Movable Type Scripts

Vincenty formula for distance between two Latitude/Longitude points

Vincenty’s formula as it is used in the script:

Where:

• s is the distance (in the same units as a & b)
• α₁ is the initial bearing, or forward azimuth
• α₂ is the final bearing (in direction p₁→p₂)

Note: Vincenty observes that eqn. (18) becomes indeterminate over equatorial lines (since cos²α → 0); in this case, set cos2σ_m to 0 and the result is computed correctly. He also points out that the formula may have no solution between two nearly antipodal points; an iteration limit traps this case.

The most accurate and widely used globally-applicable model for the earth ellipsoid is WGS-84, used in this script. Other ellipsoids offering a better fit to the local geoid include Airy (1830) in the UK, International 1924 in much of Europe, Clarke (1880) in Africa, and GRS-67 in South America. America (NAD83) and Australia (GDA) use GRS-80, functionally equivalent to the WGS-84 ellipsoid.

Note that to locate latitude/longitude points on these ellipses, they are associated with specific datums: for instance, OSGB36 for Airy in the UK, ED50 for Int’l 1924 in Europe; WGS-84 defines a datum as well as an ellipse. See my convert coordinates page for converting points between different datums.

Some of the terms involved are explained in Ed Williams’ notes on Spheroid Geometry.

[Formulation updated Dec 05 to make it closer to Vincenty’s original and computationally more efficient.]

Notes:

• Trig functions take arguments in radians, so latitude, longitude, and bearings in degrees (either decimal or degrees/minutes/seconds) need to be converted to radians, rad = π.deg/180. When converting radians back to degrees (deg = 180.rad/π), West is negative if using signed decimal degrees. For bearings, values in the range -π to +π [-180° to +180°] need to be converted to 0 to +2π [0°–360°]; this can be done by (brng+2.π)%2.π [brng+360)%360] where % is the modulo operator. View page source to see JavaScript functions to handle these conversions.
• The atan2() function used here takes two arguments, atan2(y, x), and computes the arc tangent of the ratio y/x. It is more flexible than atan(y/x), since it handles x=0, and it also returns values in all 4 quadrants -π to +π (the atan function returns values in the range -π/2 to +π/2).
• If you implement any formula involving atan2 in Microsoft Excel, you will need to reverse the arguments, as Excel has them the opposite way around from JavaScript – conventional order is atan2(y, x), but Excel uses atan2(x, y). To use atan2 in a (VBA) macro, you can use WorksheetFunction.Atan2(). You will also have to rename the variables A, B / a, b, as VBA is case-insensitive.
• All bearings are with respect to true north, 0°=N, 90°=E, etc; if you are working from a compass, magnetic north varies from true north in a complex way around the earth, and the difference has to be compensated for by variances indicated on local maps.
• For miles, divide km by 1.609344
• For nautical miles, divide km by 1.852

JavaScript functions are shown below. You are welcome to re-use these scripts [without any warranty express or implied] provided you retain my copyright notice and when possible a link to my website (under a LGPL license). If you have any queries or find any problems, please contact me.
© 2002-2008 Chris Veness

/*
 * Calculate geodesic distance (in m) between two points specified by latitude/longitude (in numeric degrees)
 * using Vincenty inverse formula for ellipsoids
 */
function distVincenty(lat1, lon1, lat2, lon2) {
  var a = 6378137, b = 6356752.3142,  f = 1/298.257223563;  // WGS-84 ellipsiod
  var L = (lon2-lon1).toRad();
  var U1 = Math.atan((1-f) * Math.tan(lat1.toRad()));
  var U2 = Math.atan((1-f) * Math.tan(lat2.toRad()));
  var sinU1 = Math.sin(U1), cosU1 = Math.cos(U1);
  var sinU2 = Math.sin(U2), cosU2 = Math.cos(U2);
  
  var lambda = L, lambdaP = 2*Math.PI;
  var iterLimit = 20;
  while (Math.abs(lambda-lambdaP) > 1e-12 && --iterLimit>0) {
    var sinLambda = Math.sin(lambda), cosLambda = Math.cos(lambda);
    var sinSigma = Math.sqrt((cosU2*sinLambda) * (cosU2*sinLambda) + 
      (cosU1*sinU2-sinU1*cosU2*cosLambda) * (cosU1*sinU2-sinU1*cosU2*cosLambda));
    if (sinSigma==0) return 0;  // co-incident points
    var cosSigma = sinU1*sinU2 + cosU1*cosU2*cosLambda;
    var sigma = Math.atan2(sinSigma, cosSigma);
    var sinAlpha = cosU1 * cosU2 * sinLambda / sinSigma;
    var cosSqAlpha = 1 - sinAlpha*sinAlpha;
    var cos2SigmaM = cosSigma - 2*sinU1*sinU2/cosSqAlpha;
    if (isNaN(cos2SigmaM)) cos2SigmaM = 0;  // equatorial line: cosSqAlpha=0 (§6)
    var C = f/16*cosSqAlpha*(4+f*(4-3*cosSqAlpha));
    lambdaP = lambda;
    lambda = L + (1-C) * f * sinAlpha *
      (sigma + C*sinSigma*(cos2SigmaM+C*cosSigma*(-1+2*cos2SigmaM*cos2SigmaM)));
  }
  if (iterLimit==0) return NaN  // formula failed to converge

  var uSq = cosSqAlpha * (a*a - b*b) / (b*b);
  var A = 1 + uSq/16384*(4096+uSq*(-768+uSq*(320-175*uSq)));
  var B = uSq/1024 * (256+uSq*(-128+uSq*(74-47*uSq)));
  var deltaSigma = B*sinSigma*(cos2SigmaM+B/4*(cosSigma*(-1+2*cos2SigmaM*cos2SigmaM)-
    B/6*cos2SigmaM*(-3+4*sinSigma*sinSigma)*(-3+4*cos2SigmaM*cos2SigmaM)));
  var s = b*A*(sigma-deltaSigma);
  
  s = s.toFixed(3); // round to 1mm precision
  return s;
}


// extend String object with method for parsing degrees or lat/long values to numeric degrees
//
// this is very flexible on formats, allowing signed decimal degrees, or deg-min-sec suffixed by 
// compass direction (NSEW). A variety of separators are accepted (eg 3º 37' 09"W) or fixed-width 
// format without separators (eg 0033709W). Seconds and minutes may be omitted. (Minimal validation 
// is done).

String.prototype.parseDeg = function() {
  if (!isNaN(this)) return Number(this);                 // signed decimal degrees without NSEW

  var degLL = this.replace(/^-/,'').replace(/[NSEW]/i,'');  // strip off any sign or compass dir'n
  var dms = degLL.split(/[^0-9.]+/);                     // split out separate d/m/s
  for (var i in dms) if (dms[i]=='') dms.splice(i,1);    // remove empty elements (see note below)
  switch (dms.length) {                                  // convert to decimal degrees...
    case 3:                                              // interpret 3-part result as d/m/s
      var deg = dms[0]/1 + dms[1]/60 + dms[2]/3600; break;
    case 2:                                              // interpret 2-part result as d/m
      var deg = dms[0]/1 + dms[1]/60; break;
    case 1:                                              // decimal or non-separated dddmmss
      if (/[NS]/i.test(this)) degLL = '0' + degLL;       // - normalise N/S to 3-digit degrees
      var deg = dms[0].slice(0,3)/1 + dms[0].slice(3,5)/60 + dms[0].slice(5)/3600; break;
    default: return NaN;
  }
  if (/^-/.test(this) || /[WS]/i.test(this)) deg = -deg; // take '-', west and south as -ve
  return deg;
}
// note: whitespace at start/end will split() into empty elements (except in IE)

// extend Number object with methods for converting degrees/radians

Number.prototype.toRad = function() {  // convert degrees to radians
  return this * Math.PI / 180;
}


<p>Lat 1: <input name="lat1" value="53 09 02N"> Long 1: <input name="long1" value="001 50 40W"></p>
<p>Lat 2: <input name="lat2" value="52 12 19N"> Long 2: <input name="long2" value="000 08 33W"></p>

<input type="button" value="calculate distance" 
    onClick="result.value = distVincenty(lat1.value.parseDeg(), long1.value.parseDeg(), 
                                         lat2.value.parseDeg(), long2.value.parseDeg()) + ' m'">
<input name="result" value="">