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Generally several subdomains (server names) are provided to get around browser limitations on the number of simultaneous HTTP connections to each host. Browser-based applications can thus request multiple tiles from multiple subdomains faster than from one subdomain. For example, OSM, OpenCycleMap and CloudMade servers have three subdomains (a.tile, b.tile, c.tile),MapQuest has four (otile1, otile2, otile3, otile4), all pointing to the same CDN.

That all comes before the /zoom/x/y.png tail.

Here are some examples:

NameURL templatezoomlevels

OSM 'standard' stylehttp://[abc].tile.openstreetmap.org/zoom/x/y.png0-19

OpenCycleMaphttp://[abc].tile.opencyclemap.org/cycle/zoom/x/y.png0-18

OpenCycleMap Transport (experimental)http://[abc].tile2.opencyclemap.org/transport/zoom/x/y.png0-18

MapQuesthttp://otile[1234].mqcdn.com/tiles/1.0.0/osm/zoom/x/y.jpg0-19

MapQuest Open Aerialhttp://otile[1234].mqcdn.com/tiles/1.0.0/sat/zoom/x/y.jpg0-11 globally, 12+ in the U.S.

Migurski's Terrainhttp://tile.stamen.com/terrain-background/zoom/x/y.jpg4-18, US-only (for now)

Further tilesets are available from various '3rd party' sources.

Zoom levels

The zoom parameter is an integer between 0 (zoomed out) and 18 (zoomed in). 18 is normally the maximum, but some tile servers might go beyond that.

zoom leveltile coveragenumber of tilestile size in degrees

01 tile covers whole world1 tile360° x 170.1022°

12 × 2 tiles4 tiles180° x 85.0511°

24 × 4 tiles16 tiles90° x 42.5256°

n2n × 2n tiles22n tiles360/2n° x 170.1022/2n°

124096 x 4096 tiles16 777 2160.0879° x 0.0415°

16232 = 4 294 967 296 tiles

1717 179 869 184 tiles

1868 719 476 736 tiles

19Maximum zoom for Mapnik layer274 877 906 944 tiles

See Zoom levels for more details

X and Y

X goes from 0 (left edge is 180 °W) to 2zoom ? 1 (right edge is 180 °E)

Y goes from 0 (top edge is 85.0511 °N) to 2zoom ? 1 (bottom edge is 85.0511 °S) in a Mercator projection

For the curious, the number 85.0511 is the result of arctan(sinh(π)). By using this bound, the entire map becomes a (very large) square.

Derivation of tile names

Reproject the coordinates to the Mercator projection (from EPSG:4326 to EPSG:3857):x = lon

y = arsinh(tan(lat)) = log[tan(lat) + sec(lat)]

(lat and lon are in radians)

Transform range of x and y to 0 – 1 and shift origin to top left corner:x = [1 + (x / π)] / 2

y = [1 ? (y / π)] / 2

Calculate the number of tiles across the map, n, using 2zoom

Multiply x and y by n. Round results down to give tilex and tiley.

Implementations

Pseudo-code

For those who like pseudo-code, here's some hints:

sec = 1/cosarsinh(x) = log(x + (x^2 + 1)^0.5)sec^2(x) = tan^2(x) + 1→ arsinh(tan(x)) = log(tan(x) + sec(x))Please note that "log" represents the natural logarithm (also known as ln or loge), not decimal logarithm (log10), as used on some calculators.

Lon./lat. to tile numbers

n = 2 ^ zoomxtile = n * ((lon_deg + 180) /360)ytile = n * (1 - (log(tan(lat_rad) + sec(lat_rad)) / π)) / 2Tile numbers to lon./lat.

n = 2 ^ zoomlon_deg = xtile / n *360.0 - 180.0lat_rad = arctan(sinh(π * (1 - 2 * ytile / n)))lat_deg = lat_rad * 180.0 / πMathematics

Idem with mathematic signs (lat and lon in degrees):

Python

Lon./lat. to tile numbers

import mathdef deg2num(lat_deg, lon_deg, zoom): lat_rad = math.radians(lat_deg) n = 2.0 ** zoom xtile = int((lon_deg + 180.0) /360.0 * n) ytile = int((1.0 - math.log(math.tan(lat_rad) + (1 / math.cos(lat_rad))) / math.pi) / 2.0 * n) return (xtile, ytile)

Tile numbers to lon./lat.

import mathdef num2deg(xtile, ytile, zoom): n = 2.0 ** zoom lon_deg = xtile / n *360.0 - 180.0 lat_rad = math.atan(math.sinh(math.pi * (1 - 2 * ytile / n))) lat_deg = math.degrees(lat_rad) return (lat_deg, lon_deg)

This returns the NW-corner of the square. Use the function with xtile+1 and/or ytile+1 to get the other corners. With xtile+0.5 & ytile+0.5 it will return the center of the tile.

See also tilenames.py and the

Ruby

Lon./lat. to tile numbers

def get_tile_number(lat_deg, lng_deg, zoom) lat_rad = lat_deg/180 * Math::PI n = 2.0 ** zoom x = ((lng_deg + 180.0) /360.0 * n).to_i y = ((1.0 - Math::log(Math::tan(lat_rad) + (1 / Math::cos(lat_rad))) / Math::PI) / 2.0 * n).to_i {:x => x, :y =>y}end

Tile numbers to lon./lat.

def get_lat_lng_for_number(xtile, ytile, zoom) n = 2.0 ** zoom lon_deg = xtile / n *360.0 - 180.0 lat_rad = Math::atan(Math::sinh(Math::PI * (1 - 2 * ytile / n))) lat_deg = 180.0 * (lat_rad / Math::PI) {:lat_deg => lat_deg, :lng_deg => lon_deg}end

Same as the Python implementation above, this returns the NW-corner of the square. Use the function with xtile+1 and/or ytile+1 to get the other corners. With xtile+0.5 & ytile+0.5 it will return the center of the tile.

Perl

Lon./lat. to tile numbers

Tile numbers to lon./lat.

use Math::Trig;sub Project { my ($X,$Y, $Zoom) = @_; my $Unit = 1 / (2 ** $Zoom); my $relY1 = $Y * $Unit; my $relY2 = $relY1 + $Unit; # note: $LimitY = ProjectF(degrees(atan(sinh(pi)))) = log(sinh(pi)+cosh(pi)) = pi # note: degrees(atan(sinh(pi))) = 85.051128.. #my $LimitY = ProjectF(85.0511); # so stay simple and more accurate my $LimitY = pi; my $RangeY = 2 * $LimitY; $relY1 = $LimitY - $RangeY * $relY1; $relY2 = $LimitY - $RangeY * $relY2; my $Lat1 = ProjectMercToLat($relY1); my $Lat2 = ProjectMercToLat($relY2); $Unit =360/ (2 ** $Zoom); my $Long1 = -180 + $X * $Unit; return ($Lat2, $Long1, $Lat1, $Long1 + $Unit); # S,W,N,E}sub ProjectMercToLat($){ my $MercY = shift; return rad2deg(atan(sinh($MercY)));}sub ProjectF{ my $Lat = shift; $Lat = deg2rad($Lat); my $Y = log(tan($Lat) + sec($Lat)); return $Y;}

Lon./lat. to bbox

use Math::Trig; sub getTileNumber { my ($lat,$lon,$zoom) = @_; my $xtile = int( ($lon+180)/360* 2**$zoom ) ; my $ytile = int( (1 - log(tan(deg2rad($lat)) + sec(deg2rad($lat)))/pi)/2 * 2**$zoom ) ; return ($xtile, $ytile);} sub getLonLat { my ($xtile, $ytile, $zoom) = @_; my $n = 2 ** $zoom; my $lon_deg = $xtile / $n *360.0 - 180.0; my $lat_deg = rad2deg(atan(sinh(pi * (1 - 2 * $ytile / $n)))); return ($lon_deg, $lat_deg);} # convert from permalink OSM format like:# http://www.openstreetmap.org/?lat=43.731049999999996&lon=15.79375&zoom=13&layers=M# to OSM "Export" iframe embedded bbox format like:# http://www.openstreetmap.org/export/embed.html?bbox=15.7444,43.708,15.8431,43.7541&layer=mapnik sub LonLat_to_bbox { my ($lat, $lon, $zoom) = @_; my $width = 425; my $height = 350; # note: must modify this to match your embed map width/height in pixels my $tile_size = 256; my ($xtile, $ytile) = getTileNumber ($lat, $lon, $zoom); my $xtile_s = ($xtile * $tile_size - $width/2) / $tile_size; my $ytile_s = ($ytile * $tile_size - $height/2) / $tile_size; my $xtile_e = ($xtile * $tile_size + $width/2) / $tile_size; my $ytile_e = ($ytile * $tile_size + $height/2) / $tile_size; my ($lon_s, $lat_s) = getLonLat($xtile_s, $ytile_s, $zoom); my ($lon_e, $lat_e) = getLonLat($xtile_e, $ytile_e, $zoom); my $bbox = "$lon_s,$lat_s,$lon_e,$lat_e"; return $bbox;}

PHP

Lon./lat. to tile numbers

$xtile = floor((($lon + 180) /360) * pow(2, $zoom));$ytile = floor((1 - log(tan(deg2rad($lat)) + 1 / cos(deg2rad($lat))) / pi()) /2 * pow(2, $zoom));

Tile numbers to lon./lat.

$n = pow(2, $zoom);$lon_deg = $xtile / $n *360.0 - 180.0;$lat_deg = rad2deg(atan(sinh(pi() * (1 - 2 * $ytile / $n))));

ECMAScript (JavaScript/ActionScript, etc.)

function long2tile(lon,zoom) { return (Math.floor((lon+180)/360*Math.pow(2,zoom))); } function lat2tile(lat,zoom) { return (Math.floor((1-Math.log(Math.tan(lat*Math.PI/180) + 1/Math.cos(lat*Math.PI/180))/Math.PI)/2 *Math.pow(2,zoom))); }

Inverse process:

function tile2long(x,z) { return (x/Math.pow(2,z)*360-180); } function tile2lat(y,z) { var n=Math.PI-2*Math.PI*y/Math.pow(2,z); return (180/Math.PI*Math.atan(0.5*(Math.exp(n)-Math.exp(-n)))); }

Example: Tilesname WebCalc V1.0

C/C++

int long2tilex(double lon, int z) { return (int)(floor((lon + 180.0) /360.0 * pow(2.0, z))); } int lat2tiley(double lat, int z){ return (int)(floor((1.0 - log( tan(lat * M_PI/180.0) + 1.0 / cos(lat * M_PI/180.0)) / M_PI) / 2.0 * pow(2.0, z))); } double tilex2long(int x, int z) { return x / pow(2.0, z) *360.0 - 180;} double tiley2lat(int y, int z) { double n = M_PI - 2.0 * M_PI * y / pow(2.0, z); return 180.0 / M_PI * atan(0.5 * (exp(n) - exp(-n)));}

Example : [1] Doc : [2]

import ( "math") type Tile struct { Z int X int Y int Lat float64 Long float64} type Conversion interface { deg2num(t *Tile) (x int, y int) num2deg(t *Tile) (lat float64, long float64)} func (*Tile) Deg2num(t *Tile) (x int, y int) { x = int(math.Floor((t.Long + 180.0) /360.0 * (math.Exp2(float64(t.Z))))) y = int(math.Floor((1.0 - math.Log(math.Tan(t.Lat*math.Pi/180.0)+1.0/math.Cos(t.Lat*math.Pi/180.0))/math.Pi) / 2.0 * (math.Exp2(float64(t.Z))))) return} func (*Tile) Num2deg(t *Tile) (lat float64, long float64) { n := math.Pi - 2.0*math.Pi*float64(t.Y)/math.Exp2(float64(t.Z)) lat = 180.0 / math.Pi * math.Atan(0.5*(math.Exp(n)-math.Exp(-n))) long = float64(t.X)/math.Exp2(float64(t.Z))*360.0 - 180.0 return lat, long}

Java

public class slippytest { public static void main(String[] args) { int zoom = 10; double lat = 47.968056d; double lon = 7.909167d; System.out.println("http://tile.openstreetmap.org/" + getTileNumber(lat, lon, zoom) + ".png"); } public static String getTileNumber(final double lat, final double lon, final int zoom) { int xtile = (int)Math.floor( (lon + 180) /360* (1<<zoom) ) ; int ytile = (int)Math.floor( (1 - Math.log(Math.tan(Math.toRadians(lat)) + 1 / Math.cos(Math.toRadians(lat))) / Math.PI) / 2 * (1<<zoom) ) ; if (xtile < 0) xtile=0; if (xtile >= (1<<zoom)) xtile=((1<<zoom)-1); if (ytile < 0) ytile=0; if (ytile >= (1<<zoom)) ytile=((1<<zoom)-1); return("" + zoom + "/" + xtile + "/" + ytile); } }

Tile bounding box

class BoundingBox { double north; double south; double east; double west; } BoundingBox tile2boundingBox(final int x, final int y, final int zoom) { BoundingBox bb = new BoundingBox(); bb.north = tile2lat(y, zoom); bb.south = tile2lat(y + 1, zoom); bb.west = tile2lon(x, zoom); bb.east = tile2lon(x + 1, zoom); return bb; } static double tile2lon(int x, int z) { return x / Math.pow(2.0, z) *360.0 - 180; } static double tile2lat(int y, int z) { double n = Math.PI - (2.0 * Math.PI * y) / Math.pow(2.0, z); return Math.toDegrees(Math.atan(Math.sinh(n))); }

VB.Net

Private Function CalcTileXY(ByVal lat As Single, ByVal lon As Single, ByVal zoom As Long) As Point CalcTileXY.X = CLng(Math.Floor((lon + 180) /360* 2 ^ zoom)) CalcTileXY.Y = CLng(Math.Floor((1 - Math.Log(Math.Tan(lat * Math.PI / 180) + 1 / Math.Cos(lat * Math.PI / 180)) / Math.PI) / 2 * 2 ^ zoom))End Function

public PointF WorldToTilePos(double lon, double lat, int zoom){ PointF p = new Point(); p.X = (float)((lon + 180.0) /360.0 * (1 << zoom)); p.Y = (float)((1.0 - Math.Log(Math.Tan(lat * Math.PI / 180.0) + 1.0 / Math.Cos(lat * Math.PI / 180.0)) / Math.PI) / 2.0 * (1 << zoom)); return p;} public PointF TileToWorldPos(double tile_x, double tile_y, int zoom) { PointF p = new Point(); double n = Math.PI - ((2.0 * Math.PI * tile_y) / Math.Pow(2.0, zoom)); p.X = (float)((tile_x / Math.Pow(2.0, zoom) *360.0) - 180.0); p.Y = (float)(180.0 / Math.PI * Math.Atan(Math.Sinh(n))); return p;}

XSLT

Requires math extensions from exslt.org.

<xsl:transform xmlns:xsl="http://www.w3.org/1999/XSL/Transform" xmlns:m="http://exslt.org/math" extension-element-prefixes="m" version="1.0"> <xsl:output method="text"/> <xsl:variable name="pi" select="3.14159265358979323846"/> <xsl:template name="tiley"> <xsl:param name="lat"/> <xsl:param name="zoomfact"/> <xsl:variable name="a" select="($lat * $pi) div 180.0"/> <xsl:variable name="b" select="m:log(m:tan($a) + (1.0 div m:cos($a)))"/> <xsl:variable name="c" select="(1.0 - ($b div $pi)) div 2.0"/> <xsl:value-of select="floor($c * $zoomfact)"/> </xsl:template> <xsl:template name="tilename"> <xsl:param name="lat"/> <xsl:param name="lon"/> <xsl:param name="zoom" select="10"/> <xsl:variable name="zoomfact" select="m:power(2,$zoom)"/> <xsl:variable name="x" select="floor((360.0 + ($lon * 2)) * $zoomfact div 720.0)"/> <xsl:variable name="y"> <xsl:call-template name="tiley"> <xsl:with-param name="lat" select="$lat"/> <xsl:with-param name="zoomfact" select="$zoomfact"/> </xsl:call-template> </xsl:variable> <xsl:value-of select="concat($zoom,'/',$x,'/',$y)"/> </xsl:template> <xsl:template match="/"> <xsl:call-template name="tilename"> <xsl:with-param name="lat" select="49.867731999999997"/> <xsl:with-param name="lon" select="8.6295369999999991"/> <xsl:with-param name="zoom" select="14"/> </xsl:call-template> </xsl:template></xsl:transform>

Haskell

-- https://github.com/j4/HSlippyMap long2tilex lon z = floor((lon + 180.0) /360.0 * (2.0 ** z)) lat2tiley lat z = floor((1.0 - log( tan(lat * pi/180.0) + 1.0 / cos(lat * pi/180.0)) / pi) / 2.0 * (2.0 ** z)) tilex2long x z = x / (2.0 ** z) *360.0 - 180 tiley2lat y z = 180.0 / pi * atan(0.5 * (exp(n) - exp(-n))) where n = pi - 2.0 * pi * y / (2.0 ** z) -- Examplemain = do --print $ long2tilex 2.2712 17 --print $ lat2tiley 48.8152 17 --print $ tilex2long 66362 17 --print $ tiley2lat 45115 17 putStrLn "gps: (lat=48.8152,long=2.2712)" putStrLn $ "http://tile.openstreetmap.org/17/" ++ show x ++ "/" ++ show y ++ ".png" where z = 17 x = long2tilex 2.2712 z y = lat2tiley 48.8152 z

Scala

import scala.math._ case class Tile(x: Int,y: Int, z: Short){ def toLatLon = new LatLonPoint( toDegrees(atan(sinh(Pi * (1.0 - 2.0 * y.toDouble / (1<<z))))), x.toDouble / (1<<z) *360.0 - 180.0, z) def toURI = new java.net.URI("http://tile.openstreetmap.org/"+z+"/"+x+"/"+y+".png")} case class LatLonPoint(lat: Double, lon: Double, z: Short){ def toTile = new Tile( ((lon + 180.0) /360.0 * (1<<z)).toInt, ((1 - log(tan(toRadians(lat)) + 1 / cos(toRadians(lat))) / Pi) / 2.0 * (1<<z)).toInt, z)} //Usage:val point = LatLonPoint(51.51202,0.02435,17)val tile = point.toTile// ==> Tile(65544,43582,17)val uri = tile.toURI// ==> http://tile.openstreetmap.org/17/65544/43582.png

Revolution/Transcript

function osmTileRef iLat, iLong, iZoom --> part path local n, xTile, yTile put (2 ^ iZoom) into n put (iLong + 180) /360* n into xTile multiply iLat by (pi / 180) -- convert to radians put ((1 - ln(tan(iLat) + 1 / cos(iLat)) / pi) / 2) * n into yTile return "/" & iZoom & "/" & trunc(xTile) & "/" & trunc(yTile)end osmTileReffunction osmTileCoords xTile, yTile, iZoom --> coordinates local twoPzoom, iLong, iLat, n put (2 ^ iZoom) into twoPzoom put xTile / twoPzoom *360- 180 into iLong put pi - 2 * pi * yTile / twoPzoom into n put "n1=" && n put 180 / pi * atan(0.5 * (exp(n) - exp(-n))) into iLat return iLat & comma & iLongend osmTileCoordsMathematica

Deg2Num[lat_, lon_, zoom_] := {IntegerPart[(2^(-3 + zoom)*(180 + lon))/45], IntegerPart[2^(-1 + zoom)*(1 - Log[Sec[Degree*lat] + Tan[Degree*lat]]/Pi)]}Num2Deg[xtile_,ytile_,zoom_] := {ArcTan[Sinh[Pi*(1 - 2*(ytile/2^zoom))]]/Degree, (xtile/2^zoom)*360- 180} // NTcl

First of all, you need to use the package map::slippy from Tcllib:

package require map::slippy

Lat./lon. to tile number

map::slippy geo 2tile [list $zoom $lat $lon]

Tile number to lat/lon

map::slippy tile 2geo [list $zoom $row $col]

Pascal

(translated from the Pythoncode above to Pascal)

Coordinates to tile numbers

uses {...}, Math;{...}var zoom: Integer; lat_rad, lat_deg, lon_deg, n: Real;begin lat_rad := DegToRad(lat_deg); n := Power(2, zoom); xtile := Trunc(((lon_deg + 180) /360) * n); ytile := Trunc((1 - (ln(Tan(lat_rad) + (1 /Cos(lat_rad))) / Pi)) / 2 * n);end;

Tile numbers to coordinates

uses {...}, Math;{...}var lat_rad, n: Real;begin n := Power(2, zoom); lat_rad := Arctan (Sinh (Pi * (1 - 2 * ytile / n))); lat_deg := RadtoDeg (lat_rad); lon_deg := xtile / n *360.0 - 180.0;end;

Coordinates to tile numbers

deg2num<-function(lat_deg, lon_deg, zoom){ lat_rad <- lat_deg * pi /180 n <- 2.0 ^ zoom xtile <- floor((lon_deg + 180.0) /360.0 * n) ytile = floor((1.0 - log(tan(lat_rad) + (1 / cos(lat_rad))) / pi) / 2.0 * n) return( c(xtile, ytile))# return(paste(paste("http://a.tile.openstreetmap.org", zoom, xtile, ytile, sep="/"),".png",sep=""))} # Returns data frame containing detailed info for all zoomsdeg2num.all<-function(lat_deg, lon_deg){ nums <- as.data.frame(matrix(ncol=6,nrow=21)) colnames(nums) <- c('zoom', 'x', 'y', 'mapquest_osm', 'mapquest_aerial', 'osm') rownames(nums) <- 0:20 for (zoom in 0:20) { num <- deg2num(lat_deg, lon_deg, zoom) nums[1+zoom,'zoom'] <- zoom nums[1+zoom,'x'] <- num[1] nums[1+zoom,'y'] <- num[2] nums[1+zoom,'mapquest_osm'] <- paste('http://otile1.mqcdn.com/tiles/1.0.0/map/', zoom, '/', num[1], '/', num[2], '.jpg', sep='') nums[1+zoom,'mapquest_aerial'] <- paste('http://otile1.mqcdn.com/tiles/1.0.0/sat/', zoom, '/', num[1], '/', num[2], '.jpg', sep='') nums[1+zoom,'osm'] <- paste('http://a.tile.openstreetmap.org/', zoom, '/', num[1], '/', num[2], '.png', sep='') } return(nums)}

Bourne shell with Awk

Tile numbers to lat./lon. / Coordinates to tile numbers / Sample of usage, with optional tms-format support

xtile2long(){ xtile=$1 zoom=$2 echo "${xtile} ${zoom}" | awk '{printf("%.9f", $1 / 2.0^$2 *360.0 - 180)}'} long2xtile() { long=$1 zoom=$2 echo "${long} ${zoom}" | awk '{ xtile = ($1 + 180.0) /360* 2.0^$2; xtile+=xtile<0?-0.5:0.5; printf("%d", xtile ) }'} ytile2lat(){ ytile=$1; zoom=$2; tms=$3; if [ ! -z "${tms}" ] then # from tms_numbering into osm_numbering ytile=`echo "${ytile}" ${zoom} | awk '{printf("%d\n",((2.0^$2)-1)-$1)}'`; fi lat=`echo "${ytile} ${zoom}" | awk -v PI=3.14159265358979323846 '{ num_tiles = PI - 2.0 * PI * $1 / 2.0^$2; printf("%.9f", 180.0 / PI * atan2(0.5 * (exp(num_tiles) - exp(-num_tiles)),1)); }'`; echo "${lat}";} lat2ytile() { lat=$1; zoom=$2; tms=$3; ytile=`echo "${lat} ${zoom}" | awk -v PI=3.14159265358979323846 '{ tan_x=sin($1 * PI / 180.0)/cos($1 * PI / 180.0); ytile = (1 - log(tan_x + 1/cos($1 * PI/ 180))/PI)/2 * 2.0^$2; ytile+=ytile<0?-0.5:0.5; printf("%d", ytile ) }'`; if [ ! -z "${tms}" ] then # from oms_numbering into tms_numbering ytile=`echo "${ytile}" ${zoom} | awk '{printf("%d\n",((2.0^$2)-1)-$1)}'`; fi echo "${ytile}";}# ------------------------------------# Sample of use: # Position Brandenburg Gate, Berlin# ------------------------------------LONG=13.37771496361961;LAT=52.51628011262304;ZOOM=17;TILE_X=70406;TILE_Y=42987; TILE_Y_TMS=88084;TMS=""; # when NOT empty: tms format assumed# ------------------------------------# assume input/output of y is in oms-format:LONG=$( xtile2long ${TILE_X} ${ZOOM} );LAT=$( ytile2lat ${TILE_Y} ${ZOOM} ${TMS} );# Result should be longitude[13.375854492] latitude[52.517892228]TILE_X=$( long2xtile ${LONG} ${ZOOM} );TILE_Y=$( lat2ytile ${LAT} ${ZOOM} ${TMS} );# Result should be x[70406] y_oms[42987] # ------------------------------------# assume input/output of y is in tms-format:TMS="tms";TILE_Y_TMS=$( lat2ytile ${LAT} ${ZOOM} ${TMS} );LAT_TMS=$( ytile2lat ${TILE_Y_TMS} ${ZOOM} ${TMS} );echo "Result should be y_oms[${TILE_Y}] latitude[${LAT}] ; y_tms[${TILE_Y_TMS}] latitude_tms[${LAT_TMS}] "# latitude and latitude_tms should have the same value ; y_oms and y_tms should have the given start values:# Result should be y_oms[42987] latitude[52.517892228] ; y_tms[88084] latitude_tms[52.517892228]# ------------------------------------

Tile bounding box and center

n=$(ytile2lat `expr ${TILE_Y}` ${ZOOM})s=$(ytile2lat `expr ${TILE_Y} + 1` ${ZOOM})e=$(xtile2long `expr ${TILE_X} + 1` ${ZOOM})w=$(xtile2long `expr ${TILE_X}` ${ZOOM}) echo "bbox=$w,$s,$e,$n" echo "-I-> Result should be [bbox=13.375854492,52.516220864,13.378601074,52.517892228]"; center_lat=`echo "$s $n" | awk '{printf("%.8f", ($1 + $2) / 2.0)}'`center_lon=`echo "$w $e" | awk '{printf("%.8f", ($1 + $2) / 2.0)}'` echo "center=$center_lat,$center_lon"echo "-I-> Result should be [center=52.51705655,13.37722778]";

Octave

Lon./lat. to tile numbers

% convert the degrees to radiansrho = pi/180;lon_rad = lon_deg * rho;lat_rad = lat_deg * rho; n = 2 ^ zoomxtile = n * ((lon_deg + 180) /360)ytile = n * (1 - (log(tan(lat_rad) + sec(lat_rad)) / pi)) / 2

Tile numbers to lon./lat.

n=2^zoomlon_deg = xtile / n *360.0 - 180.0lat_rad = arctan(sinh(pi * (1 - 2 * ytile / n)))lat_deg = lat_rad * 180.0 / pi

Emacs-lisp

(defun longitude2tile (lon zoom) (* (expt 2 zoom) (/ (+ lon 180)360))) (defun tile2longitude (x zoom) (- (/ (* x360) (expt 2 zoom)) 180)) (defun latitude2tile (lat zoom) (* (expt 2 zoom) (/ (- 1 (/ (log (+ (tan (/ (* lat pi) 180)) (/ 1 (cos (/ (* lat pi) 180))))) pi)) 2))) (defun sinh (value) (/ (- (exp value) (exp (- value))) 2))(defun tile2latitude (y zoom) (/ (* 180 (atan (sinh (* pi (- 1 (* 2 (/ y (expt 2 zoom)))))))) pi))

Erlang

-module(slippymap).-export([deg2num/3]).-export([num2deg/3]). deg2num(Lat,Lon,Zoom)-> X=math:pow(2, Zoom) * ((Lon + 180) /360), Sec=1/math:cos(deg2rad(Lat)), R = math:log(math:tan(deg2rad(Lat)) + Sec)/math:pi(), Y=math:pow(2, Zoom) * (1 - R) / 2, {round(X),round(Y)}. num2deg(X,Y,Zoom)-> N=math:pow(2, Zoom), Lon=X/N*360-180, Lat_rad=math:atan(math:sinh(math:pi()*(1-2*Y/N))), Lat=Lat_rad*180/math:pi(), {Lon,Lat}. deg2rad(C)-> C*math:pi()/180.

Lua

function deg2num(lon, lat, zoom) local n = 2 ^ zoom local lon_deg = tonumber(lon) local lat_rad = math.rad(lat) local xtile = math.floor(n * ((lon_deg + 180) /360)) local ytile = math.floor(n * (1 - (math.log(math.tan(lat_rad) + (1 / math.cos(lat_rad))) / math.pi)) / 2) return xtile, ytileend function num2deg(x, y, z) local n = 2 ^ z local lon_deg = x / n *360.0 - 180.0 local lat_rad = math.atan(math.sinh(math.pi * (1 - 2 * y / n))) local lat_deg = lat_rad * 180.0 / math.pi return lon_deg, lat_degend

PostgreSQL

CREATE OR REPLACE FUNCTION lon2tile(lon DOUBLE PRECISION, zoom INTEGER) RETURNS INTEGER AS$BODY$BEGIN RETURN FLOOR( (lon + 180) /360* (1 << zoom) )::INTEGER;END;$BODY$ LANGUAGE plpgsql IMMUTABLE; CREATE OR REPLACE FUNCTION lat2tile(lat DOUBLE PRECISION, zoom INTEGER) RETURNS INTEGER AS$BODY$BEGIN RETURN FLOOR( (1.0 - LN(TAN(RADIANS(lat)) + 1.0 / COS(RADIANS(lat))) / PI()) / 2.0 * (1 << zoom) )::INTEGER;END;$BODY$ LANGUAGE plpgsql IMMUTABLE; CREATE OR REPLACE FUNCTION tile2lat(y INTEGER, zoom INTEGER) RETURNS DOUBLE PRECISION AS$BODY$DECLARE n FLOAT; sinh FLOAT; E FLOAT = 2.7182818284;BEGIN n = PI() - (2.0 * PI() * y) / POWER(2.0, zoom); sinh = (1 - POWER(E, -2*n)) / (2 * POWER(E, -n)); RETURN DEGREES(ATAN(sinh));END;$BODY$ LANGUAGE plpgsql IMMUTABLE; CREATE OR REPLACE FUNCTION tile2lon(x INTEGER, zoom INTEGER) RETURNS DOUBLE PRECISION AS$BODY$BEGIN RETURN x * 1.0 / (1 << zoom) *360.0 - 180.0;END;$BODY$ LANGUAGE plpgsql IMMUTABLE;

Subtiles

If you're looking at tile x,y and want to zoom in, the subtiles are (in the next zoom-level's coordinate system):

2x, 2y2x + 1, 2y

2x, 2y + 12x + 1, 2y + 1

Similarly, zoom out by halving x and y (in the previous zoom level)

Resolution and Scale

Exact length of the equator (according to wikipedia) is 40075.016686 km in WGS-84. A horizontal tile size at zoom 0 would be 156543.03 meters:

40075.016686 * 1000 / 256 ≈ 6378137.0 * 2 * pi / 256 ≈ 156543.03Which gives us a formula to calculate resolution at any given zoom:

resolution = 156543.03 meters/pixel * cos(latitude) / (2 ^ zoomlevel)Some applications need to know a map scale, that is, how 1 cm on a screen translates to 1 cm of a map.

scale = 1 : (screen_dpi * 39.37 in/m * resolution)And here is the table to rid you of those calculations. All values are shown for equator, and you have to multiply them by cos(latitude) to adjust to a given latitude. For example, divide those by 2 for latitude 60 (Oslo, Helsinki, Saint-Petersburg).

zoomresolution, m/pxscale 96 dpi1 screen cm isscale 120 dpi

0156543.031 : 554 678 9325547 km1 : 739 571 909

178271.521 : 277 339 4662773 km1 : 369 785 954

239135.761 : 138 669 7331337 km1 : 184 892 977

319567.881 : 69 334 866693 km1 : 92 446 488

49783.941 : 34 667 433347 km1 : 46 223 244

54891.971 : 17 333 716173 km1 : 23 111 622

62445.981 : 8 666 85886.7 km1 : 11 555 811

71222.991 : 4 333 42943.3 km1 : 5 777 905

8611.501 : 2 166 71421.7 km1 : 2 888 952

9305.751 : 1 083 35710.8 km1 : 1 444 476

10152.871 : 541 6785.4 km1 : 722 238

1176.4371 : 270 8392.7 km1 : 361 119

1238.2191 : 135 4191.35 km1 : 180 559

1319.1091 : 67 709677 m1 : 90 279

149.55461 : 33 854339 m1 : 45 139

154.77731 : 16 927169 m1 : 22 569

162.38871 : 8 46384.6 m1 : 11 284

171.19431 : 4 23142.3 m1 : 5 642

180.59721 : 2 11521.2 m1 : 2 821

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