JavaScript Trilateration

trilateration.js

// Created by Derrick Cohodas (dav-)
// Based on the Python example by StackExchange user wwnick from http://gis.stackexchange.com/a/415/41129 


// Requires the Mathjs library - http://mathjs.org/
var math = require('mathjs')

/**
 * Represents a coordinate with a distance
 * @param {Number} lat  Latitude
 * @param {Number} lon  Longitude
 * @param {Number} dist Distance from [lat, lon] to some point in kilometers
 */
var Beacon = function(lat, lon, dist) {
  this.lat  = lat
  this.lon  = lon
  this.dist = dist

  return this
}

/**
 * Perform a trilateration calculation to determine a location
 * based on 3 beacons and their respective distances (in kilometers) to the desired point.
 *
 * @param  {Array} beacons Array of 3 Beacon objects
 * @return {Array}         Array of the format [latitude, longitude]
 */
var trilaterate = function(beacons) {

  // #assuming elevation = 0
  var earthR = 6371e3
    , rad = function(deg) {
      return deg * (math.pi/180)
    }
    , deg = function(rad) {
      return rad * (180/math.pi)
    }

  // #using authalic sphere
  // #if using an ellipsoid this step is slightly different
  // #Convert geodetic Lat/Long to ECEF xyz
  // #   1. Convert Lat/Long to radians
  // #   2. Convert Lat/Long(radians) to ECEF
  var P1 = [ earthR *(math.cos(rad(beacons[0].lat)) * math.cos(rad(beacons[0].lon)))
           , earthR *(math.cos(rad(beacons[0].lat)) * math.sin(rad(beacons[0].lon)))
           , earthR *(math.sin(rad(beacons[0].lat)))
           ]

  var P2 = [ earthR *(math.cos(rad(beacons[1].lat)) * math.cos(rad(beacons[1].lon)))
           , earthR *(math.cos(rad(beacons[1].lat)) * math.sin(rad(beacons[1].lon)))
           , earthR *(math.sin(rad(beacons[1].lat)))
           ]

  var P3 = [ earthR *(math.cos(rad(beacons[2].lat)) * math.cos(rad(beacons[2].lon)))
           , earthR *(math.cos(rad(beacons[2].lat)) * math.sin(rad(beacons[2].lon)))
           , earthR *(math.sin(rad(beacons[2].lat)))
           ]

  // #from wikipedia
  // #transform to get circle 1 at origin
  // #transform to get circle 2 on x axis
  var ex = math.divide(math.subtract(P2, P1), math.norm( math.subtract(P2, P1) ))
  var i =  math.dot(ex, math.subtract(P3, P1) )

  var ey = math.divide(
          math.subtract(
            math.subtract(P3, P1),
            math.multiply(i, ex)
          ),
          math.norm(
            math.subtract(
              math.subtract(P3, P1),
              math.multiply(i, ex)
            )
          )
       )

  var ez = math.cross(ex, ey)
  var d =  math.norm(math.subtract(P2, P1))
  var j =  math.dot(ey, math.subtract(P3, P1))

  // #from wikipedia
  // #plug and chug using above values
  var x =  (math.pow(beacons[0].dist, 2) - math.pow(beacons[1].dist,2) + math.pow(d,2))/(2*d)
  var y = ((math.pow(beacons[0].dist, 2) - math.pow(beacons[2].dist,2) + math.pow(i,2) + math.pow(j,2))/(2*j)) - ((i/j)*x)

  // # only one case shown here
  //
  // I was having problems with the number in the radical being negative,
  // so I took the absolute value. Not sure if this is always going to work
  var z = math.sqrt( math.abs(math.pow(beacons[0].dist, 2) - math.pow(x, 2) - math.pow(y, 2)) )

  // #triPt is an array with ECEF x,y,z of trilateration point
  var triPt = math.add(
            math.add(
              math.add(P1,
                math.multiply(x, ex)
              ),
              math.multiply(y, ey)
            ),
            math.multiply(z, ez)
          )

  // #convert back to lat/long from ECEF
  // #convert to degrees
  var lat = deg(math.asin(math.divide(triPt[2], earthR)))
  var lon = deg(math.atan2(triPt[1], triPt[0]))

  return [lat, lon]

}

var beacons = [ new Beacon(35.000000, -120.000000, 189.419265289145)
              , new Beacon(35.000005, -120.000010, 189.420325082156)
              , new Beacon(35.000000, -120.000020, 189.420689733286)
              ]

console.log(trilaterate(beacons))

Did you have much luck finding a solution that works well to calculate trilateration in JS

We'd used this one, works perfect. You've an error with the earth radius though. It's in kilometers, it should be in meters. Instead of using var earthR = 6371 on line 32 use -> var earthR = 6371e3
and it works like charm!

Thank you so much for sharing this. It saved us a lot of work :)

@seco35 Thanks! I've updated line 32 accordingly.