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 great-circle distances between the 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!).
Haversine formula:
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R = earth’s radius (mean radius = 6,371km)
?lat = lat2? lat1
?long = long2? long1
a = sin²(?lat/2) + cos(lat1).cos(lat2).sin²(?long/2)
c = 2.atan2(?a, ?(1?a))
d = R.c
(Note that angles need to be in radians to pass to trig functions).Code: Select all
var R = 6371; // km
var dLat = (lat2-lat1).toRad();
var dLon = (lon2-lon1).toRad();
var a = Math.sin(dLat/2) * Math.sin(dLat/2) +
Math.cos(lat1.toRad()) * Math.cos(lat2.toRad()) *
Math.sin(dLon/2) * Math.sin(dLon/2);
var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a));
var d = R * c; Spherical Law of Cosines
In fact, when Sinnott published the Haversine formula, computational precision was limited. Nowadays, JavaScript (and most modern computers & languages) use IEEE 754 64-bit floating-point numbers, which provide 15 significant figures of precision. With this precision, the simple spherical law of cosines formula gives well-conditioned results down to distances as small as around 1 metre. In view of this it is probably worth, in most situations, using either the simpler law of cosines or the more accurate ellipsoidal Vincenty formula in preference to Haversine! (bearing in mind notes below on the limitations in accuracy of the spherical model).
Spherical law of cosines:
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d = acos(sin(lat1).sin(lat2)+cos(lat1).cos(lat2).cos(long2?long1)).RCode: Select all
var R = 6371; // km
var d = Math.acos(Math.sin(lat1)*Math.sin(lat2) +
Math.cos(lat1)*Math.cos(lat2) *
Math.cos(lon2-lon1)) * R; Code: Select all
=ACOS(SIN(lat1)*SIN(lat2)+COS(lat1)*COS(lat2)*COS(lon2-lon1))*6371Bearing
In general, your current bearing will vary as you follow a great circle path (orthodrome); the final bearing will differ from the initial bearing 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 bearing of 60° and end up on a bearing of 120°!).
This is 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:
Formula:
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? = atan2( sin(?long).cos(lat2),
cos(lat1).sin(lat2) ? sin(lat1).cos(lat2).cos(?long) )Code: Select all
var y = Math.sin(dLon) * Math.cos(lat2);
var x = Math.cos(lat1)*Math.sin(lat2) -
Math.sin(lat1)*Math.cos(lat2)*Math.cos(dLon);
var brng = Math.atan2(y, x).toDeg(); Code: Select all
=ATAN2(COS(lat1)*SIN(lat2)-SIN(lat1)*COS(lat2)*COS(lon2-lon1),
SIN(lon2-lon1)*COS(lat2))
* Note that Excel reverses the arguments to ATAN2 – see notes belowFor final bearing, simply take the initial bearing from the end point to the start point and reverse it (using ? = (?+180) % 360).
Midpoint
This is the midpoint along a great circle path between the two points.
Formula:
Bx = cos(lat2).cos(?long)
By = cos(lat2).sin(?long)
latm = atan2(sin(lat1) + sin(lat2), ?((cos(lat1)+Bx)² + By²))
lonm = lon1 + atan2(By, cos(lat1)+Bx)
JavaScript:
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var Bx = Math.cos(lat2) * Math.cos(dLon);
var By = Math.cos(lat2) * Math.sin(dLon);
var lat3 = Math.atan2(Math.sin(lat1)+Math.sin(lat2),
Math.sqrt( (Math.cos(lat1)+Bx)*(Math.cos(lat1)+Bx) + By*By) );
var lon3 = lon1 + Math.atan2(By, Math.cos(lat1) + Bx); Destination point given distance and bearing from start point
Given a start point, initial bearing, and distance, this will calculate the destination point and final bearing travelling along a (shortest distance) great circle arc.
Formula:
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lat2 = asin(sin(lat1)*cos(d/R) + cos(lat1)*sin(d/R)*cos(?))
lon2 = lon1 + atan2(sin(?)*sin(d/R)*cos(lat1), cos(d/R)?sin(lat1)*sin(lat2))
d/R is the angular distance (in radians), where d is the distance travelled and R is the earth’s radiusCode: Select all
var lat2 = Math.asin( Math.sin(lat1)*Math.cos(d/R) +
Math.cos(lat1)*Math.sin(d/R)*Math.cos(brng) );
var lon2 = lon1 + Math.atan2(Math.sin(brng)*Math.sin(d/R)*Math.cos(lat1),
Math.cos(d/R)-Math.sin(lat1)*Math.sin(lat2)); Code: Select all
lat2: =ASIN(SIN(lat1)*COS(d/R) + COS(lat1)*SIN(d/R)*COS(brng))
lon2: =lon1 + ATAN2(COS(d/R)-SIN(lat1)*SIN(lat2), SIN(brng)*SIN(d/R)*COS(lat1))Intersection of two paths given start points and bearings
This is a rather more complex calculation than most others on this page, but I've been asked for it a number of times. See below for the JavaScript.
Formula:
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d12 = 2.asin( ?(sin²(?lat/2) + cos(lat1).cos(lat2).sin²(?lon/2)) )
?1 = acos( sin(lat2) ? sin(lat1).cos(d12) / sin(d12).cos(lat1) )
?2 = acos( sin(lat1) ? sin(lat2).cos(d12) / sin(d12).cos(lat2) )
if sin(lon2?lon1) > 0
?12 = ?1, ?21 = 2.? ? ?2
else
?12 = 2.? ? ?1, ?21 = ?2
?1 = (?1 ? ?12 + ?) % 2.? ? ?
?2 = (?21 ? ?2 + ?) % 2.? ? ?
?1 = |?1|
?2 = |?2|
?3 = acos( ?cos(?1).cos(?2) + sin(?1).sin(?2).cos(d12) )
d13 = atan2( sin(d12).sin(?1).sin(?2), cos(?2)+cos(?1).cos(?3) )
lat3 = asin( sin(lat1).cos(d13) + cos(lat1).sin(d13).cos(?1) )
?lon = atan2( sin(?1).sin(d13).cos(lat1), cos(d13)?sin(lat1).sin(lat3) )
lon3 = (lon1??lon+?) % 2.? ? ?
where
at1, lon1, ?1 : 1st point & bearing
lat2, lon2, ?2 : 2nd point & bearing
lat3, lon3 : intersection point
% = mod, | | = abs
note – if sin(?1)=0 and sin(?2)=0: infinite solutions
if sin(?1).sin(?2) < 0: ambiguous solution
this formulation is not always well-conditioned for meridional or equatorial lines- For each point ?,? (lat=?, lon=?), we can define a unit vector pointing to it from the centre of the earth: u{x,y,z} = [ cos?·cos?, cos?·sin?, sin? ] (taking x=0º, y=90º, z=north – note that these formulæ depend on convention used for directions and handedness)
- And for any great circle defined by two points, we can define a unit vector N normal to the plane of the circle: N(u1, u2) = (u1×u2) / ||u1×u2|| where × is the vector cross product, and ||u|| the norm (length of the vector)
- The vector representing the intersection of the two great circles is then ui = ±N( N(u1, u2), N(u3, u4) )
- We can then get the latitude and longitude of Pi by ? = atan2(uz, sqrt(ux² + uy²)), ? = atan2(uy, ux)
- The antipodal intersection point is (-?, ?+?)
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).
Formula:
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dxt = asin(sin(d13/R)*sin(?13??12)) * R
where
d13 is distance from start point to third point
?13 is (initial) bearing from start point to third point
?12 is (initial) bearing from start point to end point
R is the earth’s radiusCode: Select all
var dXt = Math.asin(Math.sin(d13/R)*Math.sin(brng13-brng12)) * R; The along-track distance, from the start point to the closest point on the path to the third point, is
Formula:
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dat = acos(cos(d13/R)/cos(dxt/R)) * R
where
d13 is distance from start point to third point
dxt is cross-track distance
R is the earth’s radiusCode: Select all
var dAt = Math.acos(Math.cos(d13/R)/Math.cos(dXt/R)) * R; And: ‘Clairaut’s formula’ will give you the maximum latitude of a great circle path, given a bearing and latitude on the great circle:
Formula:
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latmax = acos(abs(sin(?)*cos(lat)))Code: Select all
var latMax = Math.acos(Math.abs(Math.sin(brng)*Math.cos(lat)));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.
Distance/bearing
These formulæ give the distance and (constant) bearing between two points.
Formula:
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?? = ln(tan(lat2/2+?/4)/tan(lat1/2+?/4)) [= the ‘stretched’ latitude difference]
if E:W line, q = cos(lat1)
otherwise, q = ?lat/??
d = ?[?lat² + q².?lon²].R [pythagoras]
? = atan2(?lon, ??)
where ln is natural log, ?lon is taking shortest route (<180º), and R is the earth’s radiusCode: Select all
var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
// if dLon over 180° take shorter rhumb across 180° meridian:
if (Math.abs(dLon) > Math.PI) {
dLon = dLon>0 ? -(2*Math.PI-dLon) : (2*Math.PI+dLon);
}
var d = Math.sqrt(dLat*dLat + q*q*dLon*dLon) * R;
var brng = Math.atan2(dLon, dPhi); Given a start point and a distance d along constant bearing ?, 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.
Formula:
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? = d/R (angular distance)
lat2 = lat1 + ?.cos(?)
?? = ln(tan(lat2/2+?/4)/tan(lat1/2+?/4)) [= the ‘stretched’ latitude difference]
if E:W line q = cos(lat1)
otherwise q = ?lat/??
?lon = ?.sin(?)/q
lon2 = (lon1+?lon+?) % 2.? ? ?
where ln is natural log and % is modulo, ?lon is taking shortest route (<180°), and R is the earth’s radiusCode: Select all
lat2 = lat1 + d*Math.cos(brng);
var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
var dLon = d*Math.sin(brng)/q;
// check for some daft bugger going past the pole, normalise latitude if so
if (Math.abs(lat2) > Math.PI/2) lat2 = lat2>0 ? Math.PI-lat2 : -(Math.PI-lat2);
lon2 = (lon1+dLon+Math.PI)%(2*Math.PI) - Math.PI;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 = ?.deg/180. When converting radians back to degrees (deg = 180.rad/?), West is negative if using signed decimal degrees. For bearings, values in the range -? to +? [-180° to +180°] need to be converted to 0 to +2? [0°–360°]; this can be done by (brng+2.?)%2.? [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 -? to +? (the atan function returns values in the range -?/2 to +?/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().
- 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.
- 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
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(), toRad() toDeg(), and toPrecisionFixed() methods. I’ve adopted JSDoc format for the descriptions.
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/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* Latitude/longitude spherical geodesy formulae & scripts (c) Chris Veness 2002-2010 */
/* - www.movable-type.co.uk/scripts/latlong.html */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/* Note that minimal error checking is performed in this example code! */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/**
* Creates a point on the earth's surface at the supplied latitude / longitude
*
* @constructor
* @param {Number} lat: latitude in numeric degrees
* @param {Number} lon: longitude in numeric degrees
* @param {Number} [rad=6371]: radius of earth if different value is required from standard 6,371km
*/
function LatLon(lat, lon, rad) {
if (typeof rad == 'undefined') rad = 6371; // earth's mean radius in km
this._lat = lat;
this._lon = lon;
this._radius = rad;
}
/**
* Returns the distance from this point to the supplied point, in km
* (using Haversine formula)
*
* from: Haversine formula - R. W. Sinnott, "Virtues of the Haversine",
* Sky and Telescope, vol 68, no 2, 1984
*
* @param {LatLon} point: Latitude/longitude of destination point
* @param {Number} [precision=4]: no of significant digits to use for returned value
* @returns {Number} Distance in km between this point and destination point
*/
LatLon.prototype.distanceTo = function(point, precision) {
// default 4 sig figs reflects typical 0.3% accuracy of spherical model
if (typeof precision == 'undefined') precision = 4;
var R = this._radius;
var lat1 = this._lat.toRad(), lon1 = this._lon.toRad();
var lat2 = point._lat.toRad(), lon2 = point._lon.toRad();
var dLat = lat2 - lat1;
var dLon = lon2 - lon1;
var a = Math.sin(dLat/2) * Math.sin(dLat/2) +
Math.cos(lat1) * Math.cos(lat2) *
Math.sin(dLon/2) * Math.sin(dLon/2);
var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a));
var d = R * c;
return d.toPrecisionFixed(precision);
}
/**
* Returns the (initial) bearing from this point to the supplied point, in degrees
* see http://williams.best.vwh.net/avform.htm#Crs
*
* @param {LatLon} point: Latitude/longitude of destination point
* @returns {Number} Initial bearing in degrees from North
*/
LatLon.prototype.bearingTo = function(point) {
var lat1 = this._lat.toRad(), lat2 = point._lat.toRad();
var dLon = (point._lon-this._lon).toRad();
var y = Math.sin(dLon) * Math.cos(lat2);
var x = Math.cos(lat1)*Math.sin(lat2) -
Math.sin(lat1)*Math.cos(lat2)*Math.cos(dLon);
var brng = Math.atan2(y, x);
return (brng.toDeg()+360) % 360;
}
/**
* Returns final bearing arriving at supplied destination point from this point; the final bearing
* will differ from the initial bearing by varying degrees according to distance and latitude
*
* @param {LatLon} point: Latitude/longitude of destination point
* @returns {Number} Final bearing in degrees from North
*/
LatLon.prototype.finalBearingTo = function(point) {
// get initial bearing from supplied point back to this point...
var lat1 = point._lat.toRad(), lat2 = this._lat.toRad();
var dLon = (this._lon-point._lon).toRad();
var y = Math.sin(dLon) * Math.cos(lat2);
var x = Math.cos(lat1)*Math.sin(lat2) -
Math.sin(lat1)*Math.cos(lat2)*Math.cos(dLon);
var brng = Math.atan2(y, x);
// ... & reverse it by adding 180°
return (brng.toDeg()+180) % 360;
}
/**
* Returns the midpoint between this point and the supplied point.
* see http://mathforum.org/library/drmath/view/51822.html for derivation
*
* @param {LatLon} point: Latitude/longitude of destination point
* @returns {LatLon} Midpoint between this point and the supplied point
*/
LatLon.prototype.midpointTo = function(point) {
lat1 = this._lat.toRad(), lon1 = this._lon.toRad();
lat2 = point._lat.toRad();
var dLon = (point._lon-this._lon).toRad();
var Bx = Math.cos(lat2) * Math.cos(dLon);
var By = Math.cos(lat2) * Math.sin(dLon);
lat3 = Math.atan2(Math.sin(lat1)+Math.sin(lat2),
Math.sqrt( (Math.cos(lat1)+Bx)*(Math.cos(lat1)+Bx) + By*By) );
lon3 = lon1 + Math.atan2(By, Math.cos(lat1) + Bx);
return new LatLon(lat3.toDeg(), lon3.toDeg());
}
/**
* Returns the destination point from this point having travelled the given distance (in km) on the
* given initial bearing (bearing may vary before destination is reached)
*
* see http://williams.best.vwh.net/avform.htm#LL
*
* @param {Number} brng: Initial bearing in degrees
* @param {Number} dist: Distance in km
* @returns {LatLon} Destination point
*/
LatLon.prototype.destinationPoint = function(brng, dist) {
dist = dist/this._radius; // convert dist to angular distance in radians
brng = brng.toRad(); //
var lat1 = this._lat.toRad(), lon1 = this._lon.toRad();
var lat2 = Math.asin( Math.sin(lat1)*Math.cos(dist) +
Math.cos(lat1)*Math.sin(dist)*Math.cos(brng) );
var lon2 = lon1 + Math.atan2(Math.sin(brng)*Math.sin(dist)*Math.cos(lat1),
Math.cos(dist)-Math.sin(lat1)*Math.sin(lat2));
lon2 = (lon2+3*Math.PI)%(2*Math.PI) - Math.PI; // normalise to -180...+180
if (isNaN(lat2) || isNaN(lon2)) return null;
return new LatLon(lat2.toDeg(), lon2.toDeg());
}
/**
* Returns the point of intersection of two paths defined by point and bearing
*
* see http://williams.best.vwh.net/avform.htm#Intersection
*
* @param {LatLon} p1: First point
* @param {Number} brng1: Initial bearing from first point
* @param {LatLon} p2: Second point
* @param {Number} brng2: Initial bearing from second point
* @returns {LatLon} Destination point (null if no unique intersection defined)
*/
LatLon.intersection = function(p1, brng1, p2, brng2) {
lat1 = p1.lat.toRad(), lon1 = p1.lon.toRad();
lat2 = p2.lat.toRad(), lon2 = p2.lon.toRad();
brng13 = brng1.toRad(), brng23 = brng2.toRad();
dLat = lat2-lat1, dLon = lon2-lon1;
dist12 = 2*Math.asin( Math.sqrt( Math.sin(dLat/2)*Math.sin(dLat/2) +
Math.cos(lat1)*Math.cos(lat2)*Math.sin(dLon/2)*Math.sin(dLon/2) ) );
if (dist12 == 0) return null;
// initial/final bearings between points
brngA = Math.acos( ( Math.sin(lat2) - Math.sin(lat1)*Math.cos(dist12) ) / ( Math.sin(dist12)*Math.cos(lat1) ) );
if (isNaN(brngA)) brngA = 0; // protect against rounding
brngB = Math.acos( ( Math.sin(lat1) - Math.sin(lat2)*Math.cos(dist12) ) / ( Math.sin(dist12)*Math.cos(lat2) ) );
if (Math.sin(lon2-lon1) > 0) {
brng12 = brngA;
brng21 = 2*Math.PI - brngB;
} else {
brng12 = 2*Math.PI - brngA;
brng21 = brngB;
}
alpha1 = (brng13 - brng12 + Math.PI) % (2*Math.PI) - Math.PI; // angle 2-1-3
alpha2 = (brng21 - brng23 + Math.PI) % (2*Math.PI) - Math.PI; // angle 1-2-3
if (Math.sin(alpha1)==0 && Math.sin(alpha2)==0) return null; // infinite intersections
if (Math.sin(alpha1)*Math.sin(alpha2) < 0) return null; // ambiguous intersection
//alpha1 = Math.abs(alpha1);
//alpha2 = Math.abs(alpha2);
// ... Ed Williams takes abs of alpha1/alpha2, but seems to break calculation?
alpha3 = Math.acos( -Math.cos(alpha1)*Math.cos(alpha2) + Math.sin(alpha1)*Math.sin(alpha2)*Math.cos(dist12) );
dist13 = Math.atan2( Math.sin(dist12)*Math.sin(alpha1)*Math.sin(alpha2),
Math.cos(alpha2)+Math.cos(alpha1)*Math.cos(alpha3) )
lat3 = Math.asin( Math.sin(lat1)*Math.cos(dist13) + Math.cos(lat1)*Math.sin(dist13)*Math.cos(brng13) );
dLon13 = Math.atan2( Math.sin(brng13)*Math.sin(dist13)*Math.cos(lat1),
Math.cos(dist13)-Math.sin(lat1)*Math.sin(lat3) );
lon3 = lon1+dLon13;
lon3 = (lon3+Math.PI) % (2*Math.PI) - Math.PI; // normalise to -180..180º
return new LatLon(lat3.toDeg(), lon3.toDeg());
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/**
* Returns the distance from this point to the supplied point, in km, travelling along a rhumb line
*
* see http://williams.best.vwh.net/avform.htm#Rhumb
*
* @param {LatLon} point: Latitude/longitude of destination point
* @returns {Number} Distance in km between this point and destination point
*/
LatLon.prototype.rhumbDistanceTo = function(point) {
var R = this._radius;
var lat1 = this._lat.toRad(), lat2 = point._lat.toRad();
var dLat = (point._lat-this._lat).toRad();
var dLon = Math.abs(point._lon-this._lon).toRad();
var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
// if dLon over 180° take shorter rhumb across 180° meridian:
if (dLon > Math.PI) dLon = 2*Math.PI - dLon;
var dist = Math.sqrt(dLat*dLat + q*q*dLon*dLon) * R;
return dist.toPrecisionFixed(4); // 4 sig figs reflects typical 0.3% accuracy of spherical model
}
/**
* Returns the bearing from this point to the supplied point along a rhumb line, in degrees
*
* @param {LatLon} point: Latitude/longitude of destination point
* @returns {Number} Bearing in degrees from North
*/
LatLon.prototype.rhumbBearingTo = function(point) {
var lat1 = this._lat.toRad(), lat2 = point._lat.toRad();
var dLon = (point._lon-this._lon).toRad();
var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
if (Math.abs(dLon) > Math.PI) dLon = dLon>0 ? -(2*Math.PI-dLon) : (2*Math.PI+dLon);
var brng = Math.atan2(dLon, dPhi);
return (brng.toDeg()+360) % 360;
}
/**
* Returns the destination point from this point having travelled the given distance (in km) on the
* given bearing along a rhumb line
*
* @param {Number} brng: Bearing in degrees from North
* @param {Number} dist: Distance in km
* @returns {LatLon} Destination point
*/
LatLon.prototype.rhumbDestinationPoint = function(brng, dist) {
var R = this._radius;
var d = parseFloat(dist)/R; // d = angular distance covered on earth's surface
var lat1 = this._lat.toRad(), lon1 = this._lon.toRad();
brng = brng.toRad();
var lat2 = lat1 + d*Math.cos(brng);
var dLat = lat2-lat1;
var dPhi = Math.log(Math.tan(lat2/2+Math.PI/4)/Math.tan(lat1/2+Math.PI/4));
var q = (!isNaN(dLat/dPhi)) ? dLat/dPhi : Math.cos(lat1); // E-W line gives dPhi=0
var dLon = d*Math.sin(brng)/q;
// check for some daft bugger going past the pole
if (Math.abs(lat2) > Math.PI/2) lat2 = lat2>0 ? Math.PI-lat2 : -(Math.PI-lat2);
lon2 = (lon1+dLon+3*Math.PI)%(2*Math.PI) - Math.PI;
return new LatLon(lat2.toDeg(), lon2.toDeg());
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
/**
* Returns the latitude of this point; signed numeric degrees if no format, otherwise format & dp
* as per Geo.toLat()
*
* @param {String} [format]: Return value as 'd', 'dm', 'dms'
* @param {Number} [dp=0|2|4]: No of decimal places to display
* @returns {Number|String} Numeric degrees if no format specified, otherwise deg/min/sec
*/
LatLon.prototype.lat = function(format, dp) {
if (typeof format == 'undefined') return this._lat;
return Geo.toLat(this._lat, format, dp);
}
/**
* Returns the longitude of this point; signed numeric degrees if no format, otherwise format & dp
* as per Geo.toLon()
*
* @param {String} [format]: Return value as 'd', 'dm', 'dms'
* @param {Number} [dp=0|2|4]: No of decimal places to display
* @returns {Number|String} Numeric degrees if no format specified, otherwise deg/min/sec
*/
LatLon.prototype.lon = function(format, dp) {
if (typeof format == 'undefined') return this._lon;
return Geo.toLon(this._lon, format, dp);
}
/**
* Returns a string representation of this point; format and dp as per lat()/lon()
*/
LatLon.prototype.toString = function(format, dp) {
if (typeof format == 'undefined') format = 'dms';
return Geo.toLat(this._lat, format, dp) + ', ' + Geo.toLon(this._lon, format, dp);
}
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */