * this.deg2rad(lon); if (lat > 89.5) lat = 89.5; if (lat < -89.5) lat = -89.5; var temp = this.r_minor / this.r_major; var es = 1.0 - (temp * temp); var eccent = Math.sqrt(es); var phi = this.deg2rad(lat); var sinphi = Math.sin(phi); var con = eccent * sinphi; var com = .5 * eccent; var con2 = Math.pow((1.0-con)/(1.0+con), com); var ts = Math.tan(.5 * (Math.PI*0.5 - phi))/con2; var y = 0 - this.r_major * Math.log(ts); var ret={'x':x,'y':y}; return ret; }, m2ll:function(x,y) //mercator to lat lon { var lon=this.rad2deg((x/this.r_major)); var temp = this.r_minor / this.r_major; var e = Math.sqrt(1.0 - (temp * temp)); var lat=this.rad2deg(this.pj_phi2( Math.exp( 0-(y/this.r_major)), e)); var ret={'lon':lon,'lat':lat}; return ret; }, pj_phi2:function(ts, e) { var N_ITER=15; var HALFPI=Math.PI/2; var TOL=0.0000000001; var eccnth, Phi, con, dphi; var i; var eccnth = .5 * e; Phi = HALFPI - 2. * Math.atan (ts); i = N_ITER; do { con = e * Math.sin (Phi); dphi = HALFPI - 2. * Math.atan (ts * Math.pow((1. - con) / (1. + con), eccnth)) - Phi; Phi += dphi; } while ( Math.abs(dphi)>TOL && --i); return Phi; } }); //usage var mercator=Conv.ll2m(47.6035525, 9.770602);//output mercator.x, mercator.y var latlon=Conv.m2ll(5299424.36041, 1085840.05328);//output latlon.lat, latlon.lon </pre></div> <h3>C implementation</h3> <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>#include <math.h> /* * Mercator transformation * accounts for the fact that the earth is not a sphere, but a spheroid */ #define D_R (M_PI / 180.0) #define R_D (180.0 / M_PI) #define R_MAJOR 6378137.0 #define R_MINOR 6356752.3142 #define RATIO (R_MINOR/R_MAJOR) #define ECCENT (sqrt(1.0 - (RATIO * RATIO))) #define COM (0.5 * ECCENT) static double deg_rad (double ang) { return ang * D_R; } double merc_x (double lon) { return R_MAJOR * deg_rad (lon); } double merc_y (double lat) { lat = fmin (89.5, fmax (lat, -89.5)); double phi = deg_rad(lat); double sinphi = sin(phi); double con = ECCENT * sinphi; con = pow((1.0 - con) / (1.0 + con), COM); double ts = tan(0.5 * (M_PI * 0.5 - phi)) / con; return 0 - R_MAJOR * log(ts); } static double rad_deg (double ang) { return ang * R_D; } double merc_lon (double x) { return rad_deg(x) / R_MAJOR; } double merc_lat (double y) { double ts = exp ( -y / R_MAJOR); double phi = M_PI_2 - 2 * atan(ts); double dphi = 1.0; int i; for (i = 0; fabs(dphi) > 0.000000001 && i < 15; i++) { double con = ECCENT * sin (phi); dphi = M_PI_2 - 2 * atan (ts * pow((1.0 - con) / (1.0 + con), COM)) - phi; phi += dphi; } return rad_deg (phi); } </pre></div> To compile in Visual Studio / MS Windows OS, I had to add these definitions <a href="/wiki/User:Ewmjc" title="User:Ewmjc">MikeCollinson</a> 14:17, 20 January 2007 (UTC): <div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>// Add this line before including math.h: #define _USE_MATH_DEFINES // Additions for MS Windows compilation: #ifndef M_PI #define M_PI acos(-1.0) #endif #ifndef M_PI_2 #define M_PI_2 1.57079632679489661922 #endif inline double fmin(double x, double y) { return(x < y ? x : y); } inline double fmax(double x, double y) { return(x > y ? x : y); } </pre></div> <h3>C# implementation</h3> C# Implementation by Florian Müller, based on the C code published above, 14:50, 20.6.2008; updated to static functions by David Schmitt, 23.4.2010 <div class="mw-highlight mw-highlight-lang-csharp mw-content-ltr" dir="ltr"><pre>using System; public static class MercatorProjection { private static readonly double R_MAJOR = 6378137.0; private static readonly double R_MINOR = 6356752.3142; private static readonly double RATIO = R_MINOR / R_MAJOR; private static readonly double ECCENT = Math.Sqrt(1.0 - (RATIO * RATIO)); private static readonly double COM = 0.5 * ECCENT; private static readonly double DEG2RAD = Math.PI / 180.0; private static readonly double RAD2Deg = 180.0 / Math.PI; private static readonly double PI_2 = Math.PI / 2.0; public static double[] toPixel(double lon, double lat) { return new double[] { lonToX(lon), latToY(lat) }; } public static double[] toGeoCoord(double x, double y) { return new double[] { xToLon(x), yToLat(y) }; } public static double lonToX(double lon) { return R_MAJOR * DegToRad(lon); } public static double latToY(double lat) { lat = Math.Min(89.5, Math.Max(lat, -89.5)); double phi = DegToRad(lat); double sinphi = Math.Sin(phi); double con = ECCENT * sinphi; con = Math.Pow(((1.0 - con) / (1.0 + con)), COM); double ts = Math.Tan(0.5 * ((Math.PI * 0.5) - phi)) / con; return 0 - R_MAJOR * Math.Log(ts); } public static double xToLon(double x) { return RadToDeg(x) / R_MAJOR; } public static double yToLat(double y) { double ts = Math.Exp(-y / R_MAJOR); double phi = PI_2 - 2 * Math.Atan(ts); double dphi = 1.0; int i = 0; while ((Math.Abs(dphi) > 0.000000001) && (i < 15)) { double con = ECCENT * Math.Sin(phi); dphi = PI_2 - 2 * Math.Atan(ts * Math.Pow((1.0 - con) / (1.0 + con), COM)) - phi; phi += dphi; i++; } return RadToDeg(phi); } private static double RadToDeg(double rad) { return rad * RAD2Deg; } private static double DegToRad(double deg) { return deg * DEG2RAD; } } </pre></div> <h3>Java implementation</h3> Java Implementation by Moshe Sayag, based on the JavaScript code published above, 17:11, 15.1.2008 <div class="mw-highlight mw-highlight-lang-java mw-content-ltr" dir="ltr"><pre>public class Mercator { final private static double R_MAJOR = 6378137.0; final private static double R_MINOR = 6356752.3142; public double[] merc(double x, double y) { return new double[] {mercX(x), mercY(y)}; } private double mercX(double lon) { return R_MAJOR * Math.toRadians(lon); } private double mercY(double lat) { if (lat > 89.5) { lat = 89.5; } if (lat < -89.5) { lat = -89.5; } double temp = R_MINOR / R_MAJOR; double es = 1.0 - (temp * temp); double eccent = Math.sqrt(es); double phi = Math.toRadians(lat); double sinphi = Math.sin(phi); double con = eccent * sinphi; double com = 0.5 * eccent; con = Math.pow(((1.0-con)/(1.0+con)), com); double ts = Math.tan(0.5 * ((Math.PI*0.5) - phi))/con; double y = 0 - R_MAJOR * Math.log(ts); return y; } } </pre></div> <h3>PHP implementation</h3> Php Code by Erhan Baris 19:19, 01.09.2007 <div class="mw-highlight mw-highlight-lang-php mw-content-ltr" dir="ltr"><pre>function merc_x($lon) { $r_major = 6378137.000; return $r_major * deg2rad($lon); } function merc_y($lat) { if ($lat > 89.5) $lat = 89.5; if ($lat < -89.5) $lat = -89.5; $r_major = 6378137.000; $r_minor = 6356752.3142; $temp = $r_minor / $r_major; $es = 1.0 - ($temp * $temp); $eccent = sqrt($es); $phi = deg2rad($lat); $sinphi = sin($phi); $con = $eccent * $sinphi; $com = 0.5 * $eccent; $con = pow((1.0-$con)/(1.0+$con), $com); $ts = tan(0.5 * ((M_PI*0.5) - $phi))/$con; $y = - $r_major * log($ts); return $y; } function merc($x,$y) { return array('x'=>merc_x($x),'y'=>merc_y($y)); } $array = merc(122,11); </pre></div> <h3>Python implementation</h3> Python Implementation by Paulo Silva, based on all code published above, 13:32, 15.2.2008 <div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>import math def merc_x(lon): r_major=6378137.000 return r_major*math.radians(lon) def merc_y(lat): if lat>89.5:lat=89.5 if lat<-89.5:lat=-89.5 r_major=6378137.000 r_minor=6356752.3142 temp=r_minor/r_major eccent=math.sqrt(1-temp**2) phi=math.radians(lat) sinphi=math.sin(phi) con=eccent*sinphi com=eccent/2 con=((1.0-con)/(1.0+con))**com ts=math.tan((math.pi/2-phi)/2)/con y=0-r_major*math.log(ts) return y </pre></div> <h3>sdlBasic implementation</h3> sdlBasic Implementation by Paulo Silva, based on all code published above, 12:33, 18.2.2008 <div class="mw-highlight mw-highlight-lang-basic mw-content-ltr" dir="ltr"><pre>function deg_rad(ang): tang=ang pi=3.14159265359 deg_rad=tang*(pi/180.0) end function function merc_x(lon): tlon=lon r_major=6378137.0 merc_x=r_major*deg_rad(tlon) end function function merc_y(lat): tlat=lat pi=3.14159265359 if tlat>89.5 then:tlat=89.5: end if if tlat<-89.5 then:tlat=-89.5: end if r_major=6378137.000 r_minor=6356752.3142 temp=r_minor/r_major es=1-(temp*temp) eccent=sqr(es) phi=(tlat*pi)/180 sinphi=sin(phi) con=eccent*sinphi com=.5*eccent con=((1.0-con)/(1.0+con))^com ts=tan(.5*((pi*0.5)-phi))/con y=0-r_major*log(ts) merc_y=y end function </pre></div> <h3>bash-script (bc) implementation</h3> bash-script by Frank Baettermann, 18.05.2008 <div class="mw-highlight mw-highlight-lang-bash mw-content-ltr" dir="ltr"><pre>#!/bin/bash # Compute Mercator tile-coordinates if [ $# -ne 3 ]; then echo "Usage: $0 LATITUDE LONGITUDE ZOOM" exit 1 fi if [ -z "`which bc 2> /dev/null`" ]; then echo "ERROR: Could not find bc! Try to install it: sudo apt install bc" exit 1 fi LATITUDE=$1 LONGITUDE=$2 ZOOM=$3 # total width and height in tiles MAP_SIZE=$((2**$ZOOM)) echo "map_size=$MAP_SIZE" # longitude -> x echo "tile_x=`echo "($LONGITUDE + 180) * $MAP_SIZE / 360" | bc`" # latitude -> y echo "tile_y=`echo "\ scale=10; \ phi=$LATITUDE; \ mapsize=$MAP_SIZE; \ pi=3.1415926535; \ phi_rad=phi*2*pi/360; \ mercator=l((1+s(phi_rad))/(1-s(phi_rad)))/(2); \ tile=(1-mercator/pi)*(mapsize/2); \ scale=0; tile/1" \ | bc -l`" </pre></div> <h3>VBA for Office implementation</h3> Copy into a module to use as an Excel function Based on JavaScript Code above by Benjamin Judd <div class="mw-highlight mw-highlight-lang-vbnet mw-content-ltr" dir="ltr"><pre>Function deg_rad(deg As Double) As Double deg_rad = deg * Excel.WorksheetFunction.Pi / 180 End Function Function merc_x(lon As Double) As Double Dim r_major As Double r_major = 6378137 merc_x = r_major * deg_rad(lon) End Function Function merc_y(lat As Double) As Double If (lat > 89.5) Then lat = 89.5 If (lat < -89.5) Then lat = -89.5 Dim r_major As Double, r_minor As Double, temp As Double, es As Double, eccent As Double Dim phi As Double, sinphi As Double, con As Double, com As Double, ts As Double r_major = 6378137 r_minor = 6356752.3142 temp = r_minor / r_major es = 1 - temp ^ 2 eccent = Math.Sqr(es) phi = deg_rad(lat) sinphi = Math.Sin(phi) con = eccent * sinphi com = 0.5 * eccent con = (1 - con / 1 + con) ^ com ts = Math.Tan(0.5 * ((Excel.WorksheetFunction.Pi * 0.5) - phi)) / con merc_y = 0 - r_major * Math.Log(ts) End Function </pre></div> <h2>Links</h2> <ul><li><a rel="nofollow" class="external text" href="http://mal2.ch/stuff/latlontomercator.html">Lat/Lon to Mercator online converter</a></li></ul>    </div> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://wiki.openstreetmap.org/w/index.php?title=Mercator&oldid=2745139">https://wiki.openstreetmap.org/w/index.php?title=Mercator&oldid=2745139</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Category</a>: <ul><li><a href="/wiki/Category:Map_projections" title="Category:Map projections">Map projections</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Pages_using_deprecated_source_tags" title="Category:Pages using deprecated source tags">Pages using deprecated source tags</a></li><li><a href="/wiki/Category:Pages_with_syntax_highlighting_errors" title="Category:Pages with syntax highlighting errors">Pages with syntax highlighting errors</a></li><li><a href="/wiki/Category:Pages_unavailable_in_German" title="Category:Pages unavailable in German">Pages unavailable in German</a></li><li><a href="/wiki/Category:Pages_unavailable_in_Spanish" title="Category:Pages unavailable in Spanish">Pages unavailable in Spanish</a></li><li><a href="/wiki/Category:Pages_unavailable_in_French" title="Category:Pages unavailable in French">Pages unavailable in French</a></li><li><a href="/wiki/Category:Pages_unavailable_in_Italian" title="Category:Pages unavailable in Italian">Pages unavailable in Italian</a></li><li><a href="/wiki/Category:Pages_unavailable_in_Dutch" title="Category:Pages unavailable in Dutch">Pages unavailable in Dutch</a></li><li><a href="/wiki/Category:Pages_unavailable_in_Russian" title="Category:Pages unavailable in Russian">Pages unavailable in Russian</a></li><li><a href="/wiki/Category:Pages_unavailable_in_Japanese" title="Category:Pages unavailable in Japanese">Pages unavailable in Japanese</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigation menu</h2> <div id="mw-head"> <nav id="p-personal" class="vector-menu mw-portlet mw-portlet-personal vector-user-menu-legacy" aria-labelledby="p-personal-label" role="navigation" > <h3 id="p-personal-label" class="vector-menu-heading " > Personal tools </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"><li id="pt-uls" class="mw-list-item active"><a class="uls-trigger" href="#">English</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Mercator" title="You are encouraged to create an account and log in; 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