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I'm using the following function to generate random geo coordinates within a specified radius from a seed point:
function randomGeo(center, radius) {
var y0 = center.latitude;
var x0 = center.longitude;
var rd = radius / 111300;
var u = Math.random();
var v = Math.random();
var w = rd * Math.sqrt(u);
var t = 2 * Math.PI * v;
var x = w * Math.cos(t);
var y = w * Math.sin(t);
var xp = x / Math.cos(y0);
return {
'latitude': y + y0,
'longitude': xp + x0
};
}
I do this in a loop, several times, using a 2000m radius and the following seed point:
location: { // Oxford
latitude: 51.73213,
longitude: -1.20631
}
I'd expect all of these results to be within 2000m; instead, I'm seeing values upwards of 10000m:
[ { latitude: 51.73256540025445, longitude: -1.3358092771716716 }, // 3838.75070783092
{ latitude: 51.7214165686511, longitude: -1.1644147572878725 }, // 3652.1890457730474
{ latitude: 51.71721400063117, longitude: -1.2082082568884593 }, // 8196.861603477768
{ latitude: 51.73583824510363, longitude: -1.0940424351649711 }, // 5104.820455873758
{ latitude: 51.74017571473442, longitude: -1.3150742602532257 }, // 4112.3279147866215
{ latitude: 51.73496163915278, longitude: -1.0379454413532996 }, // 9920.01459343298
{ latitude: 51.73582333121239, longitude: -1.0939302282840453 }, // 11652.160906253064
{ latitude: 51.72145745285658, longitude: -1.2491630482776055 }, // 7599.550622138115
{ latitude: 51.73036335927129, longitude: -1.3516902043395063 }, // 8348.276271205428
{ latitude: 51.748104753808924, longitude: -1.2669212014250266 }, // 8880.760669882042
{ latitude: 51.72010719621805, longitude: -1.327161328951446 }, // 8182.466715589904
{ latitude: 51.725727610071125, longitude: -1.0691503599266818 } ] // 2026.3687763449955
Given that I (shamelessly!) plagiarized this solution from elsewhere (albeit I've seen several similar implementations), I can't seem to figure out where the math is going wrong.
(Also, in case you want it, this is how I'm calculating the distance. Pretty sure this is correct.)
function distance(lat1, lon1, lat2, lon2) {
var R = 6371000;
var a = 0.5 - Math.cos((lat2 - lat1) * Math.PI / 180) / 2 + Math.cos(lat1 * Math.PI / 180) * Math.cos(lat2 * Math.PI / 180) * (1 - Math.cos((lon2 - lon1) * Math.PI / 180)) / 2;
return R * 2 * Math.asin(Math.sqrt(a));
}
581
The problem seems to stem from the fact that this is just an inaccurate calculation depending on which center point you are using. Particularly this line:
var xp = x / Math.cos(y0);
Removing this line and changing longitude to
'longitude': x + x0
Seems to keep all of the points within the specified radius, although without this line it seems the points will not completely fill out east to west in some cases.
Anyway, I found someone experiencing a similar issue here with someone elses Matlab code as a possible solution. Depends on how uniformly spread out you need the random points if you wanted to work with a different formula.
Here is a google maps visualization of what's going on with your provided formula:
<!doctype html>
<html>
<head>
<script type="text/javascript" src="//maps.google.com/maps/api/js?sensor=false"></script>
<script type="text/javascript" src="//ajax.googleapis.com/ajax/libs/jquery/2.1.4/jquery.min.js"></script>
<script>
var distanceLimit = 2000; //in meters
var numberRandomPoints = 200;
var mapZoomLevel = 11;
var locationindex = 0;
var locations = [
{'name': 'Oxford, England', 'latitude': 51.73213, 'longitude': -1.20631},
{'name': 'Quito, Ecuador', 'latitude': -0.2333, 'longitude': -78.5167},
{'name': 'Ushuaia, Argentina', 'latitude': -54.8000, 'longitude': -68.3000},
{'name': 'McMurdo Station, Antartica', 'latitude': -77.847281, 'longitude': 166.667942},
{'name': 'Norilsk, Siberia', 'latitude': 69.3333, 'longitude': 88.2167},
{'name': 'Greenwich, England', 'latitude': 51.4800, 'longitude': 0.0000},
{'name': 'Suva, Fiji', 'latitude': -18.1416, 'longitude': 178.4419},
{'name': 'Tokyo, Japan', 'latitude': 35.6833, 'longitude': 139.6833},
{'name': 'Mumbai, India', 'latitude': 18.9750, 'longitude': 72.8258},
{'name': 'New York, USA', 'latitude': 40.7127, 'longitude': -74.0059},
{'name': 'Moscow, Russia', 'latitude': 55.7500, 'longitude': 37.6167},
{'name': 'Cape Town, South Africa', 'latitude': -33.9253, 'longitude': 18.4239},
{'name': 'Cairo, Egypt', 'latitude': 30.0500, 'longitude': 31.2333},
{'name': 'Sydney, Australia', 'latitude': -33.8650, 'longitude': 151.2094},
];
</script>
</head>
<body>
<div id="topbar">
<select id="location_switch">
<script>
for (i=0; i<locations.length; i++) {
document.write('<option value="' + i + '">' + locations[i].name + '</option>');
}
</script>
</select>
<img src="http://google.com/mapfiles/ms/micons/ylw-pushpin.png" style="height:15px;"> = Center
<img src="https://maps.gstatic.com/mapfiles/ms2/micons/red.png" style="height:15px;"> = No Longitude Adjustment
<img src="https://maps.gstatic.com/mapfiles/ms2/micons/pink.png" style="height:15px;"> = With Longitude Adjustment (var xp = x / Math.cos(y0);)
</div>
<div id="map_canvas" style="position:absolute; top:30px; left:0px; height:100%; height:calc(100% - 30px); width:100%;overflow:hidden;"></div>
<script>
var markers = [];
var currentcircle;
//Create the default map
var mapcenter = new google.maps.LatLng(locations[locationindex].latitude, locations[locationindex].longitude);
var myOptions = {
zoom: mapZoomLevel,
scaleControl: true,
center: mapcenter
};
var map = new google.maps.Map(document.getElementById('map_canvas'), myOptions);
//Draw default items
var centermarker = addCenterMarker(mapcenter, locations[locationindex].name + '<br>' + locations[locationindex].latitude + ', ' + locations[locationindex].longitude);
var mappoints = generateMapPoints(locations[locationindex], distanceLimit, numberRandomPoints);
drawRadiusCircle(map, centermarker, distanceLimit);
createRandomMapMarkers(map, mappoints);
//Create random lat/long coordinates in a specified radius around a center point
function randomGeo(center, radius) {
var y0 = center.latitude;
var x0 = center.longitude;
var rd = radius / 111300; //about 111300 meters in one degree
var u = Math.random();
var v = Math.random();
var w = rd * Math.sqrt(u);
var t = 2 * Math.PI * v;
var x = w * Math.cos(t);
var y = w * Math.sin(t);
//Adjust the x-coordinate for the shrinking of the east-west distances
var xp = x / Math.cos(y0);
var newlat = y + y0;
var newlon = x + x0;
var newlon2 = xp + x0;
return {
'latitude': newlat.toFixed(5),
'longitude': newlon.toFixed(5),
'longitude2': newlon2.toFixed(5),
'distance': distance(center.latitude, center.longitude, newlat, newlon).toFixed(2),
'distance2': distance(center.latitude, center.longitude, newlat, newlon2).toFixed(2),
};
}
//Calc the distance between 2 coordinates as the crow flies
function distance(lat1, lon1, lat2, lon2) {
var R = 6371000;
var a = 0.5 - Math.cos((lat2 - lat1) * Math.PI / 180) / 2 + Math.cos(lat1 * Math.PI / 180) * Math.cos(lat2 * Math.PI / 180) * (1 - Math.cos((lon2 - lon1) * Math.PI / 180)) / 2;
return R * 2 * Math.asin(Math.sqrt(a));
}
//Generate a number of mappoints
function generateMapPoints(centerpoint, distance, amount) {
var mappoints = [];
for (var i=0; i<amount; i++) {
mappoints.push(randomGeo(centerpoint, distance));
}
return mappoints;
}
//Add a unique center marker
function addCenterMarker(centerposition, title) {
var infowindow = new google.maps.InfoWindow({
content: title
});
var newmarker = new google.maps.Marker({
icon: 'http://google.com/mapfiles/ms/micons/ylw-pushpin.png',
position: mapcenter,
map: map,
title: title,
zIndex: 3
});
google.maps.event.addListenerOnce(map, 'tilesloaded', function() {
infowindow.open(map,newmarker);
});
markers.push(newmarker);
return newmarker;
}
//Draw a circle on the map
function drawRadiusCircle (map, marker, distance) {
currentcircle = new google.maps.Circle({
map: map,
radius: distance
});
currentcircle.bindTo('center', marker, 'position');
}
//Create markers for the randomly generated points
function createRandomMapMarkers(map, mappoints) {
for (var i = 0; i < mappoints.length; i++) {
//Map points without the east/west adjustment
var newmappoint = new google.maps.LatLng(mappoints[i].latitude, mappoints[i].longitude);
var marker = new google.maps.Marker({
position:newmappoint,
map: map,
title: mappoints[i].latitude + ', ' + mappoints[i].longitude + ' | ' + mappoints[i].distance + 'm',
zIndex: 2
});
markers.push(marker);
//Map points with the east/west adjustment
var newmappoint = new google.maps.LatLng(mappoints[i].latitude, mappoints[i].longitude2);
var marker = new google.maps.Marker({
icon: 'https://maps.gstatic.com/mapfiles/ms2/micons/pink.png',
position:newmappoint,
map: map,
title: mappoints[i].latitude + ', ' + mappoints[i].longitude2 + ' | ' + mappoints[i].distance2 + 'm',
zIndex: 1
});
markers.push(marker);
}
}
//Destroy all markers
function clearMarkers() {
for (var i = 0; i < markers.length; i++) {
markers[i].setMap(null);
}
markers = [];
}
$('#location_switch').change(function() {
var newlocation = $(this).val();
clearMarkers();
mapcenter = new google.maps.LatLng(locations[newlocation].latitude, locations[newlocation].longitude);
map.panTo(mapcenter);
centermarker = addCenterMarker(mapcenter, locations[newlocation].name + '<br>' + locations[newlocation].latitude + ', ' + locations[newlocation].longitude);
mappoints = generateMapPoints(locations[newlocation], distanceLimit, numberRandomPoints);
//Draw default items
currentcircle.setMap(null);
drawRadiusCircle(map, centermarker, distanceLimit);
createRandomMapMarkers(map, mappoints);
});
</script>
</body>
</html>You can generate points with a random bearing and distance from the center by moving some distance using vincenty distances (see this stackoverflow answer). In Python, for example, you could use the geopy package.
import random
from geopy import Point
from geopy.distance import geodesic
def generate_point(center: Point, radius: int) -> Point:
radius_in_kilometers = radius * 1e-3
random_distance = random.random() * radius_in_kilometers
random_bearing = random.random() * 360
return geodesic(kilometers=random_distance).destination(center, random_bearing)
radius = 2000
center = Point(51.73213, -1.20631)
points = [generate_point(center, radius) for _ in range(3000)]
Distances are confirmed with:
assert all(geodesic(center, point).meters <= radius for point in points)
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