CAMS Scripts

This page presents a variety of calculations for latitude/longitude points, with the formulæ and code fragments for implementing them.

Enter the co-ordinates into the text boxes to try out the calculations. A variety of formats are accepted, principally:

• deg-min-sec suffixed with N/S/E/W or +/-(e.g. 40°44′55″N, 73 59 11W), or
• deg min sec (using spaces instead)
• signed decimal degrees without compass direction, where negative indicates west/south
  (e.g. 40.7486, -73.9864):

All these formulæ are for calculations on the basis of a spherical earth (ignoring ellipsoidal effects) – which is accurate enough^* for most purposes… [In fact, the earth is very slightly ellipsoidal; using a spherical model gives errors typically up to 0.3% – see notes for further details].

Distance

This uses the "haversine" formula to calculate the great-circle distance between two points – that is, the shortest distance over the earth’s surface – giving an "as-the-crow-flies" distance between the points (ignoring any hills they fly over, of course!).

var R = 6378; // km
var φ1 = lat1.toRadians();
var φ2 = lat2.toRadians();
var Δφ = (lat2-lat1).toRadians();
var Δλ = (lon2-lon1).toRadians();

var a = Math.sin(Δφ/2) * Math.sin(Δφ/2) +
		Math.cos(φ1) * Math.cos(φ2) *
		Math.sin(Δλ/2) * Math.sin(Δλ/2);
		
var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a));

var d = R * c;

Note in these scripts, I generally use lat/lon for latitude/longitude in degrees, and φ/λ for latitude/longitude in radians – having found that mixing degrees & radians is often the easiest route to head-scratching bugs...

The haversine formula¹ "remains particularly well-conditioned for numerical computa­tion even at small distances" – unlike calculations based on the spherical law of cosines. The "versed sine" is 1−cosθ, and the "half-versed-sine" is (1−cosθ)/2 = sin²(θ/2) as used above. Once widely used by navigators, it was described by Roger Sinnott in Sky & Telescope magazine in 1984 (“Virtues of the Haversine”): Sinnott explained that the angular separation between Mizar and Alcor in Ursa Major – 0°11′49.69″ – could be accurately calculated on a TRS-80 using the haversine.

For the curious, c is the angular distance in radians, and a is the square of half the chord length between the points. A (remarkably marginal) performance improvement can be obtained, of course, by factoring out the terms which get squared.

Spherical Law of Cosines

In fact, JavaScript (and most modern computers & languages) use "IEEE 754" 64-bit floating-point numbers, which provide 15 significant figures of precision. By my estimate, with this precision, the simple spherical law of cosines formula (cos c = cos a cos b + sin a sin b cos C) gives well-conditioned results down to distances as small as a few metres on the Earth’s surface. (Note that the geodetic form of the law of cosines is rearranged from the canonical one so that the latitude can be used directly, rather than the colatitude).

This makes the simpler law of cosines a reasonable 1-line alternative to the haversine formula for many purposes. The choice may be driven by coding context, available trig functions (in different languages), etc – and, for very small distances an equirectangualar approximation may be more suitable.

var φ1 = lat1.toRadians(), φ2 = lat2.toRadians(), Δλ = (lon2-lon1).toRadians(), R = 6371; // gives d in km
var d = Math.acos( Math.sin(φ1)*Math.sin(φ2) + Math.cos(φ1)*Math.cos(φ2) * Math.cos(Δλ) ) * R;

Equirectangular approximation

If performance is an issue and accuracy less important, for small distances Pythagoras’ theorem can be used on an equirectangular projection:^*

var x = (λ2-λ1) * Math.cos((φ1+φ2)/2);
var y = (φ2-φ1);
var d = Math.sqrt(x*x + y*y) * R;

This uses just one trig and one sqrt function – as against half-a-dozen trig functions for cos law, and 7 trigs + 2 sqrts for haversine. Accuracy is somewhat complex: along meridians there are no errors, otherwise they depend on distance, bearing, and latitude, but are small enough for many purposes^* (and often trivial compared with the spherical approximation itself).

Alternatively, the polar coordinate flat-Earth formula can be used: using the co-latitudes θ₁ = π/2−φ₁ and θ₂ = π/2−φ₂, then d = R ⋅ √θ₁² + θ₂² − 2 ⋅ θ₁ ⋅ θ₂ ⋅ cos Δλ. I’ve not compared accuracy.

Bearing

In general, your current heading will vary as you follow a great circle path (orthodrome); the final heading will differ from the initial heading by varying degrees according to distance and latitude (if you were to go from say 35°N,45°E (≈ Baghdad) to 35°N,135°E (≈ Osaka), you would start on a heading of 60° and end up on a heading of 120°!).

This formula is for the initial bearing (sometimes referred to as forward azimuth) which if followed in a straight line along a great-circle arc will take you from the start point to the end point:¹

var y = Math.sin(λ2-λ1) * Math.cos(φ2);
var x = Math.cos(φ1)*Math.sin(φ2) -
        Math.sin(φ1)*Math.cos(φ2)*Math.cos(λ2-λ1);
var brng = Math.atan2(y, x).toDegrees();

Since atan2 returns values in the range -π ... +π (that is, -180° ... +180°), to normalise the result to a compass bearing (in the range 0° ... 360°, with −ve values transformed into the range 180° ... 360°), convert to degrees and then use (θ+360) % 360, where % is modulo.

For final bearing, simply take the initial bearing from the end point to the start point and reverse it (using θ = (θ+180) % 360).

Midpoint

This is the half-way point along a great circle path between the two points.¹

var Bx = Math.cos(φ2) * Math.cos(λ2-λ1);
var By = Math.cos(φ2) * Math.sin(λ2-λ1);
var φ3 = Math.atan2(Math.sin(φ1) + Math.sin(φ2),
                    Math.sqrt( (Math.cos(φ1)+Bx)*(Math.cos(φ1)+Bx) + By*By ) );
var λ3 = λ1 + Math.atan2(By, Math.cos(φ1) + Bx);

Just as the initial bearing may vary from the final bearing, the midpoint may not be located half-way between latitudes/longitudes; the midpoint between 35°N,45°E and 35°N,135°E is around 45°N,90°E.

var φ2 = Math.asin( Math.sin(φ1)*Math.cos(d/R) +
                    Math.cos(φ1)*Math.sin(d/R)*Math.cos(brng) );
var λ2 = λ1 + Math.atan2(Math.sin(brng)*Math.sin(d/R)*Math.cos(φ1),
                         Math.cos(d/R)-Math.sin(φ1)*Math.sin(φ2));

For final bearing, simply take the initial bearing from the end point to the start point and reverse it (using θ = (θ+180) % 360).

This is a lot simpler using vectors rather than spherical trigonometry: see latlong-vectors.html.

Cross-track distance

Here’s a new one: I’ve sometimes been asked about distance of a point from a great-circle path (sometimes called cross track error).

var dXt = Math.asin(Math.sin(d13/R)*Math.sin(θ13-θ12)) * R;

Here, the great-circle path is identified by a start point and an end point – depending on what initial data you’re working from, you can use the formulæ above to obtain the relevant distance and bearings. The sign of d_xt tells you which side of the path the third point is on.

The along-track distance, from the start point to the closest point on the path to the third point, is

var dAt = Math.acos(Math.cos(d13/R)/Math.cos(dXt/R)) * R;

Closest point to the poles

And: "Clairaut’s formula" will give you the maximum latitude of a great circle path, given a bearing θ and latitude φ on the great circle:

var φMax = Math.acos(Math.abs(Math.sin(θ)*Math.cos(φ)));

Rhumb lines

A "rhumb line" (or loxodrome) is a path of constant bearing, which crosses all meridians at the same angle.

Sailors used to (and sometimes still) navigate along rhumb lines since it is easier to follow a constant compass bearing than to be continually adjusting the bearing, as is needed to follow a great circle. Rhumb lines are straight lines on a Mercator Projection map (also helpful for navigation).

Rhumb lines are generally longer than great-circle (orthodrome) routes. For instance, London to New York is 4% longer along a rhumb line than along a great circle – important for aviation fuel, but not particularly to sailing vessels. New York to Beijing – close to the most extreme example possible (though not sailable!) – is 30% longer along a rhumb line.

Key to calculations of rhumb lines is the inverse Gudermannian function¹, which gives the height on a Mercator projection map of a given latitude: ln(tanφ + secφ) or ln( tan(π/4+φ/2) ). This of course tends to infinity at the poles (in keeping with the Mercator projection). For obsessives, there is even an ellipsoidal version, the "isometric latitude": ln( tan(π/4+φ/2) / [ (1−e⋅sinφ) / (1+e⋅sinφ) ]^e/2).

The formulæ to derive Mercator projection easting and northing coordinates from spherical latitude and longitude are then¹

The following formulæ are from Ed Williams’ aviation formulary.¹

Distance

Since a rhumb line is a straight line on a Mercator projection, the distance between two points along a rhumb line is the length of that line (by Pythagoras); but the distortion of the projection needs to be compensated for.

On a constant latitude course (travelling east-west), this compensation is simply cosφ; in the general case, it is Δφ/Δψ where Δψ = ln( tan(π/4 + φ₂/2) / tan(π/4 + φ₁/2) ) (the "projected" latitude difference)

var Δψ = Math.log(Math.tan(Math.PI/4+φ2/2)/Math.tan(Math.PI/4+φ1/2));
var q = Math.abs(Δψ) > 10e-12 ? Δφ/Δψ : Math.cos(φ1); // E-W course becomes ill-conditioned with 0/0

// if dLon over 180° take shorter rhumb across anti-meridian:
if (Math.abs(Δλ) > Math.PI) Δλ = Δλ>0 ? -(2*Math.PI-Δλ) : (2*Math.PI+Δλ);

var dist = Math.sqrt(Δφ*Δφ + q*q*Δλ*Δλ) * R;

Bearing

A rhumb line is a straight line on a Mercator projection, with an angle on the projection equal to the compass bearing.

var Δψ = Math.log(Math.tan(Math.PI/4+φ2/2)/Math.tan(Math.PI/4+φ1/2));
var q = Math.abs(Δψ) > 10e-12 ? Δφ/Δψ : Math.cos(φ1); // E-W course becomes ill-conditioned with 0/0

// if dLon over 180° take shorter rhumb across anti-meridian:
if (Math.abs(Δλ) > Math.PI) Δλ = Δλ>0 ? -(2*Math.PI-Δλ) : (2*Math.PI+Δλ);

var brng = Math.atan2(Δλ, Δψ).toDegrees();

Destination

Given a start point and a distance d along constant bearing θ, this will calculate the destination point. If you maintain a constant bearing along a rhumb line, you will gradually spiral in towards one of the poles.

var Δφ = δ*Math.cos(θ);
var φ2 = φ1 + Δλ;
var Δψ = Math.log(Math.tan(φ2/2+Math.PI/4)/Math.tan(φ1/2+Math.PI/4));
var q = Δψ > 10e-12 ? Δφ / Δψ : Math.cos(φ1); // E-W course becomes ill-conditioned with 0/0
var Δλ = δ*Math.sin(θ)/q;

// check for some daft bugger going past the pole, normalise latitude if so
if (Math.abs(φ2) > Math.PI/2) φ2 = φ2>0 ? Math.PI-φ2 : -Math.PI-φ2;

λ2 = (λ1+dLon+Math.PI)%(2*Math.PI) - Math.PI;

Mid-point

This formula for calculating the "loxodromic midpoint", the point half-way along a rhumb line between two points, is due to Robert Hill and Clive Tooth¹ (thx Axel!).

if (Math.abs(λ2-λ1) > Math.PI) λ1 += 2*Math.PI; // crossing anti-meridian

var φ3 = (φ1+φ2)/2;
var f1 = Math.tan(Math.PI/4 + φ1/2);
var f2 = Math.tan(Math.PI/4 + φ2/2);
var f3 = Math.tan(Math.PI/4 + φ3/2);
var λ3 = ( (λ2-λ1)*Math.log(f3) + λ1*Math.log(f2) - λ2*Math.log(f1) ) / Math.log(f2/f1);

if (!isFinite(λ3)) λ3 = (λ1+λ2)/2; // parallel of latitude

λ3 = (λ3+3*Math.PI) % (2*Math.PI) - Math.PI;  // normalise to -180..+180º
    

Using the scripts in web pages

Using these scripts in web pages would be something like the following:

<script src="latlon.js">/* Latitude/Longitude formulae */</script>
<script src="geo.js">/* Geodesy representation conversions */</script>
...
<form>
  Lat1: <input type="text" name="lat1" id="lat1"> Lon1: <input type="text" name="lon1" id="lon1">
  Lat2: <input type="text" name="lat2" id="lat2"> Lon2: <input type="text" name="lon2" id="lon2">
  <button onClick="var p1 = new LatLon(Geo.parseDMS(f.lat1.value), Geo.parseDMS(f.lon1.value));
                   var p2 = new LatLon(Geo.parseDMS(f.lat2.value), Geo.parseDMS(f.lon2.value));
                   alert(p1.distanceTo(p2));">Calculate distance</button>
</form>

If you use jQuery, the code can be separated from the HTML:

<script src="//ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<script src="latlon.js">/* Latitude/Longitude formulae */</script>
<script src="geo.js">/* Geodesy representation conversions */</script>
<script>
  $(document).ready(function() {
    $('#calc-dist').click(function() {
      var p1 = new LatLon(Geo.parseDMS($('#lat1').val()), Geo.parseDMS($('#lon1').val()));
      var p2 = new LatLon(Geo.parseDMS($('#lat2').val()), Geo.parseDMS($('#lon2').val()));
      $('#result-distance').html(p1.distanceTo(p2));
    });
  });
</script>
...
<form>
  Lat1: <input type="text" name="lat1" id="lat1"> Lon1: <input type="text" name="lon1" id="lon1">
  Lat2: <input type="text" name="lat2" id="lat2"> Lon2: <input type="text" name="lon2" id="lon2">
  <button id="calc-dist">Calculate distance</button>
  <output id="result-distance"></output> km
</form>

Convert between degrees-minutes-seconds & decimal degrees

No, I’ve not included decimal minutes: a decimal system is easy, a sexagesimal system has merits, but mixing the two is a complete sow’s ear. Switch off the option on your GPS!

Notes:

• Accuracy: since the earth is not quite a sphere, there are small errors in using spherical geometry; the earth is actually roughly ellipsoidal (or more precisely, oblate spheroidal) with a radius varying between about 6,378km (equatorial) and 6,357km (polar), and local radius of curvature varying from 6,336km (equatorial meridian) to 6,399km (polar). 6,371 km is the generally accepted value for the Earth’s mean radius. This means that errors from assuming spherical geometry might be up to 0.55% crossing the equator, though generally below 0.3%, depending on latitude and direction of travel. An accuracy of better than 3m in 1km is mostly good enough for me, but if you want greater accuracy, you could use the Vincenty formula for calculating geodesic distances on ellipsoids, which gives results accurate to within 1mm. (Out of sheer perversity – I’ve never needed such accuracy – I looked up this formula and discovered the JavaScript implementation was simpler than I expected).
• 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 [or brng+360)%360] where % is the modulo operator.
• The atan2() function widely 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).
• 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.
• 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().
• If you are using Google Maps, several of these functions are now provided in the Google Maps API V3 "spherical" library (computeDistanceBetween(), computeHeading(), computeOffset(), interpolate(), etc; note they use a default Earth radius of 6,378,137 meters).
• If you use Ordnance Survey Grid References, I have implemented a script for converting between Lat/Long & OS Grid References.
• I learned a lot from the US Census Bureau GIS FAQ which is no longer available, so I’ve made a copy.
• Thanks to Ed Williams’ Aviation Formulary for many of the formulæ.
• For miles, divide km by 1.609344
• For nautical miles, divide km by 1.852

See below for the source code of the JavaScript implementation, also available on GitHub. These functions should be simple to translate into other languages if required.

Update January 2010: I have revised the scripts to be structured as methods of a LatLon object. Of course, JavaScript is a prototype-based rather than class-based language, so this is only nominally a class, but isolating code into a separate namespace is good JavaScript practice, and this approach may also make it clearer to implement these functions in other languages. If you’re not familiar with JavaScript syntax, LatLon.prototype.distanceTo = function(point) { ... }, for instance, defines a "distanceTo" method of the LatLon object (/class) which takes a LatLon object as a parameter (and returns a number). The Geo namespace acts as a static class for geodesy formatting / parsing / conversion functions. I have extended (polluted, if you like) the base JavaScript object prototypes with trim(), toRadians() toDegrees(), and toPrecisionFixed() methods. I’ve adopted JSDoc format for the descriptions.

I have also created a page illustrating the use of the spherical law of cosines for selecting points from a database within a specified bounding circle – the example is based on MySQL+PDO, but should be extensible to other DBMS platforms.

Several people have asked about example Excel spreadsheets, so I have implemented the distance & bearing and the destination point formulæ as spreadsheets, in a form which breaks down the all stages involved to illustrate the operation.

I offer these formulæ & scripts for free use and adaptation as my contribution to the open-source info-sphere from which I have received so much. You are welcome to re-use these scripts [under a simple attribution license, without any warranty express or implied] provided solely that you retain my copyright notice and a reference to this page.

If you would like to show your appreciation and support continued development of these scripts, I would most gratefully accept donations.

If you need any advice or development work done, I am available for consultancy.

If you have any queries or find any problems, contact me at ku.oc.epyt-elbavom@oeg-stpircs.

© 2002-2014 Chris Veness