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Hi,

This site

http://www.jstott.me.uk/jscoord/

has a great script to convert utm coordinates to lat/lng. However, I need to modify function below (taken from the script which can be download from above), to do mercator - lat/lng conversion.

Thanks for the help!!!

This site

http://www.jstott.me.uk/jscoord/

has a great script to convert utm coordinates to lat/lng. However, I need to modify function below (taken from the script which can be download from above), to do mercator - lat/lng conversion.

Thanks for the help!!!

```
function UTMRefToLatLng() {
var wgs84 = new RefEll(6378137, 6356752.314);
var UTM_F0 = 0.9996;
var a = wgs84.maj;
var eSquared = wgs84.ecc;
var ePrimeSquared = eSquared / (1.0 - eSquared);
var e1 = (1 - Math.sqrt(1 - eSquared)) / (1 + Math.sqrt(1 - eSquared));
var x = this.easting - 500000.0;;
var y = this.northing;
var zoneNumber = this.lngZone;
var zoneLetter = this.latZone;
var longitudeOrigin = (zoneNumber - 1.0) * 6.0 - 180.0 + 3.0;
// Correct y for southern hemisphere
if ((ord(zoneLetter) - ord("N")) < 0) {
y -= 10000000.0;
}
var m = y / UTM_F0;
var mu =
m
/ (a
* (1.0
- eSquared / 4.0
- 3.0 * eSquared * eSquared / 64.0
- 5.0
* Math.pow(eSquared, 3.0)
/ 256.0));
var phi1Rad =
mu
+ (3.0 * e1 / 2.0 - 27.0 * Math.pow(e1, 3.0) / 32.0) * Math.sin(2.0 * mu)
+ (21.0 * e1 * e1 / 16.0 - 55.0 * Math.pow(e1, 4.0) / 32.0)
* Math.sin(4.0 * mu)
+ (151.0 * Math.pow(e1, 3.0) / 96.0) * Math.sin(6.0 * mu);
var n =
a
/ Math.sqrt(1.0 - eSquared * Math.sin(phi1Rad) * Math.sin(phi1Rad));
var t = Math.tan(phi1Rad) * Math.tan(phi1Rad);
var c = ePrimeSquared * Math.cos(phi1Rad) * Math.cos(phi1Rad);
var r =
a
* (1.0 - eSquared)
/ Math.pow(
1.0 - eSquared * Math.sin(phi1Rad) * Math.sin(phi1Rad),
1.5);
var d = x / (n * UTM_F0);
var latitude = (
phi1Rad
- (n * Math.tan(phi1Rad) / r)
* (d * d / 2.0
- (5.0
+ (3.0 * t)
+ (10.0 * c)
- (4.0 * c * c)
- (9.0 * ePrimeSquared))
* Math.pow(d, 4.0)
/ 24.0
+ (61.0
+ (90.0 * t)
+ (298.0 * c)
+ (45.0 * t * t)
- (252.0 * ePrimeSquared)
- (3.0 * c * c))
* Math.pow(d, 6.0)
/ 720.0)) * (180.0 / Math.PI);
var longitude = longitudeOrigin + (
(d
- (1.0 + 2.0 * t + c) * Math.pow(d, 3.0) / 6.0
+ (5.0
- (2.0 * c)
+ (28.0 * t)
- (3.0 * c * c)
+ (8.0 * ePrimeSquared)
+ (24.0 * t * t))
* Math.pow(d, 5.0)
/ 120.0)
/ Math.cos(phi1Rad)) * (180.0 / Math.PI);
return new LatLng(latitude, longitude);
```

http://en.wikipedia.org/wiki/Mercator_projection

however, a conversion from UTM - Mercator would also work if anyone has that!!

To convert from the standard Mercator projection (x, y) coordinates to a latitude, longitude pair, do this :

lat = tan^-1(sinh(y))

lon = x + lon0

with lon0 the longitude of the center of the map. Just put that in JavaScript ...

lat = tan raised to the power of (-1 * (sin (h??) (y)))

what is h? is that right?

sinh is the hyperbolic sine - I'm not sure if that exists in JavaScript, but if not, you can use this equality :

sinh(x) = (exp(x) - exp(-x)) / 2

below is the javascript, but it does not seem to give the correct results. Im particularly curious about the forumla for lon -- how can lon be the mercator X coords + center lon? a typical mercator coordinate is

(x,y)

1851604, 331835

```
var sinh = (Math.exp(coordsY) - Math.exp(-coordsY)) / 2;
var degrees = centerLatLng.substring(0,centerLatLng.indexOf("Â°"))/1;
var minutes = centerLatLng.substring(centerLatLng.indexOf("Â°")+1,centerLatLng.indexOf("."))/ 60;
var seconds = centerLatLng.substring(centerLatLng.indexOf("."),centerLatLng.indexOf("'")) / 3600;
var centerLng = degrees + minutes + seconds;
var lat = Math.atan(sinh);
var lng = coordsX + centerLng;
```

You have to use the same units for the x,y and lat,lon values of course. I would prefer radians. (if they're not the same, you'll have to convert them).

That way, the center longitude of the Mercator map added to the Mercator x coordinate would then obviously give you the longitude.

>> 1851604, 331835

What's the unit ? That doesn't look like standard Mercator coordinates ...

Finally, also check what units the standard JavaScript functions expect ... Probably radians, but I'm not sure.

i think you're right about the radians things, Math.exp(coordsY) returns infinity....

That's for sure for such a high value of y lol.

>> ok, i'm a little rusty here, those coords are in meters.

That's not the standard Mercator projection afaik.

Luckily it's easy to convert meters (m) to radians (rad) :

rad = m / R

with R the radius of the Earth (average here) :

R = 6372795.477598 m

```
var rad = 6372795.477598;
var sinh = (Math.exp(coordsY/rad) - Math.exp(-coordsY/rad)) / 2;
alert(sinh);
var lat = Math.atan(sinh);
var degrees = centerLatLng.substring(0,centerLatLng.indexOf("Â°"))/1;
var minutes = centerLatLng.substring(centerLatLng.indexOf("Â°")+1,centerLatLng.indexOf("."))/ 60;
var seconds = centerLatLng.substring(centerLatLng.indexOf("."),centerLatLng.indexOf("'")) / 3600;
var centerLng = degrees + minutes + seconds;
var lng = (coordsX/rad) + (centerLng);
```

Depending on the input, that can be right ... What was the input ? Remember that the result is in radians, not degrees !!

>> the Math.atan() doesn't seem to change this number either-

For low values, that's normal.

Also :

>> var centerLng = degrees + minutes + seconds;

That has to be in radians too !! Convert degrees, minutes and seconds to radians.

For the example input you posted earlier :

>> 1851604, 331835

The result should be (with center longitude = 0) :

lon = 0.29054816 radians = 16.6471833 degrees

lat = 0.0520470412 radians = 2.9820758 degrees

here are the coords of the center of my map in mercator (this is western ny on my map)

1143031.34, 835295.19

the center longitude is 81.16691694444445 (and the correct center lat should be 42. etc)

the new code is below. none of these values correspond to what im expecting. when i reproject the map im using to utm, and use the above script, i get the correct lat/lng, so i think the mercator coords are correct.

```
var rad = 6372795.477598;
var sinh = (Math.exp(coordsY/rad) - Math.exp(-coordsY/rad)) / 2;
var lat = Math.atan(sinh)*(180/Math.PI);
var lngrad = (coordsX/rad) + ((centerLng)*(Math.PI/180))
var lng = (lngrad)*(180/Math.PI);
```

??? The center of the map should be the (0,0) point, and it should have a corresponding (lat,lon) pair, usually, the equator and the 0-meridian.

>> the center longitude is 81.16691694444445 (and the correct center lat should be 42. etc)

Are you sure this is a Mercator map ?

>> when i reproject the map im using to utm

Which is it now ... is it UTM as I suspected earlier ? Or is it a basic Mercator projection ?

yes, i am working with a gis internet mapping application, and it returns enough information for me to calculate the correct projected coodinates; it also returns the center lat/lng (but nothing more). In the GIS, the projection is indeed mercator. When i reproject the map to use UTM, the coords are very different. are there different versions of mercator? is there any way to check whether the coords im using represent what you think of as mercator?

A Mercator projection is just a type of projection. The original one is sometimes still used for maps

>> is there any way to check whether the coords im using represent what you think of as mercator?

Is this a world map ? If not, what does the map contain ? What are the maximum and minimum (Mercator) coordinates ? What's the center latitude/longitude ?

Can you also give a known correct pair of Mercator coordinates and the corresponding latitude/longitude ?

mercator

World Geodetic 1984 (WGS84) Auto

Center Latitude 0.00

Center Longitude 0.00

Local offset -9367812.99, 4954053.105

local scale 0.298582119, .29858213

scale correction 1.00, 1.00, 1.00

false easting/northing 0.00, 0.00

here are some coordinate pairs

1059121.38, 855806.52 || -81.3117, 42.3236

1661408.43, 1124318.39 || -79.6963, 42.8539

```
var rad = 6372795.477598;
var centerLng = (-9367812.99732577/rad);
var X = ((coordsX*0.2985821196282)-9367812.99732577);
var Y = coordsY*0.2985821374044+(4954053.105471344);
var sinh = (Math.exp(Y/rad) - Math.exp(-Y/rad)) / 2;
var lat = Math.atan(sinh)*(180/Math.PI);
var lngrad = (X/rad) + centerLng;
var lng = (lngrad)*(180/Math.PI);
```

I'm not sure, but I would suspect that that's due to rounding errors, either in the input data, or in the calculations. Or maybe the verification values are a bit off ?

http://en.wikipedia.org/wiki/Reference_ellipsoid

now its accurate to 4 decimal places!

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This is the offset to the center of the map, and with the earlier formulae, you get this corresponding latitude, longitude pair :

lat0 = atan(sinh(4954053.105/R)) = 0.70916 rad = 40.63195Â°

lon0 = -9367812.99/R + 0.0 = -1.46997 rad = -84.22303Â°

So, that's our center.

Now the coordinate pair examples you mentioned :

>> 1059121.38, 855806.52 || -81.3117, 42.3236

lat = atan(sinh(855806.52/R)) + 0.70916 = 0.84305 rad = 48.30321Â°

lon = 1059121.38/R - 1.46997 = -1.30378 rad = -74.70085Â°

>> 1661408.43, 1124318.39 || -79.6963, 42.8539

lat = atan(sinh(1124318.39/R)) + 0.70916 = 0.88468 rad = 50.68823Â°

lon = 1661408.43/R - 1.46997 = -1.20927 rad = -69.28588Â°

As you see, they're not a perfect match, so it's probably not standard Mercator. This seems to confirm it :

>> World Geodetic 1984 (WGS84) Auto