Find a point in a polyline which is closest to a latlng

Question

i have a polyine which i have drawn with latlngs obtained from google maps directions service. Now i want to find a point on the polyline that is closest to a given point.

The obvious way (to me) is to kind of loop through all the points in the polyline and find the distance between them and the given point, however this is inefficient because the points on the polyline can potentially be large.

I would be glad to hear any alternatives of doing this. Thanks in advance.

Answer 1

See Bill Chadwick's example here:

http://www.bdcc.co.uk/Gmaps/BdccGmapBits.htm

above example ported to v3 (code at bottom of this answer)

on his page under:

DISTANCE POINT TO POLYLINE OR POLYGON

from that post:

There is a similar, better demo here http://wtp2.appspot.com/cSnapToRouteDemo.html

It is finding the closest point on the line to the mouse. Also note that it is a Google Maps API v2 example (but the principle with v3 would be the same).

// Code to find the distance in metres between a lat/lng point and a polyline of lat/lng points
// All in WGS84. Free for any use.
//
// Bill Chadwick 2007
// updated to Google Maps API v3, Lawrence Ross 2014

        // Construct a bdccGeo from its latitude and longitude in degrees
        function bdccGeo(lat, lon) 
        {
            var theta = (lon * Math.PI / 180.0);
            var rlat = bdccGeoGeocentricLatitude(lat * Math.PI / 180.0);
            var c = Math.cos(rlat); 
            this.x = c * Math.cos(theta);
            this.y = c * Math.sin(theta);
            this.z = Math.sin(rlat);        
        }
        bdccGeo.prototype = new bdccGeo();

        // internal helper functions =========================================

        // Convert from geographic to geocentric latitude (radians).
        function bdccGeoGeocentricLatitude(geographicLatitude) 
        {
            var flattening = 1.0 / 298.257223563;//WGS84
            var f = (1.0 - flattening) * (1.0 - flattening);
            return Math.atan((Math.tan(geographicLatitude) * f));
        }

         // Returns the two antipodal points of intersection of two great
         // circles defined by the arcs geo1 to geo2 and
         // geo3 to geo4. Returns a point as a Geo, use .antipode to get the other point
        function bdccGeoGetIntersection( geo1,  geo2,  geo3,  geo4) 
        {
            var geoCross1 = geo1.crossNormalize(geo2);
            var geoCross2 = geo3.crossNormalize(geo4);
            return geoCross1.crossNormalize(geoCross2);
        }

        //from Radians to Meters
        function bdccGeoRadiansToMeters(rad)
        {
            return rad * 6378137.0; // WGS84 Equatorial Radius in Meters
        }

        //from Meters to Radians
        function bdccGeoMetersToRadians(m)
        {
            return m / 6378137.0; // WGS84 Equatorial Radius in Meters
        }

        // properties =================================================


        bdccGeo.prototype.getLatitudeRadians = function() 
        {
            return (bdccGeoGeographicLatitude(Math.atan2(this.z,
                Math.sqrt((this.x * this.x) + (this.y * this.y)))));
        }

        bdccGeo.prototype.getLongitudeRadians = function() 
        {
            return (Math.atan2(this.y, this.x));
        }

        bdccGeo.prototype.getLatitude = function() 
        {
            return this.getLatitudeRadians()  * 180.0 / Math.PI;
        }

        bdccGeo.prototype.getLongitude = function() 
        {
            return this.getLongitudeRadians()  * 180.0 / Math.PI ;
        }

        // Methods =================================================

        //Maths
        bdccGeo.prototype.dot = function( b) 
        {
            return ((this.x * b.x) + (this.y * b.y) + (this.z * b.z));
        }

        //More Maths
        bdccGeo.prototype.crossLength = function( b) 
        {
            var x = (this.y * b.z) - (this.z * b.y);
            var y = (this.z * b.x) - (this.x * b.z);
            var z = (this.x * b.y) - (this.y * b.x);
            return Math.sqrt((x * x) + (y * y) + (z * z));
        }

      //More Maths
        bdccGeo.prototype.scale = function( s) 
        {
            var r = new bdccGeo(0,0);
            r.x = this.x * s;
            r.y = this.y * s;
            r.z = this.z * s;
            return r;
        }

        // More Maths
        bdccGeo.prototype.crossNormalize = function( b) 
        {
            var x = (this.y * b.z) - (this.z * b.y);
            var y = (this.z * b.x) - (this.x * b.z);
            var z = (this.x * b.y) - (this.y * b.x);
            var L = Math.sqrt((x * x) + (y * y) + (z * z));
            var r = new bdccGeo(0,0);
            r.x = x / L;
            r.y = y / L;
            r.z = z / L;
            return r;
        }

      // point on opposite side of the world to this point
        bdccGeo.prototype.antipode = function() 
        {
            return this.scale(-1.0);
        }






        //distance in radians from this point to point v2
        bdccGeo.prototype.distance = function( v2) 
        {
            return Math.atan2(v2.crossLength(this), v2.dot(this));
        }

      //returns in meters the minimum of the perpendicular distance of this point from the line segment geo1-geo2
      //and the distance from this point to the line segment ends in geo1 and geo2 
        bdccGeo.prototype.distanceToLineSegMtrs = function(geo1, geo2)
        {            

            //point on unit sphere above origin and normal to plane of geo1,geo2
            //could be either side of the plane
            var p2 = geo1.crossNormalize(geo2); 

            // intersection of GC normal to geo1/geo2 passing through p with GC geo1/geo2
            var ip = bdccGeoGetIntersection(geo1,geo2,this,p2); 

            //need to check that ip or its antipode is between p1 and p2
            var d = geo1.distance(geo2);
            var d1p = geo1.distance(ip);
            var d2p = geo2.distance(ip);
            //window.status = d + ", " + d1p + ", " + d2p;
            if ((d >= d1p) && (d >= d2p)) 
                return bdccGeoRadiansToMeters(this.distance(ip));
            else
            {
                ip = ip.antipode(); 
                d1p = geo1.distance(ip);
                d2p = geo2.distance(ip);
            }
            if ((d >= d1p) && (d >= d2p)) 
                return bdccGeoRadiansToMeters(this.distance(ip)); 
            else 
                return bdccGeoRadiansToMeters(Math.min(geo1.distance(this),geo2.distance(this))); 
        }

        // distance in meters from GLatLng point to GPolyline or GPolygon poly
        function bdccGeoDistanceToPolyMtrs(poly, point)
        {
            var d = 999999999;
            var i;
            var p = new bdccGeo(point.lat(),point.lng());
            for(i=0; i<(poly.getPath().getLength()-1); i++)
                 {
                    var p1 = poly.getPath().getAt(i);
                    var l1 = new bdccGeo(p1.lat(),p1.lng());
                    var p2 = poly.getPath().getAt(i+1);
                    var l2 = new bdccGeo(p2.lat(),p2.lng());
                    var dp = p.distanceToLineSegMtrs(l1,l2);
                    if(dp < d)
                        d = dp;    
                 }
             return d;
        }

        // get a new GLatLng distanceMeters away on the compass bearing azimuthDegrees
        // from the GLatLng point - accurate to better than 200m in 140km (20m in 14km) in the UK

        function bdccGeoPointAtRangeAndBearing (point, distanceMeters, azimuthDegrees) 
        {
             var latr = point.lat() * Math.PI / 180.0;
             var lonr = point.lng() * Math.PI / 180.0;

             var coslat = Math.cos(latr); 
             var sinlat = Math.sin(latr); 
             var az = azimuthDegrees* Math.PI / 180.0;
             var cosaz = Math.cos(az); 
             var sinaz = Math.sin(az); 
             var dr = distanceMeters / 6378137.0; // distance in radians using WGS84 Equatorial Radius
             var sind = Math.sin(dr); 
             var cosd = Math.cos(dr);

            return new google.maps.LatLng(Math.asin((sinlat * cosd) + (coslat * sind * cosaz)) * 180.0 / Math.PI,
            (Math.atan2((sind * sinaz), (coslat * cosd) - (sinlat * sind * cosaz)) + lonr) * 180.0 / Math.PI); 
        }

Answer 2

I needed a cleaner version that was ported to V3, so here it is:

/**
*   Snap marker to closest point on a line.
*
*   Based on Distance to line example by 
*   Marcelo, maps.forum.nu - http://maps.forum.nu/gm_mouse_dist_to_line.html 
*   Then 
*   @ work of Björn Brala - Swis BV who wrapped the algorithm in a class operating on GMap Objects
*   And now 
*   Bill Chadwick, who factored the basic algorithm out of the class (removing much intermediate storage of results)
*       and added distance along line to nearest point calculation
*   Followed by
*   Robert Crowe, who ported it to v3 of the Google Maps API and factored out the marker to make it more general.
*
*   Usage:
*
*   Create the class
*       var oSnap = new cSnapToRoute();
*
*   Initialize the subjects
*       oSnap.init(oMap, oPolyline);
*
**/

function cSnapToRoute() {

    this.routePoints = Array();
    this.routePixels = Array();
    this._oMap;
    this._oPolyline;

    /**
    *   @desc Initialize the objects.
    *   @param Map object
    *   @param GPolyline object - the 'route'
    **/
    this.init = function (oMap, oPolyline) {
        this._oMap = oMap;
        this._oPolyline = oPolyline;

        this.loadRouteData();   // Load needed data for point calculations
    }

    /**
    *   @desc internal use only, Load route points into RoutePixel array for calculations, do this whenever zoom changes 
    **/
    this.loadRouteData = function () {
        this.routePixels = new Array();
        var proj = this._oMap.getProjection();
        for (var i = 0; i < this._oPolyline.getPath().getLength(); i++) {
            var Px = proj.fromLatLngToPoint(this._oPolyline.getPath().getAt(i));
            this.routePixels.push(Px);
        }
    }

    /**
    *   @desc Get closest point on route to test point
    *   @param GLatLng() the test point
    *   @return new GLatLng();
    **/
    this.getClosestLatLng = function (latlng) {
        var r = this.distanceToLines(latlng);
        var proj = this._oMap.getProjection();
        return proj.fromPointToLatLng(new google.maps.Point(r.x, r.y));
    }

    /**
    *   @desc Get distance along route in meters of closest point on route to test point
    *   @param GLatLng() the test point
    *   @return distance in meters;
    **/
    this.getDistAlongRoute = function (latlng) {
        var r = this.distanceToLines(latlng);
        return this.getDistToLine(r.i, r.fTo);
    }

    /**
    *   @desc internal use only, gets test point xy and then calls fundamental algorithm
    **/
    this.distanceToLines = function (thisLatLng) {
        var tm = this._oMap;
        var proj = this._oMap.getProjection();
        var thisPx = proj.fromLatLngToPoint(thisLatLng);
        var routePixels = this.routePixels;
        return getClosestPointOnLines(thisPx, routePixels);
    }

    /**
    *   @desc internal use only, find distance along route to point nearest test point
    **/
    this.getDistToLine = function (iLine, fTo) {

        var routeOverlay = this._oPolyline;
        var d = 0;
        for (var n = 1 ; n < iLine ; n++) {
            d += routeOverlay.getPath().getAt(n - 1).distanceFrom(routeOverlay.getPath().getAt(n));
        }
        d += routeOverlay.getPath().getAt(iLine - 1).distanceFrom(routeOverlay.getPath().getAt(iLine)) * fTo;

        return d;
    }


}

/* desc Static function. Find point on lines nearest test point
   test point pXy with properties .x and .y
   lines defined by array aXys with nodes having properties .x and .y 
   return is object with .x and .y properties and property i indicating nearest segment in aXys 
   and property fFrom the fractional distance of the returned point from aXy[i-1]
   and property fTo the fractional distance of the returned point from aXy[i]   */


function getClosestPointOnLines(pXy, aXys) {

    var minDist;
    var fTo;
    var fFrom;
    var x;
    var y;
    var i;
    var dist;

    if (aXys.length > 1) {

        for (var n = 1 ; n < aXys.length ; n++) {

            if (aXys[n].x != aXys[n - 1].x) {
                var a = (aXys[n].y - aXys[n - 1].y) / (aXys[n].x - aXys[n - 1].x);
                var b = aXys[n].y - a * aXys[n].x;
                dist = Math.abs(a * pXy.x + b - pXy.y) / Math.sqrt(a * a + 1);
            }
            else
                dist = Math.abs(pXy.x - aXys[n].x)

            // length^2 of line segment 
            var rl2 = Math.pow(aXys[n].y - aXys[n - 1].y, 2) + Math.pow(aXys[n].x - aXys[n - 1].x, 2);

            // distance^2 of pt to end line segment
            var ln2 = Math.pow(aXys[n].y - pXy.y, 2) + Math.pow(aXys[n].x - pXy.x, 2);

            // distance^2 of pt to begin line segment
            var lnm12 = Math.pow(aXys[n - 1].y - pXy.y, 2) + Math.pow(aXys[n - 1].x - pXy.x, 2);

            // minimum distance^2 of pt to infinite line
            var dist2 = Math.pow(dist, 2);

            // calculated length^2 of line segment
            var calcrl2 = ln2 - dist2 + lnm12 - dist2;

            // redefine minimum distance to line segment (not infinite line) if necessary
            if (calcrl2 > rl2)
                dist = Math.sqrt(Math.min(ln2, lnm12));

            if ((minDist == null) || (minDist > dist)) {
                if (calcrl2 > rl2) {
                    if (lnm12 < ln2) {
                        fTo = 0;//nearer to previous point
                        fFrom = 1;
                    }
                    else {
                        fFrom = 0;//nearer to current point
                        fTo = 1;
                    }
                }
                else {
                    // perpendicular from point intersects line segment
                    fTo = ((Math.sqrt(lnm12 - dist2)) / Math.sqrt(rl2));
                    fFrom = ((Math.sqrt(ln2 - dist2)) / Math.sqrt(rl2));
                }
                minDist = dist;
                i = n;
            }
        }

        var dx = aXys[i - 1].x - aXys[i].x;
        var dy = aXys[i - 1].y - aXys[i].y;

        x = aXys[i - 1].x - (dx * fTo);
        y = aXys[i - 1].y - (dy * fTo);

    }

    return { 'x': x, 'y': y, 'i': i, 'fTo': fTo, 'fFrom': fFrom };
}

Answer 3

I do not think you can avoid checking all the points. What if the not checked point is the nearest one?

If you have to do this operation many times, you can choose a data structure that is optimized for such a search, quadtree for example. Note that you should not use lat lng as Descartes coordinates.

See also Finding nearest point in an efficient way That is for the 2D plane, and not for lat lng, but you can approximate: https://stackoverflow.com/a/16271669/59019

Answer 4

Inspired by jmihalicza's answer, i came up with this function to find the closest point in an array of LatLngs to a given LatLng.

function closest takes a LatLng(llng) and an array of LatLngs (listData) and finds the distance between each latlng in the array and the given latlng, it then finds the least distance and returns the Latlng from the list which provided that distance.

function closest(llng, listData) {
    var arr     = listData;
    var pnt     = llng;
    var distArr = [];
    var dist    = google.maps.geometry.spherical.computeDistanceBetween;


    for (index in arr)
        distArr.push([arr[index], dist(pnt, arr[index])]);

    return distArr.sort(function(a,b){
        return a[1]-b[1];
    })[0][0];
}

EDIT

If you don't have access to the array of LatLngs which make up the polyline, but have access to the polyline itself, you can use polyline's getPath method to get the path which is an MVC array so you can use .getArray() to return an array of LatLngs to use with the above function (closest).