gc_distance Google Sheets Script

README.md

gc_distance

Add the following three files to the script for a Google Sheet to get a gc_distance function.

Usage

gc_distance("JFK","SFO")

Very simple… just use the function like any other Google Sheets function.

The result should be entirely accurate on the order of meters — it's using the more advanced calculation for distance which takes into account the earth's non-circular shape.

Credits

The vast majority of the underlying code is (c) Chris Veness. I am not the author of the GC calculation algorithm, I just made his implementation usable in Google Sheets and added functionality to recognize airport codes.

Code.gs

function _tester() {
  Logger.log("New test run…")
  var result = gc_distance('sfo', 'jfk');
  Logger.log(result)
}

function location_of(airport) {
  return data[airport.toUpperCase()];
}

function gc_distance(from, to) {
  var from_loc = location_of(from), to_loc = location_of(to);
  var from_ll = new LatLon(from_loc[1], from_loc[0], LatLon.datum.WGS84);
  var to_ll = new LatLon(to_loc[1], to_loc[0], LatLon.datum.WGS84);
  var d = from_ll.distanceTo(to_ll) / 1609.344; // m -> mi

  return d;
}

data.js.gs

var data = {"UTK":[169.852005,11.222],"OCA":[-80.274803161621,25.325399398804],"PQS":[-162.899994,61.934601],"CSE":[-106.928341,38.851918],"JCY":[-98.62249755859999,30.251800537100003],"PMX":[-72.31140136719999,42.223300933800004],"WLR":[-131.636993408,55.6012992859],"NUP":[-162.440454,60.905591],"PTC":[-133.597,55.803],"ICY":[-141.662002563,59.96900177],"PPV":[-133.61000061035,56.328800201416],"KKK":[-156.820392609,64.4166256967],"MHS":[-122.272003,41.263199],"ELW":[-146.704038,60.893818],"LVD":[-155.440002441,61.3591003418],"FIV":[-133.630996704,57.26990127559999],"HGZ":[-155.6690063,66.2161026],"OTN":[-87.4997024536,38.851398468],"TLF":[-153.26899719238,63.393901824951],"BZT":[-95.579696655273,28.982200622559],"HBH":[-133.43848,57.412201],"FAK":[-135.212997437,57.5321998596],"BYW":[-122.825996399,48.578998565700005],"DRF":[-152.162002563,60.588901519800004],"BDF":[-89.61569976810001,41.230899810800004],"VRS":[-92.875198364258,38.427799224854],"GDH":[-158.875,59.7470016479],"ATT":[-162.272995,60.866699],"LIV":[-148.6534,65.467],"PDB":[-154.12399292,59.7896003723],"KOZ":[-152.496715,57.925362],"TNK":[-165.27200317383,60.575500488281],"EVA":[-94.07350158690001,30.3209991455],"WHD":[-130.009975,55.903324],"TKI":[-133.32699585,55.9370994568],"WKK":[-158.617996216,59.2826004028],"WFB":[-131.677002,55.349899],"NNK":[-157.02000427246,58.732898712158],"BKF":[-155.77699279785,58.554798126221],"BCS":[-90.0222015381,29.8661003113],"SPQ":[-118.275001526,33.7491989136],"TSS":[-73.97209930419922,40.74259948730469],"QNY":[-73.9729,40.734001],"BWL":[-97.3170013428,36.7958984375],"WYB":[-131.800994873,55.9163017273],"CWS":[-122.832000732,48.4901008606],"TEK":[-146.690297,60.871449],"DUF":[-75.7884979248,36.2535018921],"SSW":[-123.17600250244,48.672901153564],"FOB":[-123.79599762,39.4743003845],"PUL":[-122.64700317383,47.734001159668],"CCD":[-118.417999268,34.060798645],"WMK":[-132.255004883,55.7396011353],"AXB":[-75.90034,44.312002],"REE":[-102.03700256348,33.590301513672],"WDN":[-123.017997742,48.7117996216],"WWP":[-133.121994,56.116299],"CHU":[-159.216003,61.579102],"UGS":[-157.399002075,57.5279006958],"PSQ":[-75.299598693848,39.859001159668],"KLL":[-156.85899353,59.128101348899996],"SGW":[-134.158004761,56.8862991333],"WTL":[-162.667007446,60.3353004456],"TWA":[-160.27299499512,59.07559967041],"KCQ":[-158.774993896,56.2550010681],"AAD":[46.6375,6.095802],"ABP":[141.095277778,-6.0655555555600005],"ABW":[148.7389,-10.1956],"ALV":[1.533551,42.511174],"ADC":[145.744722222,-7.13722222222],"ADV":[26.1186,11.4023],"DJH":[55.023796,24.988967],"AEE":[32.9594444444,10.0536111111],"AEI":[-5.441118,36.12888],"AEK":[146.19386673,-7.35080485552],"URZ":[66.630897522,32.9029998779],"OLR":[69.9561004639,33.363800048799995],"AFK":[81.644,7.284],"AFR":[148.390833333,-9.142222222220001],"AFT":[160.948611111,-9.19138888889],"RNA":[161.979546547,-9.86054358262],"ATD":[161.011002,-8.87333],"VEV":[156.70599365234375,-7.912779808044434],"GEF":[156.5970001220703,-7.575829982757568],"AGG":[144.073888889,-4.05583333333],"AKS":[160.682007,-8.70257],"BNY":[159.798333333,-11.302222222200001],"CHY":[156.396111,-6.711944],"BAS":[155.886656,-6.990745],"FRE":[159.576996,-8.1075],"HIR":[160.05499267578,-9.4280004501343],"MBU":[159.83900451660156,-9.7475004196167],"AVU":[160.410555556,-9.86833333333],"IRA":[161.897994995,-10.449700355500001],"SCZ":[165.7949981689453,-10.72029972076416],"MUA":[157.26300048828125,-8.327969551086426],"GZO":[156.863998413,-8.09778022766],"MNY":[155.56500244140625,-7.416940212249756],"PRS":[161.4250030517578,-9.641670227050781],"OTV":[159.527777778,-5.515],"RNL":[160.06300354003906,-11.533900260925293],"EGM":[157.87600708007812,-8.578889846801758],"NNB":[162.454108,-10.847994],"RUS":[160.824996948,-9.861669540409999],"VAO":[158.7310028076172,-7.585559844970703],"XYA":[159.21841,-9.092816],"AGK":[143.853611111,-6.396388888890001],"KGE":[157.585,-7.3305],"KUE":[156.94783,-8.0262],"KWS":[160.775127,-8.360508],"AGL":[149.155555556,-9.3375],"NAZ":[162.2049,-10.6758],"RIN":[157.14300537109375,-8.12639045715332],"RBV":[157.64300537109375,-8.168060302734375],"TLG":[160.149166667,-9.108],"AGY":[128.9245,-16.514],"AHJ":[102.35224,32.53154],"AHT":[179.259166667,51.3777777778],"AHY":[45.535,-20.015833333299998],"AIE":[144.7307003,-5.145699978],"AIH":[141.2675,-7.342777777779999],"AIC":[168.8257,7.279422],"CEX":[-146.04699707,65.0518035889],"SOL":[-164.4457,64.5605],"HED":[-160.899002075,55.801399231],"TWE":[-164.798995972,65.6792984009],"KTH":[-158.477005005,59.9631996155],"NKI":[-133.227994,55.849602],"LNI":[-153.2420044,70.91069794],"CDL":[-161.925994873,65.907699585],"AOS":[-153.815002441,57.4715003967],"BSZ":[-157.352005005,58.2162017822],"BSW":[-146.145996094,60.4230995178],"AKM":[19.8172222222,10.8902777778],"TGE":[-85.7755966187,32.4919013977],"ALZ":[-154.248001099,56.8995018005],"PPE":[-113.305177,31.351987],"AME":[37.6813888889,-15.6102777778],"AMF":[141.67,-4.10111111111],"AMU":[141.214333333,-3.586],"AMY":[45.623888888900005,-17.686666666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gc_lib.js.gs

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
/* Vector handling functions                                          (c) Chris Veness 2011-2016  */
/*                                                                                   MIT Licence  */
/* www.movable-type.co.uk/scripts/geodesy/docs/module-vector3d.html                               */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

'use strict';


/**
 * Library of 3-d vector manipulation routines.
 *
 * In a geodesy context, these vectors may be used to represent:
 *  - n-vector representing a normal to point on Earth's surface
 *  - earth-centered, earth fixed vector (≡ Gade’s ‘p-vector’)
 *  - great circle normal to vector (on spherical earth model)
 *  - motion vector on Earth's surface
 *  - etc
 *
 * Functions return vectors as return results, so that operations can be chained.
 * @example var v = v1.cross(v2).dot(v3) // ≡ v1×v2⋅v3
 *
 * @module vector3d
 */


/**
 * Creates a 3-d vector.
 *
 * The vector may be normalised, or use x/y/z values for eg height relative to the sphere or
 * ellipsoid, distance from earth centre, etc.
 *
 * @constructor
 * @param {number} x - X component of vector.
 * @param {number} y - Y component of vector.
 * @param {number} z - Z component of vector.
 */
function Vector3d(x, y, z) {
    // allow instantiation without 'new'
    if (!(this instanceof Vector3d)) return new Vector3d(x, y, z);

    this.x = Number(x);
    this.y = Number(y);
    this.z = Number(z);
}


/**
 * Adds supplied vector to ‘this’ vector.
 *
 * @param   {Vector3d} v - Vector to be added to this vector.
 * @returns {Vector3d} Vector representing sum of this and v.
 */
Vector3d.prototype.plus = function(v) {
    if (!(v instanceof Vector3d)) throw new TypeError('v is not Vector3d object');

    return new Vector3d(this.x + v.x, this.y + v.y, this.z + v.z);
};


/**
 * Subtracts supplied vector from ‘this’ vector.
 *
 * @param   {Vector3d} v - Vector to be subtracted from this vector.
 * @returns {Vector3d} Vector representing difference between this and v.
 */
Vector3d.prototype.minus = function(v) {
    if (!(v instanceof Vector3d)) throw new TypeError('v is not Vector3d object');

    return new Vector3d(this.x - v.x, this.y - v.y, this.z - v.z);
};


/**
 * Multiplies ‘this’ vector by a scalar value.
 *
 * @param   {number}   x - Factor to multiply this vector by.
 * @returns {Vector3d} Vector scaled by x.
 */
Vector3d.prototype.times = function(x) {
    x = Number(x);

    return new Vector3d(this.x * x, this.y * x, this.z * x);
};


/**
 * Divides ‘this’ vector by a scalar value.
 *
 * @param   {number}   x - Factor to divide this vector by.
 * @returns {Vector3d} Vector divided by x.
 */
Vector3d.prototype.dividedBy = function(x) {
    x = Number(x);

    return new Vector3d(this.x / x, this.y / x, this.z / x);
};


/**
 * Multiplies ‘this’ vector by the supplied vector using dot (scalar) product.
 *
 * @param   {Vector3d} v - Vector to be dotted with this vector.
 * @returns {number} Dot product of ‘this’ and v.
 */
Vector3d.prototype.dot = function(v) {
    if (!(v instanceof Vector3d)) throw new TypeError('v is not Vector3d object');

    return this.x*v.x + this.y*v.y + this.z*v.z;
};


/**
 * Multiplies ‘this’ vector by the supplied vector using cross (vector) product.
 *
 * @param   {Vector3d} v - Vector to be crossed with this vector.
 * @returns {Vector3d} Cross product of ‘this’ and v.
 */
Vector3d.prototype.cross = function(v) {
    if (!(v instanceof Vector3d)) throw new TypeError('v is not Vector3d object');

    var x = this.y*v.z - this.z*v.y;
    var y = this.z*v.x - this.x*v.z;
    var z = this.x*v.y - this.y*v.x;

    return new Vector3d(x, y, z);
};


/**
 * Negates a vector to point in the opposite direction
 *
 * @returns {Vector3d} Negated vector.
 */
Vector3d.prototype.negate = function() {
    return new Vector3d(-this.x, -this.y, -this.z);
};


/**
 * Length (magnitude or norm) of ‘this’ vector
 *
 * @returns {number} Magnitude of this vector.
 */
Vector3d.prototype.length = function() {
    return Math.sqrt(this.x*this.x + this.y*this.y + this.z*this.z);
};


/**
 * Normalizes a vector to its unit vector
 * – if the vector is already unit or is zero magnitude, this is a no-op.
 *
 * @returns {Vector3d} Normalised version of this vector.
 */
Vector3d.prototype.unit = function() {
    var norm = this.length();
    if (norm == 1) return this;
    if (norm == 0) return this;

    var x = this.x/norm;
    var y = this.y/norm;
    var z = this.z/norm;

    return new Vector3d(x, y, z);
};


/**
 * Calculates the angle between ‘this’ vector and supplied vector.
 *
 * @param   {Vector3d} v
 * @param   {Vector3d} [n] - Plane normal: if supplied, angle is -π..+π, signed +ve if this->v is
 *     clockwise looking along n, -ve in opposite direction (if not supplied, angle is always 0..π).
 * @returns {number} Angle (in radians) between this vector and supplied vector.
 */
Vector3d.prototype.angleTo = function(v, n) {
    if (!(v instanceof Vector3d)) throw new TypeError('v is not Vector3d object');
    if (!(n instanceof Vector3d || n == undefined)) throw new TypeError('n is not Vector3d object');

    var sign = n==undefined ? 1 : Math.sign(this.cross(v).dot(n));
    var sinθ = this.cross(v).length() * sign;
    var cosθ = this.dot(v);

    return Math.atan2(sinθ, cosθ);
};


/**
 * Rotates ‘this’ point around an axis by a specified angle.
 *
 * @param   {Vector3d} axis - The axis being rotated around.
 * @param   {number}   theta - The angle of rotation (in radians).
 * @returns {Vector3d} The rotated point.
 */
Vector3d.prototype.rotateAround = function(axis, theta) {
    if (!(axis instanceof Vector3d)) throw new TypeError('axis is not Vector3d object');

    // en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
    // en.wikipedia.org/wiki/Quaternions_and_spatial_rotation#Quaternion-derived_rotation_matrix
    var p1 = this.unit();
    var p = [ p1.x, p1.y, p1.z ]; // the point being rotated
    var a = axis.unit();          // the axis being rotated around
    var s = Math.sin(theta);
    var c = Math.cos(theta);
    // quaternion-derived rotation matrix
    var q = [
        [ a.x*a.x*(1-c) + c,     a.x*a.y*(1-c) - a.z*s, a.x*a.z*(1-c) + a.y*s ],
        [ a.y*a.x*(1-c) + a.z*s, a.y*a.y*(1-c) + c,     a.y*a.z*(1-c) - a.x*s ],
        [ a.z*a.x*(1-c) - a.y*s, a.z*a.y*(1-c) + a.x*s, a.z*a.z*(1-c) + c     ],
    ];
    // multiply q × p
    var qp = [ 0, 0, 0 ];
    for (var i=0; i<3; i++) {
        for (var j=0; j<3; j++) {
            qp[i] += q[i][j] * p[j];
        }
    }
    var p2 = new Vector3d(qp[0], qp[1], qp[2]);
    return p2;
    // qv en.wikipedia.org/wiki/Rodrigues'_rotation_formula...
};


/**
 * String representation of vector.
 *
 * @param   {number} [precision=3] - Number of decimal places to be used.
 * @returns {string} Vector represented as [x,y,z].
 */
Vector3d.prototype.toString = function(precision) {
    var p = (precision === undefined) ? 3 : Number(precision);

    var str = '[' + this.x.toFixed(p) + ',' + this.y.toFixed(p) + ',' + this.z.toFixed(p) + ']';

    return str;
};


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

/** Polyfill Math.sign for old browsers / IE */
if (Math.sign === undefined) {
    Math.sign = function(x) {
        x = +x; // convert to a number
        if (x === 0 || isNaN(x)) return x;
        return x > 0 ? 1 : -1;
    };
}

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
if (typeof module != 'undefined' && module.exports) module.exports = Vector3d; // ≡ export default Vector3d


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
/* Geodesy tools for an ellipsoidal earth model                       (c) Chris Veness 2005-2016  */
/*                                                                                   MIT Licence  */
/* www.movable-type.co.uk/scripts/latlong-convert-coords.html                                     */
/* www.movable-type.co.uk/scripts/geodesy/docs/module-latlon-ellipsoidal.html                     */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

'use strict';
if (typeof module!='undefined' && module.exports) var Vector3d = require('./vector3d.js'); // ≡ import Vector3d from 'vector3d.js'
if (typeof module!='undefined' && module.exports) var Dms = require('./dms.js');           // ≡ import Dms from 'dms.js'


/**
 * Library of geodesy functions for operations on an ellipsoidal earth model.
 *
 * Includes ellipsoid parameters and datums for different coordinate systems, and methods for
 * converting between them and to cartesian coordinates.
 *
 * q.v. Ordnance Survey ‘A guide to coordinate systems in Great Britain’ Section 6
 * www.ordnancesurvey.co.uk/docs/support/guide-coordinate-systems-great-britain.pdf.
 *
 * @module   latlon-ellipsoidal
 * @requires dms
 */


/**
 * Creates lat/lon (polar) point with latitude & longitude values, on a specified datum.
 *
 * @constructor
 * @param {number}       lat - Geodetic latitude in degrees.
 * @param {number}       lon - Longitude in degrees.
 * @param {LatLon.datum} [datum=WGS84] - Datum this point is defined within.
 *
 * @example
 *     var p1 = new LatLon(51.4778, -0.0016, LatLon.datum.WGS84);
 */
function LatLon(lat, lon, datum) {
    // allow instantiation without 'new'
    if (!(this instanceof LatLon)) return new LatLon(lat, lon, datum);

    if (datum === undefined) datum = LatLon.datum.WGS84;

    this.lat = Number(lat);
    this.lon = Number(lon);
    this.datum = datum;
}


/**
 * Ellipsoid parameters; major axis (a), minor axis (b), and flattening (f) for each ellipsoid.
 */
LatLon.ellipsoid = {
    WGS84:         { a: 6378137,     b: 6356752.314245, f: 1/298.257223563 },
    Airy1830:      { a: 6377563.396, b: 6356256.909,    f: 1/299.3249646   },
    AiryModified:  { a: 6377340.189, b: 6356034.448,    f: 1/299.3249646   },
    Bessel1841:    { a: 6377397.155, b: 6356078.962818, f: 1/299.1528128   },
    Clarke1866:    { a: 6378206.4,   b: 6356583.8,      f: 1/294.978698214 },
    Clarke1880IGN: { a: 6378249.2,   b: 6356515.0,      f: 1/293.466021294 },
    GRS80:         { a: 6378137,     b: 6356752.314140, f: 1/298.257222101 },
    Intl1924:      { a: 6378388,     b: 6356911.946,    f: 1/297           }, // aka Hayford
    WGS72:         { a: 6378135,     b: 6356750.5,      f: 1/298.26        },
};

/**
 * Datums; with associated ellipsoid, and Helmert transform parameters to convert from WGS 84 into
 * given datum.
 *
 * Note that precision of various datums will vary, and WGS-84 (original) is not defined to be
 * accurate to better than ±1 metre. No transformation should be assumed to be accurate to better
 * than a meter; for many datums somewhat less.
 */
LatLon.datum = {
    // transforms: t in metres, s in ppm, r in arcseconds                    tx       ty        tz       s        rx       ry       rz
    ED50:       { ellipsoid: LatLon.ellipsoid.Intl1924,      transform: [   89.5,    93.8,    123.1,    -1.2,     0.0,     0.0,     0.156  ] },
    Irl1975:    { ellipsoid: LatLon.ellipsoid.AiryModified,  transform: [ -482.530, 130.596, -564.557,  -8.150,  -1.042,  -0.214,  -0.631  ] },
    NAD27:      { ellipsoid: LatLon.ellipsoid.Clarke1866,    transform: [    8,    -160,     -176,       0,       0,       0,       0      ] },
    NAD83:      { ellipsoid: LatLon.ellipsoid.GRS80,         transform: [    1.004,  -1.910,   -0.515,  -0.0015,  0.0267,  0.00034, 0.011  ] },
    NTF:        { ellipsoid: LatLon.ellipsoid.Clarke1880IGN, transform: [  168,      60,     -320,       0,       0,       0,       0      ] },
    OSGB36:     { ellipsoid: LatLon.ellipsoid.Airy1830,      transform: [ -446.448, 125.157, -542.060,  20.4894, -0.1502, -0.2470, -0.8421 ] },
    Potsdam:    { ellipsoid: LatLon.ellipsoid.Bessel1841,    transform: [ -582,    -105,     -414,      -8.3,     1.04,    0.35,   -3.08   ] },
    TokyoJapan: { ellipsoid: LatLon.ellipsoid.Bessel1841,    transform: [  148,    -507,     -685,       0,       0,       0,       0      ] },
    WGS72:      { ellipsoid: LatLon.ellipsoid.WGS72,         transform: [    0,       0,     -4.5,      -0.22,    0,       0,       0.554  ] },
    WGS84:      { ellipsoid: LatLon.ellipsoid.WGS84,         transform: [    0.0,     0.0,      0.0,     0.0,     0.0,     0.0,     0.0    ] },
};
/* sources:
 * - ED50:          www.gov.uk/guidance/oil-and-gas-petroleum-operations-notices#pon-4
 * - Irl1975:       www.osi.ie/wp-content/uploads/2015/05/transformations_booklet.pdf
 *   ... note: many sources have opposite sign to rotations - to be checked!
 * - NAD27:         en.wikipedia.org/wiki/Helmert_transformation
 * - NAD83: (2009); www.uvm.edu/giv/resources/WGS84_NAD83.pdf
 *   ... note: functionally ≡ WGS84 - if you *really* need to convert WGS84<->NAD83, you need more knowledge than this!
 * - NTF:           Nouvelle Triangulation Francaise geodesie.ign.fr/contenu/fichiers/Changement_systeme_geodesique.pdf
 * - OSGB36:        www.ordnancesurvey.co.uk/docs/support/guide-coordinate-systems-great-britain.pdf
 * - Potsdam:       kartoweb.itc.nl/geometrics/Coordinate%20transformations/coordtrans.html
 * - TokyoJapan:    www.geocachingtoolbox.com?page=datumEllipsoidDetails
 * - WGS72:         www.icao.int/safety/pbn/documentation/eurocontrol/eurocontrol wgs 84 implementation manual.pdf
 *
 * more transform parameters are available from earth-info.nga.mil/GandG/coordsys/datums/NATO_DT.pdf,
 * www.fieldenmaps.info/cconv/web/cconv_params.js
 */


/**
 * Converts ‘this’ lat/lon coordinate to new coordinate system.
 *
 * @param   {LatLon.datum} toDatum - Datum this coordinate is to be converted to.
 * @returns {LatLon} This point converted to new datum.
 *
 * @example
 *     var pWGS84 = new LatLon(51.4778, -0.0016, LatLon.datum.WGS84);
 *     var pOSGB = pWGS84.convertDatum(LatLon.datum.OSGB36); // 51.4773°N, 000.0000°E
 */
LatLon.prototype.convertDatum = function(toDatum) {
    var oldLatLon = this;
    var transform = null;

    if (oldLatLon.datum == LatLon.datum.WGS84) {
        // converting from WGS 84
        transform = toDatum.transform;
    }
    if (toDatum == LatLon.datum.WGS84) {
        // converting to WGS 84; use inverse transform (don't overwrite original!)
        transform = [];
        for (var p=0; p<7; p++) transform[p] = -oldLatLon.datum.transform[p];
    }
    if (transform == null) {
        // neither this.datum nor toDatum are WGS84: convert this to WGS84 first
        oldLatLon = this.convertDatum(LatLon.datum.WGS84);
        transform = toDatum.transform;
    }

    var oldCartesian = oldLatLon.toCartesian();                // convert polar to cartesian...
    var newCartesian = oldCartesian.applyTransform(transform); // ...apply transform...
    var newLatLon = newCartesian.toLatLonE(toDatum);           // ...and convert cartesian to polar

    return newLatLon;
};


/**
 * Converts ‘this’ point from (geodetic) latitude/longitude coordinates to (geocentric) cartesian
 * (x/y/z) coordinates.
 *
 * @returns {Vector3d} Vector pointing to lat/lon point, with x, y, z in metres from earth centre.
 */
LatLon.prototype.toCartesian = function() {
    var φ = this.lat.toRadians(), λ = this.lon.toRadians();
    var h = 0; // height above ellipsoid - not currently used
    var a = this.datum.ellipsoid.a, f = this.datum.ellipsoid.f;

    var sinφ = Math.sin(φ), cosφ = Math.cos(φ);
    var sinλ = Math.sin(λ), cosλ = Math.cos(λ);

    var eSq = 2*f - f*f;                      // 1st eccentricity squared ≡ (a²-b²)/a²
    var ν = a / Math.sqrt(1 - eSq*sinφ*sinφ); // radius of curvature in prime vertical

    var x = (ν+h) * cosφ * cosλ;
    var y = (ν+h) * cosφ * sinλ;
    var z = (ν*(1-eSq)+h) * sinφ;

    var point = new Vector3d(x, y, z);

    return point;
};


/**
 * Converts ‘this’ (geocentric) cartesian (x/y/z) point to (ellipsoidal geodetic) latitude/longitude
 * coordinates on specified datum.
 *
 * Uses Bowring’s (1985) formulation for μm precision in concise form.
 *
 * @param {LatLon.datum.transform} datum - Datum to use when converting point.
 */
Vector3d.prototype.toLatLonE = function(datum) {
    var x = this.x, y = this.y, z = this.z;
    var a = datum.ellipsoid.a, b = datum.ellipsoid.b, f = datum.ellipsoid.f;

    var e2 = 2*f - f*f;   // 1st eccentricity squared ≡ (a²-b²)/a²
    var ε2 = e2 / (1-e2); // 2nd eccentricity squared ≡ (a²-b²)/b²
    var p = Math.sqrt(x*x + y*y); // distance from minor axis
    var R = Math.sqrt(p*p + z*z); // polar radius

    // parametric latitude (Bowring eqn 17, replacing tanβ = z·a / p·b)
    var tanβ = (b*z)/(a*p) * (1+ε2*b/R);
    var sinβ = tanβ / Math.sqrt(1+tanβ*tanβ);
    var cosβ = sinβ / tanβ;

    // geodetic latitude (Bowring eqn 18: tanφ = z+ε²bsin³β / p−e²cos³β)
    var φ = isNaN(cosβ) ? 0 : Math.atan2(z + ε2*b*sinβ*sinβ*sinβ, p - e2*a*cosβ*cosβ*cosβ);

    // longitude
    var λ = Math.atan2(y, x);

    // height above ellipsoid (Bowring eqn 7) [not currently used]
    var sinφ = Math.sin(φ), cosφ = Math.cos(φ);
    var ν = a/Math.sqrt(1-e2*sinφ*sinφ); // length of the normal terminated by the minor axis
    var h = p*cosφ + z*sinφ - (a*a/ν);

    var point = new LatLon(φ.toDegrees(), λ.toDegrees(), datum);

    return point;
};

/**
 * Applies Helmert transform to ‘this’ point using transform parameters t.
 *
 * @private
 * @param   {number[]} t - Transform to apply to this point.
 * @returns {Vector3} Transformed point.
 */
Vector3d.prototype.applyTransform = function(t)   {
    // this point
    var x1 = this.x, y1 = this.y, z1 = this.z;

    // transform parameters
    var tx = t[0];                    // x-shift
    var ty = t[1];                    // y-shift
    var tz = t[2];                    // z-shift
    var s1 = t[3]/1e6 + 1;            // scale: normalise parts-per-million to (s+1)
    var rx = (t[4]/3600).toRadians(); // x-rotation: normalise arcseconds to radians
    var ry = (t[5]/3600).toRadians(); // y-rotation: normalise arcseconds to radians
    var rz = (t[6]/3600).toRadians(); // z-rotation: normalise arcseconds to radians

    // apply transform
    var x2 = tx + x1*s1 - y1*rz + z1*ry;
    var y2 = ty + x1*rz + y1*s1 - z1*rx;
    var z2 = tz - x1*ry + y1*rx + z1*s1;

    return new Vector3d(x2, y2, z2);
};


/**
 * Returns a string representation of ‘this’ point, formatted as degrees, degrees+minutes, or
 * degrees+minutes+seconds.
 *
 * @param   {string} [format=dms] - Format point as 'd', 'dm', 'dms'.
 * @param   {number} [dp=0|2|4] - Number of decimal places to use - default 0 for dms, 2 for dm, 4 for d.
 * @returns {string} Comma-separated latitude/longitude.
 */
LatLon.prototype.toString = function(format, dp) {
    return Dms.toLat(this.lat, format, dp) + ', ' + Dms.toLon(this.lon, format, dp);
};


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

/** Extend Number object with method to convert numeric degrees to radians */
if (Number.prototype.toRadians === undefined) {
    Number.prototype.toRadians = function() { return this * Math.PI / 180; };
}

/** Extend Number object with method to convert radians to numeric (signed) degrees */
if (Number.prototype.toDegrees === undefined) {
    Number.prototype.toDegrees = function() { return this * 180 / Math.PI; };
}

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
if (typeof module != 'undefined' && module.exports) module.exports = LatLon, module.exports.Vector3d = Vector3d; // ≡ export { LatLon as default, Vector3d }




/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
/* Vincenty Direct and Inverse Solution of Geodesics on the Ellipsoid (c) Chris Veness 2002-2017  */
/*                                                                                   MIT Licence  */
/* www.movable-type.co.uk/scripts/latlong-vincenty.html                                           */
/* www.movable-type.co.uk/scripts/geodesy/docs/module-latlon-vincenty.html                        */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

'use strict';
if (typeof module!='undefined' && module.exports) var LatLon = require('./latlon-ellipsoidal.js'); // ≡ import LatLon from 'latlon-ellipsoidal.js'


/**
 * Direct and inverse solutions of geodesics on the ellipsoid using Vincenty formulae.
 *
 * From: T Vincenty, "Direct and Inverse Solutions of Geodesics on the Ellipsoid with application of
 *       nested equations", Survey Review, vol XXIII no 176, 1975.
 *       www.ngs.noaa.gov/PUBS_LIB/inverse.pdf.
 *
 * @module  latlon-vincenty
 * @extends latlon-ellipsoidal
 */
/** @class LatLon */


/**
 * Returns the distance between ‘this’ point and destination point along a geodesic, using Vincenty
 * inverse solution.
 *
 * Note: the datum used is of ‘this’ point; distance is on the surface of the ellipsoid (height is
 * ignored).
 *
 * @param   {LatLon} point - Latitude/longitude of destination point.
 * @returns (Number} Distance in metres between points or NaN if failed to converge.
 *
 * @example
 *   var p1 = new LatLon(50.06632, -5.71475);
 *   var p2 = new LatLon(58.64402, -3.07009);
 *   var d = p1.distanceTo(p2); // 969,954.166 m
 */
LatLon.prototype.distanceTo = function(point) {
    if (!(point instanceof LatLon)) throw new TypeError('point is not LatLon object');

    try {
        return Number(this.inverse(point).distance.toFixed(3)); // round to 1mm precision
    } catch (e) {
        return NaN; // failed to converge
    }
};


/**
 * Returns the initial bearing (forward azimuth) to travel along a geodesic from ‘this’ point to the
 * specified point, using Vincenty inverse solution.
 *
 * Note: the datum used is of ‘this’ point.
 *
 * @param   {LatLon} point - Latitude/longitude of destination point.
 * @returns {number}  initial Bearing in degrees from north (0°..360°) or NaN if failed to converge.
 *
 * @example
 *   var p1 = new LatLon(50.06632, -5.71475);
 *   var p2 = new LatLon(58.64402, -3.07009);
 *   var b1 = p1.initialBearingTo(p2); // 9.1419°
 */
LatLon.prototype.initialBearingTo = function(point) {
    if (!(point instanceof LatLon)) throw new TypeError('point is not LatLon object');

    try {
        return Number(this.inverse(point).initialBearing.toFixed(9)); // round to 0.00001″ precision
    } catch (e) {
        return NaN; // failed to converge
    }
};


/**
 * Returns the final bearing (reverse azimuth) having travelled along a geodesic from ‘this’ point
 * to the specified point, using Vincenty inverse solution.
 *
 * Note: the datum used is of ‘this’ point.
 *
 * @param   {LatLon} point - Latitude/longitude of destination point.
 * @returns {number}  Initial bearing in degrees from north (0°..360°) or NaN if failed to converge.
 *
 * @example
 *   var p1 = new LatLon(50.06632, -5.71475);
 *   var p2 = new LatLon(58.64402, -3.07009);
 *   var b2 = p1.finalBearingTo(p2); // 11.2972°
 */
LatLon.prototype.finalBearingTo = function(point) {
    if (!(point instanceof LatLon)) throw new TypeError('point is not LatLon object');

    try {
        return Number(this.inverse(point).finalBearing.toFixed(9)); // round to 0.00001″ precision
    } catch (e) {
        return NaN; // failed to converge
    }
};


/**
 * Returns the destination point having travelled the given distance along a geodesic given by
 * initial bearing from ‘this’ point, using Vincenty direct solution.
 *
 * Note: the datum used is of ‘this’ point; distance is on the surface of the ellipsoid (height is
 * ignored).
 *
 * @param   {number} distance - Distance travelled along the geodesic in metres.
 * @param   {number} initialBearing - Initial bearing in degrees from north.
 * @returns {LatLon} Destination point.
 *
 * @example
 *   var p1 = new LatLon(-37.95103, 144.42487);
 *   var p2 = p1.destinationPoint(54972.271, 306.86816); // 37.6528°S, 143.9265°E
 */
LatLon.prototype.destinationPoint = function(distance, initialBearing) {
    return this.direct(Number(distance), Number(initialBearing)).point;
};


/**
 * Returns the final bearing (reverse azimuth) having travelled along a geodesic given by initial
 * bearing for a given distance from ‘this’ point, using Vincenty direct solution.
 *
 * Note: the datum used is of ‘this’ point; distance is on the surface of the ellipsoid (height is
 * ignored).
 *
 * @param   {number} distance - Distance travelled along the geodesic in metres.
 * @param   {LatLon} initialBearing - Initial bearing in degrees from north.
 * @returns {number} Final bearing in degrees from north (0°..360°).
 *
 * @example
 *   var p1 = new LatLon(-37.95103, 144.42487);
 *   var b2 = p1.finalBearingOn(306.86816, 54972.271); // 307.1736°
 */
LatLon.prototype.finalBearingOn = function(distance, initialBearing) {
    return Number(this.direct(Number(distance), Number(initialBearing)).finalBearing.toFixed(9)); // round to 0.00001″ precision
};


/**
 * Vincenty direct calculation.
 *
 * @private
 * @param   {number} distance - Distance along bearing in metres.
 * @param   {number} initialBearing - Initial bearing in degrees from north.
 * @returns (Object} Object including point (destination point), finalBearing.
 * @throws  {Error}  If formula failed to converge.
 */
LatLon.prototype.direct = function(distance, initialBearing) {
    var φ1 = this.lat.toRadians(), λ1 = this.lon.toRadians();
    var α1 = initialBearing.toRadians();
    var s = distance;

    var a = this.datum.ellipsoid.a, b = this.datum.ellipsoid.b, f = this.datum.ellipsoid.f;

    var sinα1 = Math.sin(α1);
    var cosα1 = Math.cos(α1);

    var tanU1 = (1-f) * Math.tan(φ1), cosU1 = 1 / Math.sqrt((1 + tanU1*tanU1)), sinU1 = tanU1 * cosU1;
    var σ1 = Math.atan2(tanU1, cosα1);
    var sinα = cosU1 * sinα1;
    var cosSqα = 1 - sinα*sinα;
    var uSq = cosSqα * (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 cos2σM, sinσ, cosσ, Δσ;

    var σ = s / (b*A), σʹ, iterations = 0;
    do {
        cos2σM = Math.cos(2*σ1 + σ);
        sinσ = Math.sin(σ);
        cosσ = Math.cos(σ);
        Δσ = B*sinσ*(cos2σM+B/4*(cosσ*(-1+2*cos2σM*cos2σM)-
            B/6*cos2σM*(-3+4*sinσ*sinσ)*(-3+4*cos2σM*cos2σM)));
        σʹ = σ;
        σ = s / (b*A) + Δσ;
    } while (Math.abs(σ-σʹ) > 1e-12 && ++iterations<100);
    if (iterations >= 100) throw new Error('Formula failed to converge'); // not possible!

    var x = sinU1*sinσ - cosU1*cosσ*cosα1;
    var φ2 = Math.atan2(sinU1*cosσ + cosU1*sinσ*cosα1, (1-f)*Math.sqrt(sinα*sinα + x*x));
    var λ = Math.atan2(sinσ*sinα1, cosU1*cosσ - sinU1*sinσ*cosα1);
    var C = f/16*cosSqα*(4+f*(4-3*cosSqα));
    var L = λ - (1-C) * f * sinα *
        (σ + C*sinσ*(cos2σM+C*cosσ*(-1+2*cos2σM*cos2σM)));
    var λ2 = (λ1+L+3*Math.PI)%(2*Math.PI) - Math.PI;  // normalise to -180..+180

    var α2 = Math.atan2(sinα, -x);
    α2 = (α2 + 2*Math.PI) % (2*Math.PI); // normalise to 0..360

    return {
        point:        new LatLon(φ2.toDegrees(), λ2.toDegrees(), this.datum),
        finalBearing: α2.toDegrees(),
        iterations:   iterations,
    };
};


/**
 * Vincenty inverse calculation.
 *
 * @private
 * @param   {LatLon} point - Latitude/longitude of destination point.
 * @returns {Object} Object including distance, initialBearing, finalBearing.
 * @throws  {Error}  If λ > π or formula failed to converge.
 */
LatLon.prototype.inverse = function(point) {
    var p1 = this, p2 = point;
    if (p1.lon == -180) p1.lon = 180;
    var φ1 = p1.lat.toRadians(), λ1 = p1.lon.toRadians();
    var φ2 = p2.lat.toRadians(), λ2 = p2.lon.toRadians();

    var a = this.datum.ellipsoid.a, b = this.datum.ellipsoid.b, f = this.datum.ellipsoid.f;

    var L = λ2 - λ1;
    var tanU1 = (1-f) * Math.tan(φ1), cosU1 = 1 / Math.sqrt((1 + tanU1*tanU1)), sinU1 = tanU1 * cosU1;
    var tanU2 = (1-f) * Math.tan(φ2), cosU2 = 1 / Math.sqrt((1 + tanU2*tanU2)), sinU2 = tanU2 * cosU2;

    var sinλ, cosλ, sinSqσ, sinσ=0, cosσ=0, σ=0, sinα, cosSqα=0, cos2σM=0, C;

    var λ = L, λʹ, iterations = 0, antimeridian = Math.abs(L) > Math.PI;
    do {
        sinλ = Math.sin(λ);
        cosλ = Math.cos(λ);
        sinSqσ = (cosU2*sinλ) * (cosU2*sinλ) + (cosU1*sinU2-sinU1*cosU2*cosλ) * (cosU1*sinU2-sinU1*cosU2*cosλ);
        if (sinSqσ == 0) break; // co-incident points
        sinσ = Math.sqrt(sinSqσ);
        cosσ = sinU1*sinU2 + cosU1*cosU2*cosλ;
        σ = Math.atan2(sinσ, cosσ);
        sinα = cosU1 * cosU2 * sinλ / sinσ;
        cosSqα = 1 - sinα*sinα;
        cos2σM = (cosSqα != 0) ? (cosσ - 2*sinU1*sinU2/cosSqα) : 0; // equatorial line: cosSqα=0 (§6)
        C = f/16*cosSqα*(4+f*(4-3*cosSqα));
        λʹ = λ;
        λ = L + (1-C) * f * sinα * (σ + C*sinσ*(cos2σM+C*cosσ*(-1+2*cos2σM*cos2σM)));
        var iterationCheck = antimeridian ? Math.abs(λ)-Math.PI : Math.abs(λ);
        if (iterationCheck > Math.PI) throw new Error('λ > π');
    } while (Math.abs(λ-λʹ) > 1e-12 && ++iterations<1000);
    if (iterations >= 1000) throw new Error('Formula failed to converge');

    var uSq = cosSqα * (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 Δσ = B*sinσ*(cos2σM+B/4*(cosσ*(-1+2*cos2σM*cos2σM)-
        B/6*cos2σM*(-3+4*sinσ*sinσ)*(-3+4*cos2σM*cos2σM)));

    var s = b*A*(σ-Δσ);

    var α1 = Math.atan2(cosU2*sinλ,  cosU1*sinU2-sinU1*cosU2*cosλ);
    var α2 = Math.atan2(cosU1*sinλ, -sinU1*cosU2+cosU1*sinU2*cosλ);

    α1 = (α1 + 2*Math.PI) % (2*Math.PI); // normalise to 0..360
    α2 = (α2 + 2*Math.PI) % (2*Math.PI); // normalise to 0..360

    return {
        distance:       s,
        initialBearing: s==0 ? NaN : α1.toDegrees(),
        finalBearing:   s==0 ? NaN : α2.toDegrees(),
        iterations:     iterations,
    };
};


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

/** Extend Number object with method to convert numeric degrees to radians */
if (Number.prototype.toRadians === undefined) {
    Number.prototype.toRadians = function() { return this * Math.PI / 180; };
}

/** Extend Number object with method to convert radians to numeric (signed) degrees */
if (Number.prototype.toDegrees === undefined) {
    Number.prototype.toDegrees = function() { return this * 180 / Math.PI; };
}

/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
if (typeof module != 'undefined' && module.exports) module.exports = LatLon; // ≡ export default LatLon




/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
/* Geodesy representation conversion functions                        (c) Chris Veness 2002-2017  */
/*                                                                                   MIT Licence  */
/* www.movable-type.co.uk/scripts/latlong.html                                                    */
/* www.movable-type.co.uk/scripts/geodesy/docs/module-dms.html                                    */
/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */

'use strict';
/* eslint no-irregular-whitespace: [2, { skipComments: true }] */


/**
 * Latitude/longitude points may be represented as decimal degrees, or subdivided into sexagesimal
 * minutes and seconds.
 *
 * @module dms
 */


/**
 * Functions for parsing and representing degrees / minutes / seconds.
 * @class Dms
 */
var Dms = {};

// note Unicode Degree = U+00B0. Prime = U+2032, Double prime = U+2033


/**
 * Parses string representing degrees/minutes/seconds into numeric degrees.
 *
 * This is very flexible on formats, allowing signed decimal degrees, or deg-min-sec optionally
 * suffixed by compass direction (NSEW). A variety of separators are accepted (eg 3° 37′ 09″W).
 * Seconds and minutes may be omitted.
 *
 * @param   {string|number} dmsStr - Degrees or deg/min/sec in variety of formats.
 * @returns {number} Degrees as decimal number.
 *
 * @example
 *     var lat = Dms.parseDMS('51° 28′ 40.12″ N');
 *     var lon = Dms.parseDMS('000° 00′ 05.31″ W');
 *     var p1 = new LatLon(lat, lon); // 51.4778°N, 000.0015°W
 */
Dms.parseDMS = function(dmsStr) {
    // check for signed decimal degrees without NSEW, if so return it directly
    if (typeof dmsStr == 'number' && isFinite(dmsStr)) return Number(dmsStr);

    // strip off any sign or compass dir'n & split out separate d/m/s
    var dms = String(dmsStr).trim().replace(/^-/, '').replace(/[NSEW]$/i, '').split(/[^0-9.,]+/);
    if (dms[dms.length-1]=='') dms.splice(dms.length-1);  // from trailing symbol

    if (dms == '') return NaN;

    // and convert to decimal degrees...
    var deg;
    switch (dms.length) {
        case 3:  // interpret 3-part result as d/m/s
            deg = dms[0]/1 + dms[1]/60 + dms[2]/3600;
            break;
        case 2:  // interpret 2-part result as d/m
            deg = dms[0]/1 + dms[1]/60;
            break;
        case 1:  // just d (possibly decimal) or non-separated dddmmss
            deg = dms[0];
            // check for fixed-width unseparated format eg 0033709W
            //if (/[NS]/i.test(dmsStr)) deg = '0' + deg;  // - normalise N/S to 3-digit degrees
            //if (/[0-9]{7}/.test(deg)) deg = deg.slice(0,3)/1 + deg.slice(3,5)/60 + deg.slice(5)/3600;
            break;
        default:
            return NaN;
    }
    if (/^-|[WS]$/i.test(dmsStr.trim())) deg = -deg; // take '-', west and south as -ve

    return Number(deg);
};


/**
 * Separator character to be used to separate degrees, minutes, seconds, and cardinal directions.
 *
 * Set to '\u202f' (narrow no-break space) for improved formatting.
 *
 * @example
 *   var p = new LatLon(51.2, 0.33);  // 51°12′00.0″N, 000°19′48.0″E
 *   Dms.separator = '\u202f';        // narrow no-break space
 *   var pʹ = new LatLon(51.2, 0.33); // 51° 12′ 00.0″ N, 000° 19′ 48.0″ E
 */
Dms.separator = '';


/**
 * Converts decimal degrees to deg/min/sec format
 *  - degree, prime, double-prime symbols are added, but sign is discarded, though no compass
 *    direction is added.
 *
 * @private
 * @param   {number} deg - Degrees to be formatted as specified.
 * @param   {string} [format=dms] - Return value as 'd', 'dm', 'dms' for deg, deg+min, deg+min+sec.
 * @param   {number} [dp=0|2|4] - Number of decimal places to use – default 0 for dms, 2 for dm, 4 for d.
 * @returns {string} Degrees formatted as deg/min/secs according to specified format.
 */
Dms.toDMS = function(deg, format, dp) {
    if (isNaN(deg)) return null;  // give up here if we can't make a number from deg

    // default values
    if (format === undefined) format = 'dms';
    if (dp === undefined) {
        switch (format) {
            case 'd':    case 'deg':         dp = 4; break;
            case 'dm':   case 'deg+min':     dp = 2; break;
            case 'dms':  case 'deg+min+sec': dp = 0; break;
            default:    format = 'dms'; dp = 0;  // be forgiving on invalid format
        }
    }

    deg = Math.abs(deg);  // (unsigned result ready for appending compass dir'n)

    var dms, d, m, s;
    switch (format) {
        default: // invalid format spec!
        case 'd': case 'deg':
            d = deg.toFixed(dp);                       // round/right-pad degrees
            if (d<100) d = '0' + d;                    // left-pad with leading zeros (note may include decimals)
            if (d<10) d = '0' + d;
            dms = d + '°';
            break;
        case 'dm': case 'deg+min':
            d = Math.floor(deg);                       // get component deg
            m = ((deg*60) % 60).toFixed(dp);           // get component min & round/right-pad
            if (m == 60) { m = 0; d++; }               // check for rounding up
            d = ('000'+d).slice(-3);                   // left-pad with leading zeros
            if (m<10) m = '0' + m;                     // left-pad with leading zeros (note may include decimals)
            dms = d + '°'+Dms.separator + m + '′';
            break;
        case 'dms': case 'deg+min+sec':
            d = Math.floor(deg);                       // get component deg
            m = Math.floor((deg*3600)/60) % 60;        // get component min
            s = (deg*3600 % 60).toFixed(dp);           // get component sec & round/right-pad
            if (s == 60) { s = (0).toFixed(dp); m++; } // check for rounding up
            if (m == 60) { m = 0; d++; }               // check for rounding up
            d = ('000'+d).slice(-3);                   // left-pad with leading zeros
            m = ('00'+m).slice(-2);                    // left-pad with leading zeros
            if (s<10) s = '0' + s;                     // left-pad with leading zeros (note may include decimals)
            dms = d + '°'+Dms.separator + m + '′'+Dms.separator + s + '″';
            break;
    }

    return dms;
};


/**
 * Converts numeric degrees to deg/min/sec latitude (2-digit degrees, suffixed with N/S).
 *
 * @param   {number} deg - Degrees to be formatted as specified.
 * @param   {string} [format=dms] - Return value as 'd', 'dm', 'dms' for deg, deg+min, deg+min+sec.
 * @param   {number} [dp=0|2|4] - Number of decimal places to use – default 0 for dms, 2 for dm, 4 for d.
 * @returns {string} Degrees formatted as deg/min/secs according to specified format.
 */
Dms.toLat = function(deg, format, dp) {
    var lat = Dms.toDMS(deg, format, dp);
    return lat===null ? '–' : lat.slice(1)+Dms.separator + (deg<0 ? 'S' : 'N');  // knock off initial '0' for lat!
};


/**
 * Convert numeric degrees to deg/min/sec longitude (3-digit degrees, suffixed with E/W)
 *
 * @param   {number} deg - Degrees to be formatted as specified.
 * @param   {string} [format=dms] - Return value as 'd', 'dm', 'dms' for deg, deg+min, deg+min+sec.
 * @param   {number} [dp=0|2|4] - Number of decimal places to use – default 0 for dms, 2 for dm, 4 for d.
 * @returns {string} Degrees formatted as deg/min/secs according to specified format.
 */
Dms.toLon = function(deg, format, dp) {
    var lon = Dms.toDMS(deg, format, dp);
    return lon===null ? '–' : lon+Dms.separator + (deg<0 ? 'W' : 'E');
};


/**
 * Converts numeric degrees to deg/min/sec as a bearing (0°..360°)
 *
 * @param   {number} deg - Degrees to be formatted as specified.
 * @param   {string} [format=dms] - Return value as 'd', 'dm', 'dms' for deg, deg+min, deg+min+sec.
 * @param   {number} [dp=0|2|4] - Number of decimal places to use – default 0 for dms, 2 for dm, 4 for d.
 * @returns {string} Degrees formatted as deg/min/secs according to specified format.
 */
Dms.toBrng = function(deg, format, dp) {
    deg = (Number(deg)+360) % 360;  // normalise -ve values to 180°..360°
    var brng =  Dms.toDMS(deg, format, dp);
    return brng===null ? '–' : brng.replace('360', '0');  // just in case rounding took us up to 360°!
};


/**
 * Returns compass point (to given precision) for supplied bearing.
 *
 * @param   {number} bearing - Bearing in degrees from north.
 * @param   {number} [precision=3] - Precision (1:cardinal / 2:intercardinal / 3:secondary-intercardinal).
 * @returns {string} Compass point for supplied bearing.
 *
 * @example
 *   var point = Dms.compassPoint(24);    // point = 'NNE'
 *   var point = Dms.compassPoint(24, 1); // point = 'N'
 */
Dms.compassPoint = function(bearing, precision) {
    if (precision === undefined) precision = 3;
    // note precision could be extended to 4 for quarter-winds (eg NbNW), but I think they are little used

    bearing = ((bearing%360)+360)%360; // normalise to range 0..360°

    var cardinals = [
        'N', 'NNE', 'NE', 'ENE',
        'E', 'ESE', 'SE', 'SSE',
        'S', 'SSW', 'SW', 'WSW',
        'W', 'WNW', 'NW', 'NNW' ];
    var n = 4 * Math.pow(2, precision-1); // no of compass points at req’d precision (1=>4, 2=>8, 3=>16)
    var cardinal = cardinals[Math.round(bearing*n/360)%n * 16/n];

    return cardinal;
};


/* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  */
if (typeof module != 'undefined' && module.exports) module.exports = Dms; // ≡ export default Dms