I work quite a lot with location based information. More often than not (don't ask why) I get given geographical references in Northings and Eastings (known as Cartesian) rather than latitude and longitude.
After a long time a friend of mine sent me over the calculations to convert cartesian northings and eastings into latitude and longitude in PHP.
Here's the class I use to convert these values:
PHP Code:
<?php
/**
* App_Coordinates_From_Cartesian, a geographical class
*
* ******************************************************************
*
* Converts Cartesian geographical values into geographic coordinates
*
* ******************************************************************
*
* @author Roger E Thomas
* @package App_
* @subpackage App_Coordinates_From_
* @copyright 2012 Roger E Thomas | http://www.rogerethomas.com
*
* ******************************************************************
*
*
*/
class App_Coordinates_From_Cartesian {
/**
* Holds the final Latitude
* @var Integer
*/
public $latitude;
/**
* Holds the final Longitude
* @var Integer
*/
public $longitude;
/**
* Holds the initial easting
* @var Integer
*/
public $easting;
/**
* Holds the initial northing
* @var Integer
*/
public $northing;
function __construct($easting, $northing)
{
if (is_int($easting) && is_int($northing)) {
$this->easting = $easting;
$this->northing = $northing;
}
else {
$this->easting = $easting;
$this->northing = $northing;
}
}
/**
* Uses the $this->northing and $this->easting set in __construct to return the
* latitude and longitude of the final point.
* @return array
*/
function Convert() {
$East = $this->easting;
$North = $this->northing;
if ($East == "" || $East == 0 || $North == "" || $North == 0) {
$this->latitude = 0;
$this->longitude = 0;
return array('latitude' => 0, 'longitude' => 0);
}
$a = 6377563.396; // Semi-major axis, a
$b = 6356256.910; //Semi-minor axis, b
$e0 = 400000.000; //True origin Easting, E0
$n0 = -100000.000; //True origin Northing, N0
$f0 = 0.999601271700; //Central Meridan Scale, F0
$PHI0 = 49.0; // True origin latitude, j0
$LAM0 = -2.0; // True origin longitude, l0
//Convert angle measures to radians
$RadPHI0 = $PHI0 * (M_PI / 180);
$RadLAM0 = $LAM0 * (M_PI / 180);
//Compute af0, bf0, e squared (e2), n and Et
$af0 = $a * $f0;
$bf0 = $b * $f0;
$e2 = ($af0*$af0 - $bf0*$bf0 ) / ($af0*$af0);
$n = ($af0 - $bf0) / ($af0 + $bf0);
$Et = $East - $e0;
//Compute initial value for latitude (PHI) in radians
$PHId = $this->_initialLatitude($North, $n0, $af0, $RadPHI0, $n, $bf0);
$sinPHId2 = pow(sin($PHId), 2);
$cosPHId = pow(cos($PHId), -1);
$tanPHId = tan($PHId);
$tanPHId2 = pow($tanPHId, 2);
$tanPHId4 = pow($tanPHId, 4);
$tanPHId6 = pow($tanPHId, 6);
//Compute nu, rho and eta2 using value for PHId
$nu = $af0 / (sqrt(1 - ($e2 * $sinPHId2)));
$rho = ($nu * (1 - $e2)) / (1 - $e2 * $sinPHId2);
$eta2 = ($nu / $rho) - 1;
//Compute Longitude
$X = $cosPHId / $nu;
$XI = $cosPHId / ( 6 * pow($nu, 3)) * (($nu / $rho) + 2 * $tanPHId2);
$XII = $cosPHId / ( 120 * pow($nu, 5)) * (5 + 28 * $tanPHId2 + 24 * $tanPHId4);
$XIIA = $cosPHId / (5040 * pow($nu, 7)) * (61 + 662 * $tanPHId2 + 1320 * $tanPHId4 + 720 * $tanPHId6);
$VII = $tanPHId / ( 2 * $rho * $nu);
$VIII = $tanPHId / ( 24 * $rho * pow($nu, 3)) * ( 5 + 3 * $tanPHId2 + $eta2 - 9 * $eta2 * $tanPHId2 );
$IX = $tanPHId / (720 * $rho * pow($nu, 5)) * (61 + 90 * $tanPHId2 + 45 * $tanPHId4 );
$long = (180 / M_PI) * ($RadLAM0 + ($Et * $X) - pow($Et,3) * $XI + pow($Et,5) * $XII - pow($Et,7) * $XIIA);
$lat = (180 / M_PI) * ($PHId - (pow($Et,2) * $VII) + (pow($Et, 4) * $VIII) - (pow($Et, 6) * $IX));
$this->latitude = $lat;
$this->longitude = $long;
return array('latitude' => $lat, 'longitude' => $long);
}
/**
* Helper function to compute meridional arc.
* @param $bf0 ellipsoid semi major axis multiplied by central meridian scale factor (bf0) in meters;
* @param $n n (computed from a, b and f0);
* @param $PHI0 lat of false origin
* @param $PHI initial or final latitude of point IN RADIANS.
*/
private function _meridianArc($bf0, $n, $PHI0, $PHI) {
$n2 = pow($n, 2);
$n3 = pow($n, 3);
$ans = ((1 + $n + ((5 / 4) * ($n2)) + ((5 / 4) * $n3)) * ($PHI - $PHI0));
$ans -= (((3 * $n) + (3 * $n2) + ((21 / 8) * $n3)) * (sin($PHI - $PHI0)) * (cos($PHI + $PHI0)));
$ans += ((((15 / 8) * $n2) + ((15 / 8) * $n3)) * (sin(2 * ($PHI - $PHI0))) * (cos(2 * ($PHI + $PHI0))));
$ans -= (((35 / 24) * $n3) * (sin(3 * ($PHI - $PHI0))) * (cos(3 * ($PHI + $PHI0))));
return $bf0 * $ans;
}
/**
* Helper function to compute initial value for Latitude IN RADIANS.
* @param $North northing of point
* @param $n0 northing of false origin in meters;
* @param $afo semi major axis multiplied by central meridian scale factor in meters;
* @param $PHI0 latitude of false origin IN RADIANS;
* @param $n computed from a, b and f0
* @param $bfo ellipsoid semi major axis multiplied by central meridian scale factor in meters.
*/
private function _initialLatitude($North, $n0, $afo, $PHI0, $n, $bfo) {
//First PHI value (PHI1)
$PHI1 = (($North - $n0) / $afo) + $PHI0;
//Calculate M
$M = $this->_meridianArc($bfo, $n, $PHI0, $PHI1);
//Calculate new PHI value (PHI2)
$PHI2 = (($North - $n0 - $M) / $afo) + $PHI1;
//Iterate to get final value for InitialLat
while ( abs($North - $n0 - $M) > 0.00001 ) {
$PHI2 = (($North - $n0 - $M) / $afo) + $PHI1;
$M = $this->_meridianArc($bfo, $n, $PHI0, $PHI2);
$PHI1 = $PHI2;
}
return $PHI2;
}
}
To use the class all you need to know are the northings and eastings.
PHP Code:
<?php
$class = new App_Coordinates_From_Cartesian(464343, 100366);
$result = $class->Convert();
var_dump($result);
// array(2) {
// ["latitude"]=>
// float(50.798714617948)
// ["longitude"]=>
// float(-1.0868854103489)
// }
You can see that the return from convert is an array containing 2 keys. latitude and longitude!
The returned values point to somewhere in Portsmouth. http://maps.google.co.uk/maps?q=50.798714617948+-1.0868854103489
Regardless of those northings and eastings, the fact that we get the correct latitude and longitude back is a big plus.
Let me know if this helps you out (or if you manage to simplify the calculations).
Joaquim commented on Jan 22nd 2013