Find the final latitude longitude after a movement on the globe

Question

I am using the Haversine formula to calculate the distance from two latitude-longitude pairs.

function getDistanceFromLatLonInKm(lat1,lon1,lat2,lon2) {
    var R = 6371; // Radius of the earth in km
    var dLat = deg2rad(lat2-lat1);
    var dLon = deg2rad(lon2-lon1); 
    var lat1 = deg2rad(lat1);
    var lat2 = deg2rad(lat2);

    var a = 
        Math.sin(dLat/2) * Math.sin(dLat/2) + 
        Math.sin(dLon/2) * Math.sin(dLon/2) * 
        Math.cos(lat1) * Math.cos(lat2);

    var c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1-a)); 
    var d = R * c;
    return d;
}

Given a starting point (lat1, lat2), the distance required to move on a straight line and the angle, I need to determine the endpoint (as in lat2 and lon2).

See my attempt below:

function getFinalLatLon(lat1, lon1, distance, angle) {
    var R = 6371; // Radius of the earth in km
    var c = distance/R;
    // Math.atan2(Math.sqrt(a), Math.sqrt(1-a)) = c/2
    var a = // stuck here

    // looking for this part of the code

    return [lat2, lon2];
}

Answer 1

If you are moving horizontally, you can increment the longitude by distance / (R * cos(lat)). No atan needed.

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EDIT: Since you wanted a formula for the general case, consider the following geometric derivation:

• Front view:

  

• Side view:

  

• Entire setup:

  

Notes:

• r is the unit vector of your starting position, and s is the endpoint.

  

• a, b, c are intermediate vectors to aid calculation.

• (θ, φ) are the (lat, long) coordinates.
• γ is the bearing of the direction you are going to travel in.
• δ is the angle travelled through (distance / radius R = 6400000m).

We need a, b to be perpendicular to r and also a aligned with North. This gives:

c is given by (simple trigonometry):

And thus we get s (through some very tedious algebra):

Now we can calculate the final (lat, long) coordinates of s using:

---

Code:

function deg2rad(deg) { return deg * (Math.PI / 180.0) }
function rad2deg(rad) { return rad * (180.0 / Math.PI) }

function getFinalLatLong(lat1, long1, distance, angle, radius) {
    // calculate angles
    var delta = distance / radius,
        theta = deg2rad(lat1),
        phi   = deg2rad(long1),
        gamma = deg2rad(angle);

    // calculate sines and cosines
    var c_theta = Math.cos(theta), s_theta = Math.sin(theta);
    var c_phi   = Math.cos(phi)  , s_phi   = Math.sin(phi)  ;
    var c_delta = Math.cos(delta), s_delta = Math.sin(delta);
    var c_gamma = Math.cos(gamma), s_gamma = Math.sin(gamma);

    // calculate end vector
    var x = c_delta * c_theta * c_phi - s_delta * (s_theta * c_phi * c_gamma + s_phi * s_gamma);
    var y = c_delta * c_theta * s_phi - s_delta * (s_theta * s_phi * c_gamma - c_phi * s_gamma);
    var z = s_delta * c_theta * c_gamma + c_delta * s_theta;

    // calculate end lat long
    var theta2 = Math.asin(z), phi2 = Math.atan2(y, x);

    return [rad2deg(theta2), rad2deg(phi2)];
}

Test cases:

1. Input (lat, long) = (45, 0), angle = 0, distance = radius * deg2rad(90) => (45, 180) (as I said before)

2. Input (lat, long) = (0, 0), angle = 90, distance = radius * deg2rad(90) => (0, 90) (as expected - start at equator, travel east by 90 longitude)

3. Input (lat, long) = (54, 29), angle = 36, distance = radius * deg2rad(360) => (54, 29) (as expected - start at any random position and go full-circle in any direction)

4. Interesting case: input (lat, long) = (30, 0), everything else same. => (0, 90) (we expected (30, 90)? - not starting at equator, travel by 90 degrees to North)

   The reason for this is that 90 degrees to North is not East (if you're not at the equator)! This diagram should show why:

   

   As you can see, the path of movement at 90 degrees to the North is not in the direction of East.

Answer 2

I just found a similar question here and I followed the solution to come up with a function that works for my case.

Hope it helps someone else:

function getFinalLatLon(lat1, lon1, distance, angle){ 
    function deg2rad(deg) {
        return deg * (Math.PI/180)
    }
    // dy = R*sin(theta) 
    var dy = distance * Math.sin(deg2rad(angle)) 
    var delta_latitude = dy/110574
    // One degree of latitude on the Earth's surface equals (110574 meters
    delta_latitude = parseFloat(delta_latitude.toFixed(6));

    // final latitude = start_latitude + delta_latitude
    var lat2 = lat1 + delta_latitude

    // dx = R*cos(theta) 
    var dx = distance * Math.cos(deg2rad(angle)) 
    // One degree of longitude equals 111321 meters (at the equator)
    var delta_longitude = dx/(111321*Math.cos(deg2rad(lat1))) 
    delta_longitude = parseFloat(delta_longitude.toFixed(6));

    // final longitude = start_longitude + delta_longitude
    var lon2 = lon1 + delta_longitude

    return [lat2, lon2];
}

The angle is 0 degrees for a horizontal move. You can switch that as you wish. If someone is moving north that would be 90 deg. 135 degrees for north west and so on...