Slippy map tilenames

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This article describes the file naming conventions for the Slippy Map application.

  • Tiles are 256 × 256 pixel PNG files
  • Each zoom level is a directory, each column is a subdirectory, and each tile in that column is a file
  • Filename(url) format is /zoom/x/y.png

The slippy map expects tiles to be served up at URLs following this scheme, so all tile server URLs look pretty similar.

Tile servers

The first part of the URL specifies the tile server, and perhaps other parameters which might influence the style.

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:

Name URL template zoomlevels
OSM 'standard' style http://[abc].tile.openstreetmap.org/zoom/x/y.png 0-19
OpenCycleMap http://[abc].tile.opencyclemap.org/cycle/zoom/x/y.png 0-18
OpenCycleMap Transport (experimental) http://[abc].tile2.opencyclemap.org/transport/zoom/x/y.png 0-18
MapQuest http://otile[1234].mqcdn.com/tiles/1.0.0/osm/zoom/x/y.jpg 0-19
MapQuest Open Aerial http://otile[1234].mqcdn.com/tiles/1.0.0/sat/zoom/x/y.jpg 0-11 globally, 12+ in the U.S.
Migurski's Terrain http://tile.stamen.com/terrain-background/zoom/x/y.jpg 4-18, 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 level tile coverage number of tiles tile size in degrees
0 1 tile covers whole world 1 tile 360° x 170.1022°
1 2 × 2 tiles 4 tiles 180° x 85.0511°
2 4 × 4 tiles 16 tiles 90° x 42.5256°
n 2n × 2n tiles 22n tiles 360/2n° x 170.1022/2n°
12 4096 x 4096 tiles 16 777 216 0.0879° x 0.0415°
16 232 = 4 294 967 296 tiles
17 17 179 869 184 tiles
18 68 719 476 736 tiles
19 Maximum zoom for Mapnik layer 274 877 906 944 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. See also the Osmarender bug.

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/cos
arsinh(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 ^ zoom
xtile = n * ((lon_deg + 180) / 360)
ytile = n * (1 - (log(tan(lat_rad) + sec(lat_rad)) / π)) / 2

Tile numbers to lon./lat.

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

Mathematics

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

Latlon to tile.pngTile to latlon.png

Python

Lon./lat. to tile numbers

import math
def 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 math
def 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.

Another python implementation

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

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);
}

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)));
}

Go

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

C#

 
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)
 
-- Example
main = 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 osmTileRef

function 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 & iLong
end osmTileCoords

Mathematica

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} // N

Tcl

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;

R

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 zooms
deg2num.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 radians
rho = pi/180;
lon_rad = lon_deg * rho;
lat_rad = lat_deg * rho;
 
n = 2 ^ zoom
xtile = n * ((lon_deg + 180) / 360)
ytile = n * (1 - (log(tan(lat_rad) + sec(lat_rad)) / pi)) / 2

Tile numbers to lon./lat.

n=2^zoom
lon_deg = xtile / n * 360.0 - 180.0
lat_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) (- (/ (* x 360) (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, ytile
end
 
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_deg
end

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, 2y 2x + 1, 2y
2x, 2y + 1 2x + 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 156543.034 meters:

6378137.0 * 2 * pi / 256 = 156543.034

Which gives us a formula to calculate resolution at any given zoom:

resolution = 156543.034 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).

zoom resolution, m/px scale 96 dpi 1 screen cm is scale 120 dpi
0 156543.03 1 : 554 678 932 5547 km 1 : 739 571 909
1 78271.52 1 : 277 339 466 2773 km 1 : 369 785 954
2 39135.76 1 : 138 669 733 1337 km 1 : 184 892 977
3 19567.88 1 : 69 334 866 693 km 1 : 92 446 488
4 9783.94 1 : 34 667 433 347 km 1 : 46 223 244
5 4891.97 1 : 17 333 716 173 km 1 : 23 111 622
6 2445.98 1 : 8 666 858 86.7 km 1 : 11 555 811
7 1222.99 1 : 4 333 429 43.3 km 1 : 5 777 905
8 611.50 1 : 2 166 714 21.7 km 1 : 2 888 952
9 305.75 1 : 1 083 357 10.8 km 1 : 1 444 476
10 152.87 1 : 541 678 5.4 km 1 : 722 238
11 76.437 1 : 270 839 2.7 km 1 : 361 119
12 38.219 1 : 135 419 1.4 km 1 : 180 559
13 19.109 1 : 67 709 677 m 1 : 90 279
14 9.5546 1 : 33 854 339 m 1 : 45 139
15 4.7773 1 : 16 927 169 m 1 : 22 569
16 2.3887 1 : 8 463 84.6 m 1 : 11 284
17 1.1943 1 : 4 231 42.3 m 1 : 5 642
18 0.5972 1 : 2 115 21.2 m 1 : 2 821

See also Zoom levels

Tools

References

(note: Slippy tiles and Google map tiles count tile 0,0 down from the top-left of the tile grid; the TMS spec specifies tiles count up from 0,0 in the lower-left!)