Slippy map tilenames

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. The tile coordinates are typically specified by /zoom/x/y.png tail. Some tileservers will use a directory (e.g. "/cycle/") to specify a particular stylesheet. (Historically several subdomains were often provided to get around browser limitations on the number of simultaneous HTTP connections to each host - such as a.tile, b.tile, c.tile - but this is less important with modern browsers.)

Here are some examples:

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.

(*) While the width (longitude) in degrees is constant, given a zoom level, for all tiles, this does not happen for the height. In general, tiles belonging to the same row have equal height in degrees, but it decreases moving from the equator to the poles.

See Zoom levels for more details

X and Y

• X goes from 0 (left edge is 180 °W) to 2^zoom − 1 (right edge is 180 °E)
• Y goes from 0 (top edge is 85.0511 °N) to 2^zoom − 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

The following is identical to the well-known Web Mercator projection.

• Reproject the coordinates to the Spherical 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 2^zoom
• Multiply x and y by n. Round results down to give tilex and tiley.

Implementations

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))

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

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

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.asinh(math.tan(lat_rad)) / math.pi) / 2.0 * n)
  return (xtile, ytile)

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)

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

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

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);
}

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;
}

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:
# https://www.openstreetmap.org/?lat=43.731049999999996&lon=15.79375&zoom=13&layers=M
# to OSM "Export" iframe embedded bbox format like:
# https://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;
}

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

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

<cfscript>
    function longitude2tile(longitude, zoom) {
	    return floor((longitude + 180) / 360 * (2 ^ zoom));
	    }

    function latitude2tile(latitude, zoom) {
	    return floor((1 - log(tan(latitude * pi() / 180) + 1 / cos(latitude * pi() / 180)) / pi()) / 2 * (2 ^ zoom));
	    }

    xtile = longitude2tile(longitude, zoom);
    ytile = latitude2tile(latitude, zoom);
</cfscript>

<cffunction name="longitude2tile" output="no" returntype="numeric">
	<cfargument name="longitude" type="numeric" required="yes" />
	<cfargument name="zoom" type="numeric" required="yes" />
	<cfreturn floor((arguments.longitude + 180) / 360 * (2 ^ arguments.zoom)) />
</cffunction>

<cffunction name="latitude2tile" output="no" returntype="numeric">
	<cfargument name="latitude" type="numeric" required="yes" />
	<cfargument name="zoom" type="numeric" required="yes" />
	<cfreturn floor((1 - log(tan(arguments.latitude * pi() / 180) + 1 / cos(arguments.latitude * pi() / 180)) / pi()) / 2 * (2 ^ arguments.zoom)) />
</cffunction>

<cfset xtile = longitude2tile(longitude, zoom) />
<cfset ytile = latitude2tile(latitude, zoom) />

<cfscript>
    function tile2longitude(xtile, zoom) {
	    return (xtile / (2 ^ zoom) * 360 - 180);
	    }

    function tile2latitude(ytile, zoom) {
	    var n = pi() - 2 * pi() * ytile / (2 ^ zoom);
	    return (180 / pi() * atn(0.5 * (exp(n) - exp(-n))));
	    }

    longitude = tile2longitude(xtile, zoom);
    latitude = tile2latitude(ytile, zoom);
</cfscript>

<cffunction name="tile2longitude" output="no" returntype="numeric">
	<cfargument name="xtile" type="numeric" required="yes" />
	<cfargument name="zoom" type="numeric" required="yes" />
	<cfreturn (arguments.xtile / (2 ^ arguments.zoom) * 360 - 180) />
</cffunction>

<cffunction name="tile2latitude" output="no" returntype="numeric">
	<cfargument name="ytile" type="numeric" required="yes" />
	<cfargument name="zoom" type="numeric" required="yes" />
	<cfset var n = pi() - 2 * pi() * arguments.ytile / (2 ^ arguments.zoom) />
	<cfreturn (180 / pi() * atn(0.5 * (exp(n) - exp(-n)))) />
</cffunction>

<cfset longitude = tile2longitude(xtile, zoom) />
<cfset latitude = tile2latitude(ytile, zoom) />

 function lon2tile(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))); }

 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))));
 }

var zoom        = 9;
var top_tile    = lat2tile(north_edge, zoom); // eg.lat2tile(34.422, 9);
var left_tile   = lon2tile(west_edge, zoom);
var bottom_tile = lat2tile(south_edge, zoom);
var right_tile  = lon2tile(east_edge, zoom);
var width       = Math.abs(left_tile - right_tile) + 1;
var height      = Math.abs(top_tile - bottom_tile) + 1;

// total tiles
var total_tiles = width * height; // -> eg. 377

const EARTH_CIR_METERS = 40075016.686;
const degreesPerMeter = 360 / EARTH_CIR_METERS;

function toRadians(degrees) {
  return degrees * Math.PI / 180;
};

function latLngToBounds(lat, lng, zoom, width, height){ // width and height must correspond to the iframe width/height
  const metersPerPixelEW = EARTH_CIR_METERS / Math.pow(2, zoom + 8);
  const metersPerPixelNS = EARTH_CIR_METERS / Math.pow(2, zoom + 8) * Math.cos(toRadians(lat));

  const shiftMetersEW = width/2 * metersPerPixelEW;
  const shiftMetersNS = height/2 * metersPerPixelNS;

  const shiftDegreesEW = shiftMetersEW * degreesPerMeter;
  const shiftDegreesNS = shiftMetersNS * degreesPerMeter;

  return {
    south: lat-shiftDegreesNS,
    west: lng-shiftDegreesEW,
    north: lat+shiftDegreesNS,
    east: lng+shiftDegreesEW
  }
}

// Usage Example: create the src attribute for Open Street Map:
const bb = latLngToBounds(latitude,longitude,zoom,width,height); // e.g. latLngToBounds(47,12,16,450,350)

const src = [
  "https://www.openstreetmap.org/export/embed.html?bbox=",
  bb.west,
  ",",
  bb.south,
  ",",
  bb.east,
  ",",
  bb.north,
  "&layer=mapnik&marker=",
  latitude,
  ",",
  longitude,
].join('');

int long2tilex(double lon, int z) 
{ 
	return (int)(floor((lon + 180.0) / 360.0 * (1 << z))); 
}

int lat2tiley(double lat, int z)
{ 
    double latrad = lat * M_PI/180.0;
	return (int)(floor((1.0 - asinh(tan(latrad)) / M_PI) / 2.0 * (1 << z))); 
}

double tilex2long(int x, int z) 
{
	return x / (double)(1 << z) * 360.0 - 180;
}

double tiley2lat(int y, int z) 
{
	double n = M_PI - 2.0 * M_PI * y / (double)(1 << z);
	return 180.0 / M_PI * atan(0.5 * (exp(n) - exp(-n)));
}

int long2tilex(double lon, int z)
{
    return (int)(Math.Floor((lon + 180.0) / 360.0 * (1 << z)));
}

int lat2tiley(double lat, int z)
{
    return (int)Math.Floor((1 - Math.Log(Math.Tan(ToRadians(lat)) + 1 / Math.Cos(ToRadians(lat))) / Math.PI) / 2 * (1 << z));
}

double tilex2long(int x, int z)
{
    return x / (double)(1 << z) * 360.0 - 180;
}

double tiley2lat(int y, int z)
{
    double n = Math.PI - 2.0 * Math.PI * y / (double)(1 << z);
    return 180.0 / Math.PI * Math.Atan(0.5 * (Math.Exp(n) - Math.Exp(-n)));
}

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) {
	n := math.Exp2(float64(z))
	x = int(math.Floor((lon + 180.0) / 360.0 * n))
	if float64(x) >= n {
		x = int(n - 1)
	}
	y = int(math.Floor((1.0 - math.Log(math.Tan(lat*math.Pi/180.0)+1.0/math.Cos(lat*math.Pi/180.0))/math.Pi) / 2.0 * n))
	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
}

 public class slippytest {
 public static void main(String[] args) {
   int zoom = 10;
   double lat = 47.968056d;
   double lon = 7.909167d;
   System.out.println("https://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);
   }
 }

  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)));
  }

import kotlin.math.*

fun getXYTile(lat : Double, lon: Double, zoom : Int) : Pair<Int, Int> {
        val latRad = Math.toRadians(lat)
        var xtile = floor( (lon + 180) / 360 * (1 shl zoom) ).toInt()
        var ytile = floor( (1.0 - asinh(tan(latRad)) / PI) / 2 * (1 shl zoom) ).toInt()

        if (xtile < 0) {
            xtile = 0
        }
        if (xtile >= (1 shl zoom)) {
            xtile= (1 shl zoom) - 1
        }
        if (ytile < 0) {
            ytile = 0
        }
        if (ytile >= (1 shl zoom)) {
            ytile = (1 shl zoom) - 1
        }

        return Pair(xtile, ytile)
}

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;
}

<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>

-- https://github.com/apeyroux/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 $ "https://tile.openstreetmap.org/17/" ++ show x ++ "/" ++ show y ++ ".png"
        where
                z = 17
                x = long2tilex 2.2712 z
                y = lat2tiley 48.8152 z

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("https://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
// ==> https://tile.openstreetmap.org/17/65544/43582.png

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

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

package require map::slippy

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

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

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;

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;

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("https://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('https://a.tile.openstreetmap.org/', zoom, '/', num[1], '/', num[2], '.png', sep='')
  }
  return(nums)
}

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]
# ------------------------------------

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]";

% 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

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

(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))

-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.

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

CREATE OR REPLACE FUNCTION lon2tile(lon DOUBLE PRECISION, zoom INTEGER)
  RETURNS INTEGER AS
$BODY$
    SELECT FLOOR( (lon + 180) / 360 * (1 << zoom) )::INTEGER;
$BODY$
  LANGUAGE SQL IMMUTABLE;

CREATE OR REPLACE FUNCTION lat2tile(lat double precision, zoom integer)
  RETURNS integer AS
$BODY$
    SELECT floor( (1.0 - ln(tan(radians(lat)) + 1.0 / cos(radians(lat))) / pi()) / 2.0 * (1 << zoom) )::integer;
$BODY$
  LANGUAGE sql 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
$BODY$
 SELECT CAST(x * 1.0 / (1 << zoom) * 360.0 - 180.0 AS double precision);
$BODY$
  LANGUAGE sql IMMUTABLE;

+(NSString*) transformWorldCoordinateToTilePathForZoom:(int)zoom fromLon:(double) lon  fromLat:(double) lat
{
    int tileX = (int)(floor((lon + 180.0) / 360.0 * pow(2.0, zoom)));
    int tileY = (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, zoom)));
    NSString * path = [NSString stringWithFormat:@"%d/%d/%d",zoom,tileX,tileY];
    return path;
}

func tranformCoordinate(_ latitude: Double, _ longitude: Double, withZoom zoom: Int) -> (x: Int, y: Int) {
    let tileX = Int(floor((longitude + 180) / 360.0 * pow(2.0, Double(zoom))))
    let tileY = Int(floor((1 - log( tan( latitude * Double.pi / 180.0 ) + 1 / cos( latitude * Double.pi / 180.0 )) / Double.pi ) / 2 * pow(2.0, Double(zoom))))
    
    return (tileX, tileY)
}

    func tileToLatLon(tileX : Int, tileY : Int, mapZoom: Int) -> (lat_deg : Double, lon_deg : Double) {
        let n : Double = pow(2.0, Double(mapZoom))
        let lon = (Double(tileX) / n) * 360.0 - 180.0
        let lat = atan( sinh (.pi - (Double(tileY) / n) * 2 * Double.pi)) * (180.0 / .pi)
        
        return (lat, lon)
    }

(defn tile [lat lon zoom]
  (let [zoom-shifted (bit-shift-left 1 zoom)
        lat-radians (Math/toRadians lat)
        xtile (int (Math/floor (* (/ (+ 180 lon) 360) zoom-shifted)))
        ytile (int (Math/floor (* (/ (- 1
                                        (/
                                          (Math/log (+ (Math/tan lat-radians)
                                                       (/ 1 (Math/cos lat-radians))))
                                          Math/PI))
                                     2)
                                  zoom-shifted)))]
    (str zoom
         "/"
         (cond (< xtile 0) 0
               (>= xtile zoom-shifted) (- zoom-shifted 1)
               :else xtile)
         "/"
         (cond (< ytile 0) 0
               (>= ytile zoom-shifted) (- zoom-shifted 1)
               :else ytile))))

lng2tile(lng, zoom) = floor((lng+180)/360*2^zoom)
lat2tile(lat, zoom) = floor((1-log(tan(lat*pi/180)+1/cos(lat*pi/180))/pi)/2*2^zoom)

tile2lng(x, z) = (x/2^z*360)-180
tile2lat(y, z) = 180/pi*atan(0.5*(exp(pi-2*pi*y/2^z)-exp(2*pi*y/2^z-pi)))

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):

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. At zoom 0, one pixel would equal 156543.03 meters (assuming a tile size of 256 px):

40075.016686 * 1000 / 256 ≈ 6378137.0 * 2 * pi / 256 ≈ 156543.03

Which 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 * 1/0.0254 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).

See also Zoom levels

Tools

• Javascript Example: Tilesname WebCalc V1.0
• Geo-OSM-Tiles: a Perl module that calculates tile numbers along with a script that downloads map tiles
• Kachelbrowser
• feuille de calcul openoffice (sheet)
• Geofabrik map showing tile grid and coordinates on the map
• Same as above plus Tiles preview and direct link to Bigmap

References

• http://code.google.com/apis/maps/documentation/overlays.html#Google_Maps_Coordinates
• http://cfis.savagexi.com/articles/2006/05/03/google-maps-deconstructed
• "Google Map" projection, see Spatialreference.org
• OSM mailing list refering to this page.
• Setting up TMS
• TMS specification from the OSGeo Foundation

(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!)